Recent zbMATH articles in MSC 35Rhttps://zbmath.org/atom/cc/35R2024-11-01T15:51:55.949586ZUnknown authorWerkzeugAsymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaceshttps://zbmath.org/1544.220102024-11-01T15:51:55.949586Z"Papageorgiou, Effie"https://zbmath.org/authors/?q=ai:papageorgiou.effie-gAuthor's abstract: This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for L1 initial data. In the case of the Laplace-Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as L1 asymptotic convergence without the assumption of bi-K-invariance.
Reviewer: Antonio Vitolo (Fisciano)A tensor product approach to non-local differential complexeshttps://zbmath.org/1544.300282024-11-01T15:51:55.949586Z"Hinz, Michael"https://zbmath.org/authors/?q=ai:hinz.michael"Kommer, Jörn"https://zbmath.org/authors/?q=ai:kommer.jornSummary: We study differential complexes of Kolmogorov-Alexander-Spanier type on metric measure spaces associated with unbounded non-local operators, such as operators of fractional Laplacian type. We define Hilbert complexes, observe invariance properties and obtain self-adjoint non-local analogues of Hodge Laplacians. For \(d\)-regular measures and operators of fractional Laplacian type we provide results on removable sets in terms of Hausdorff measures. We prove a Mayer-Vietoris principle and a Poincaré lemma and verify that in the compact Riemannian manifold case the deRham cohomology can be recovered.The singular strata of a free-boundary problem for harmonic measurehttps://zbmath.org/1544.310112024-11-01T15:51:55.949586Z"McCurdy, Sean"https://zbmath.org/authors/?q=ai:mccurdy.seanSummary: We obtain \textit{quantitative} estimates on the fine structure of the singular set of the mutual boundary \(\partial \Omega^{\pm}\) for pairs of complementary domains \(\Omega^+, \Omega^- \subset \mathbb{R}^n\) which arise in a class of two-sided free boundary problems for harmonic measure. These estimates give new insight into the structure of the mutual boundary \(\partial \Omega^{\pm}\).Internal aggregation models with multiple sources and obstacle problems on Sierpiński gasketshttps://zbmath.org/1544.310202024-11-01T15:51:55.949586Z"Freiberg, Uta"https://zbmath.org/authors/?q=ai:freiberg.uta-renata"Heizmann, Nico"https://zbmath.org/authors/?q=ai:heizmann.nico"Kaiser, Robin"https://zbmath.org/authors/?q=ai:kaiser.robin"Sava-Huss, Ecaterina"https://zbmath.org/authors/?q=ai:sava-huss.ecaterinaSummary: We consider the doubly infinite Sierpiński gasket graph \(\mathsf{SG}_0\), rescale it by factor \(2^{-n}\), and on the rescaled graphs \(\mathsf{SG}_n = 2^{-n} \mathsf{SG}_0\), for every \(n \in \mathbb{N}\), we investigate the limit shape of three aggregation models with initial configuration \(\sigma_n\) of particles supported on multiple vertices. The models under consideration are: \textit{divisible sandpile} in which the excess mass is distributed among the vertices until each vertex is stable and has mass less or equal to one, \textit{internal DLA} in which particles do random walks until finding an empty site, and \textit{rotor aggregation} in which particles perform deterministic counterparts of random walks until finding an empty site. We denote by \( \mathsf{SG} = \mathrm{cl} \left( \bigcup_{n=0}^\infty \mathsf{SG}_n \right)\) the infinite Sierpiński gasket, which is a closed subset of \(\mathbb{R}^2\), for which \(\mathsf{SG}_n\) represents the level-\(n\) approximating graph, and we consider a continuous function \(\sigma: \mathsf{SG} \to \mathbb{N}\). For \(\sigma\) we solve the obstacle problem, and we describe the noncoincidence set \( D \subset \mathsf{SG}\) as the solution of a free boundary problem on the fractal \(\mathsf{SG}\). If the discrete particle configurations \(\sigma_n\) on the approximating graphs \(\mathsf{SG}_n\) converge pointwise to the continuous function \(\sigma\) on the limit set \(\mathsf{SG}\), we prove that, as \(n \to \infty\), the scaling limits of the three aforementioned models on \(\mathsf{SG}_n\) starting with initial particle configuration \(\sigma_n\) converge to the deterministic solution \(D\) of the free boundary problem on the limit set \(\mathsf{SG} \subset \mathbb{R}^2\). For \(D\) we also investigate boundary regularity properties.On the Cauchy-Kovalevskaya theorem for Caputo fractional differential equationshttps://zbmath.org/1544.340112024-11-01T15:51:55.949586Z"Jornet, Marc"https://zbmath.org/authors/?q=ai:jornet.marcSummary: We aim at proving the Cauchy-Kovalevskaya theorem for systems of nonlinear fractional differential equations in the Caputo sense, not necessarily polynomial or compartmental. Essentially, the theorem states that if the input function has a Taylor series, then the solution can be locally expressed as a fractional power series. We use, in the real field, the method of majorants and the analytic version of the implicit-function theorem, in a way that circumvents difficulties associated to fractional calculus. Some corollaries on continuity are derived, with computational examples for illustration, and a discussion on fractional partial differential equations is included with a case study and counterexamples. Open problems are raised at the end.Almost sure scattering for the one dimensional nonlinear Schrödinger equationhttps://zbmath.org/1544.350012024-11-01T15:51:55.949586Z"Burq, Nicolas"https://zbmath.org/authors/?q=ai:burq.nicolas"Thomann, Laurent"https://zbmath.org/authors/?q=ai:thomann.laurentIn this book, the authors study the long time dynamics for the solutions to the Cauchy problem for the one-dimensional nonlinear Schrödinger equation \(i\partial_sU+\partial_y^2U=|U|^{p-1}U\), where \((s,y)\in\mathbb{R}\times\mathbb{R},\;p>1\), and \(U(s_0,y)=U_0\) is a random initial condition with low Sobolev regularity. As we know, on compact manifolds, many probability measures are invariant by the flow of the linear Schrödinger equation, such as Wiener measures, and it is sometimes possible to modify them suitably and get invariant (e.g. Gibbs measures) or quasi-invariant measures for the nonlinear problem. While on Euclidean space \(\mathbb{R}^d\), the large time dispersion shows that the only invariant measure is the \(\delta\)-measure on the trivial solution \(u=0\), and the good notion to track is whether the nonlinear evolution of the initial measure is well described by the linear evolution. In this work, the authors achieve this conclusion. More precisely, the authors first define measures on the space of initial data for which they can describe precisely the nontrivial evolution by the linear Schrödinger flow. Second, they prove that the nonlinear evolution of these measures is absolutely continuous with respect to their linear evolutions. Actually, they give precise and optimal bounds on the Radon-Nikodym derivatives of these measures with respect to each other and characterise their \(L^p\) regularity. And then, they get benefit from this precise description to prove the global well-posedness for \(p>1\) and almost sure scattering for \(p>3\). This is the first occurrence where the description of quasi-invariant measures allows to get quantitative asymptotics for the nonlinear evolution.
Reviewer: Jiqiang Zheng (Beijing)Mixed local and nonlocal parabolic equation: global existence, decay and blow-uphttps://zbmath.org/1544.350332024-11-01T15:51:55.949586Z"Zhao, Yanan"https://zbmath.org/authors/?q=ai:zhao.yanan"Zhang, Binlin"https://zbmath.org/authors/?q=ai:zhang.binlinSummary: In this paper, we use the modified potential well method and the Galerkin method to study the following mixed local and nonlocal parabolic equation:
\[
\begin{cases}
u_t - \Delta u+(-\Delta)^s u = |u|^{p-2}u &\text{in }\Omega\times\mathbb{R}^+,\\
u(x, 0) = u_0(x) &\text{in }\Omega,\\
u(x, t) = 0 &\text{in } \mathbb{R}^N\backslash\Omega\times\mathbb{R}_0^+,
\end{cases}
\]
where \(\Delta\) is the Laplace operator, \((-\Delta)^s\) is the fractional Laplace operator, \(\Omega\subset \mathbb{R}^N\) is a bounded domain with Lipschitz boundary \(\partial\Omega\), \(N > 2s\), \(2 < p \leq 2^\ast\) and \(s\in(0, 1)\). In the cases of low and critical initial energy, we not only prove the existence of global solutions and the decay rate of the \(L^2\) norm for global solutions, but also obtain blow-up of solutions in finite time and the lower and upper bounds of the blow-up time. In the case of high initial energy, we give sufficient conditions for the global existence and blow-up of solutions, and the lower and upper bounds on the blow-up time.Asymptotic profiles for inhomogeneous heat equations with memoryhttps://zbmath.org/1544.350442024-11-01T15:51:55.949586Z"Cortázar, Carmen"https://zbmath.org/authors/?q=ai:cortazar.carmen"Quirós, Fernando"https://zbmath.org/authors/?q=ai:quiros-gracian.fernando"Wolanski, Noemí"https://zbmath.org/authors/?q=ai:wolanski.noemi-iSummary: We study the large-time behavior in all \(L^{p}\) norms of solutions to an inhomogeneous nonlocal heat equation in \(\mathbb{R}^{N}\) involving a Caputo \(\alpha\)-time derivative and a power \(\beta\) of the Laplacian when the dimension is large, \(N > 4 \beta\). The asymptotic profiles depend strongly on the space-time scale and on the time behavior of the spatial \(L^{1}\) norm of the forcing term.Optimal decay rates and space-time analyticity of solutions to the Patlak-Keller-Segel equationshttps://zbmath.org/1544.350472024-11-01T15:51:55.949586Z"Gao, Yu"https://zbmath.org/authors/?q=ai:gao.yu"Wang, Cong"https://zbmath.org/authors/?q=ai:wang.cong.4"Xue, Xiaoping"https://zbmath.org/authors/?q=ai:xue.xiaopingSummary: Based on some new elementary estimates for the space-time derivatives of the heat kernel, we use a bootstrapping approach to establish quantitative estimates on the optimal decay rates for the \(L^q(\mathbb{R}^d)\) (\(1 \leq q \leq \infty\), \(d\in\mathbb{N}\)) norm of the space-time derivatives of solutions to the (modified) Patlak-Keller-Segel equations with initial data in \(L^1(\mathbb{R}^d)\), which implies the joint space-time analyticity of solutions. When the \(L^1(\mathbb{R}^d)\) norm of the initial datum is small, the upper bound for the decay estimates is global in time, which yields a lower bound on the growth rate of the radius of space-time analyticity in time. As a byproduct, the space analyticity is obtained for any initial data in \(L^1(\mathbb{R}^d)\). The decay estimates and space-time analyticity are also established for solutions bounded in both space and time variables. The results can be extended to a more general class of equations.Existence and regularity of global attractors for a Kirchhoff wave equation with strong damping and memoryhttps://zbmath.org/1544.350572024-11-01T15:51:55.949586Z"Yang, Bin"https://zbmath.org/authors/?q=ai:yang.bin.3"Qin, Yuming"https://zbmath.org/authors/?q=ai:qin.yuming"Miranville, Alain"https://zbmath.org/authors/?q=ai:miranville.alain-m"Wang, Ke"https://zbmath.org/authors/?q=ai:wang.ke.4|wang.ke|wang.ke.1|wang.ke.7Summary: This paper is concerned with the existence and regularity of global attractor \(\mathcal{A}\) for a Kirchhoff wave equation with strong damping and memory in \(\mathcal{H}\) and \(\mathcal{H}^1\), respectively. In order to obtain the existence of \(\mathcal{A}\), we mainly use the energy method in the priori estimations, and then verify the asymptotic compactness of the semigroup by the method of contraction function. Finally, by decomposing the weak solutions into two parts and some elaborate calculations, we prove the regularity of \(\mathcal{A}\).Generalized Hölder estimates via generalized Morrey norms for some ultraparabolic operatorshttps://zbmath.org/1544.350632024-11-01T15:51:55.949586Z"Guliyev, V. S."https://zbmath.org/authors/?q=ai:guliyev.vagif-sabirSummary: We consider a class of hypoelliptic operators of the following type
\[
\mathcal{L} = \sum_{i, j=1}^{p_0} a_{ij} \partial_{x_i x_j}^2 + \sum \limits_{i, j=1}^N b_{ij} x_i \partial_{x_j}-\partial_t,
\]
where \((a_{ij})\), \((b_{ij})\) are constant matrices and \((a_{ij})\) is symmetric positive definite on \(\mathbb{R}^{p_0}\) (\(p_0 \leq N\)). We obtain generalized Hölder estimates for \(\mathcal{L}\) on \(\mathbb{R}^{N+1}\) by establishing several estimates of singular integrals in generalized Morrey spaces.Uniqueness of continuation for semilinear elliptic equationshttps://zbmath.org/1544.350642024-11-01T15:51:55.949586Z"Choulli, Mourad"https://zbmath.org/authors/?q=ai:choulli.mouradSummary: We quantify the uniqueness of continuation from Cauchy or interior data. Our approach consists in extending the existing results in the linear case. As by product, we obtain a new stability estimate in the linear case. We also show strong uniqueness of continuation and the uniqueness of continuation from a set of positive measure. These results are derived using a linearization procedure.Regularity and convergent result of mild solution of Love equationhttps://zbmath.org/1544.350662024-11-01T15:51:55.949586Z"Bui, Duc Nam"https://zbmath.org/authors/?q=ai:bui.duc-nam"Nghia, Bui Dai"https://zbmath.org/authors/?q=ai:nghia.bui-dai"Phuong, Nguyen Duc"https://zbmath.org/authors/?q=ai:phuong.nguyen-ducSummary: In this paper, we are interested to study a mild solution of the Love equation. This type of equation has many applications in physics. Under various assumptions of the Cauchy data, we obtain the regularity of the mild solution. We also obtain the continuity of the mild solution concerning the parameter. We show that the mild solution of the Love equation converges to the mild solution of the wave equation. Our work may be the first paper on the mild solution of the Love equation.On the regularity theory for mixed local and nonlocal quasilinear parabolic equationshttps://zbmath.org/1544.350682024-11-01T15:51:55.949586Z"Garain, Prashanta"https://zbmath.org/authors/?q=ai:garain.prashanta"Kinnunen, Juha"https://zbmath.org/authors/?q=ai:kinnunen.juhaThe weak subsolutions and supersolutions of very general parabolic nonlocal equations
\[
\frac{\partial u}{\partial t}+ \mathcal{L}_p u(x,t)- \operatorname{div}\mathcal{B}_p(x,t,u,\nabla u)= g(x,t,u)
\]
are studied in space \(\times\) time cylinders \(\Omega\times(0,T)\), where \(\Omega\subset\mathbb{R}^N\). A special case is the equation with the operators
\[
\mathcal{L}_p u(x,t)= \int_{\mathbb{R}^N} \frac{|u(x,t)- u(y,t)|^{p-2}(u(x,t)- u(y,t))}{|x-y|^{N+ps}}\,dy,
\]
where \(0<s<1\), \(1<p<\infty\) (the principal value of the integral is taken), and
\[
\mathcal{B}_p(x,t,u,\nabla u)= |\nabla u|^{p-2}\nabla u.
\]
(There is a misprint in formula (1.2), defining \(\mathcal{L}_p\).)
The weak subsolutions are proved to be locally bounded. The theorem comes with an estimate; the cases \(2N/(N+2)<p\) and \(1<p\le 2N/(N+2)\) have somewhat different bounds.
The semicontinuity and pointwise behaviour of the weak supersolutions is studied for the pointwise defined representation
\[
u(x,t)=\operatorname{ess}\liminf_{\substack{(y,\tau)\to(x,t)\\ \tau<t}} u(y,\tau).
\]
The assumption \(g=0\) is required for the deeper results. Energy estimates and a technically advanced variant of De Giorgi's method are used.
Reviewer: Peter Lindqvist (Trondheim)Time-periodic traveling wave solutions of a reaction-diffusion Zika epidemic model with seasonalityhttps://zbmath.org/1544.350792024-11-01T15:51:55.949586Z"Zhao, Lin"https://zbmath.org/authors/?q=ai:zhao.linSummary: In this paper, the full information about the existence and nonexistence of a time-periodic traveling wave solution of a reaction-diffusion Zika epidemic model with seasonality, which is non-monotonic, is investigated. More precisely, if the basic reproduction number, denoted by \(R_0\), is larger than one, there exists a minimal wave speed \(c^* >0\) satisfying for each \(c>c^*\), the system admits a nontrivial time-periodic traveling wave solution with wave speed \(c\), and for \(c<c^*\), there exist no nontrivial time-periodic traveling waves such that if \(R_0 \leqslant 1\), the system admits no nontrivial time-periodic traveling waves.A comparison principle for semilinear Hamilton-Jacobi-Bellman equations in the Wasserstein spacehttps://zbmath.org/1544.350832024-11-01T15:51:55.949586Z"Daudin, Samuel"https://zbmath.org/authors/?q=ai:daudin.samuel"Seeger, Benjamin"https://zbmath.org/authors/?q=ai:seeger.benjaminIn this paper is proven the validity of the comparison principle for viscosity solutions of semi-linear Hamilton-Jacobi-Bellman equations in the Wasserstein space of probability measures with finite second-order moment and time horizon. The proof is based on doubling of variables argument and use of the square of the 2-Wasserstein norm as a penalising test function. The main novelty in the article is the applicability of this approach to equations with non-convex Hamiltonians.
Reviewer: Georgi Boyadzhiev (Sofia)Wegner estimate and localisation for alloy-type operators with minimal support assumptions on the single site potentialhttps://zbmath.org/1544.350872024-11-01T15:51:55.949586Z"Täufer, Matthias"https://zbmath.org/authors/?q=ai:taufer.matthias"Veselić, Ivan"https://zbmath.org/authors/?q=ai:veselic.ivanSummary: We prove a Wegner estimate for alloy-type models merely assuming that the single site potential is lower bounded by a characteristic function of a thick set (a particular class of sets of positive measure). The proof exploits on one hand recently proven unique continuation principles or uncertainty relations for linear combinations of eigenfunctions of the Laplacian on cubes and on the other hand the well developed machinery for proving Wegner estimates. We obtain a Wegner estimate with optimal volume dependence at all energies, and localization near the minimum of the spectrum, even for some non-stationary random potentials. We complement the result by showing that a lower bound on the potential by the characteristic function of a thick set is necessary for a Wegner estimate to hold. Hence, we have identified a sharp condition on the size for the support of random potentials that is sufficient and necessary for the validity of Wegner estimates.A nonlocal reaction-diffusion-advection model with free boundarieshttps://zbmath.org/1544.350992024-11-01T15:51:55.949586Z"Tang, Yaobin"https://zbmath.org/authors/?q=ai:tang.yaobin"Dai, Binxiang"https://zbmath.org/authors/?q=ai:dai.binxiangSummary: A nonlocal diffusion single population model with advection and free boundaries is considered. Our aim is to discuss how the advection rate affects dynamic behaviors of species under the case of small advection. Firstly, the well-posed global solution is obtained. Secondly, we apply the eigenvalue problem of integro-differential operator to obtain the dichotomy and sharp criteria for spreading and vanishing, which is determined by initial habitat and initial density. Further, the asymptotic spreading speed of species is estimated when spreading happens. Namely, we get the exact asymptotic spreading speed and find that if kernel function satisfies the certain condition, then the leftward asymptotic spreading speed is less than the rightward one due to the impact of advection rate. Meanwhile, we also observe that accelerated spreading happens if the certain condition does not be satisfied.The forward self-similar solution of fractional incompressible Navier-Stokes system: the critical casehttps://zbmath.org/1544.351162024-11-01T15:51:55.949586Z"Lai, Baishun"https://zbmath.org/authors/?q=ai:lai.baishunSummary: In this paper, we study the regularity and pointwise estimates of forward self-similar solutions of fractional Navier-Stokes system under the critical case. By employing a Caffarelli, Kohn and Nirenberg-type iteration, \(L^{\infty}\) estimates of the self-similar solution's profile are established, which is a key ingredient to ensure that the global weighted energy estimate procedure used in [\textit{B. Lai} et al., Trans. Am. Math. Soc. 374, No. 10, 7449--7497 (2021; Zbl 1479.35622)] is performed under the critical case. As a product, its natural pointwise bounds are recovered. Moreover, to obtain the optimal spatial decay estimate of self-similar solution's profile, a new technique is required due to lack of the related regularity theory.Partial regularity and nonlinear potential estimates for Stokes systems with super-quadratic growthhttps://zbmath.org/1544.351262024-11-01T15:51:55.949586Z"Ma, Lingwei"https://zbmath.org/authors/?q=ai:ma.lingwei"Zhang, Zhenqiu"https://zbmath.org/authors/?q=ai:zhang.zhenqiuSummary: This paper builds a bridge between partial regularity theory and nonlinear potential theory for the following generalized stationary Stokes system with super-quadratic growth and continuous coefficients:
\[
-\operatorname{div} \mathcal{A}(x, D \boldsymbol{u}) + \nabla \pi = \boldsymbol{f} \quad \operatorname{div} \boldsymbol{u} = 0,
\]
where \(D\boldsymbol{u}\) is the symmetric part of the gradient \(\nabla \boldsymbol{u}\). We first establish an \(\varepsilon\)-regularity criterion involving both the excess functional of the symmetric gradient \(D\boldsymbol{u}\) and Wolff potentials of the nonhomogeneous term \(\boldsymbol{f}\) to guarantee the local vanishing mean oscillation (VMO)-regularity of \(D\boldsymbol{u}\) in an open subset \(\Omega_{\boldsymbol{u}}\) of \(\Omega\) with full measure. Such an \(\varepsilon\)-regularity criterion leads to a pointwise Wolff potential estimate of \(D\boldsymbol{u}\), which immediately infers that \(D\boldsymbol{u}\) is partially \(C^0\)-regular under appropriate assumptions. Finally, we give a local continuous modulus estimate of \(D\boldsymbol{u}\).On the analytical soliton approximations to fractional forced Korteweg-de Vries equation arising in fluids and plasmas using two novel techniqueshttps://zbmath.org/1544.351412024-11-01T15:51:55.949586Z"Alhejaili, Weaam"https://zbmath.org/authors/?q=ai:alhejaili.weaam"Az-Zo'bi, Emad A."https://zbmath.org/authors/?q=ai:az-zobi.emad-a"Shah, Rasool"https://zbmath.org/authors/?q=ai:shah.rasool"El-Tantawy, S. A."https://zbmath.org/authors/?q=ai:el-tantawy.s-a(no abstract)Nonlinear dynamic wave characteristics of optical soliton solutions in ion-acoustic wavehttps://zbmath.org/1544.351682024-11-01T15:51:55.949586Z"Zaman, U. H. M."https://zbmath.org/authors/?q=ai:zaman.u-h-m"Arefin, Mohammad Asif"https://zbmath.org/authors/?q=ai:arefin.mohammad-asif"Hossain, Md. Akram"https://zbmath.org/authors/?q=ai:hossain.md-akram"Akbar, M. Ali"https://zbmath.org/authors/?q=ai:ali-akbar.m"Uddin, M. Hafiz"https://zbmath.org/authors/?q=ai:hafiz-uddin.mSummary: Traveling wave solutions are utilized to depict reaction-diffusion and analyze electrical signal transmission and propagation in space-time nonlinear fractional order partial differential equations like the space-time fractional Telegraph and Kolmogorov-Petrovsky Piskunov equations, and that are used in physical science to model combustion, biological research to model nerve impulse propagation, chemical dynamics to model concentration in order wave propagation, and plasma to model the progression of a set of duffing oscillators. In this study, the new generalized \((G^\prime / G)\)-expansion technique was employed to construct some novel and more universal closed-form traveling wave solutions in the sense of conformable derivatives which explain the above-stated phenomena properly. By utilizing complex fractional transformation, the ordinary differential equations are generated from fractional order differential equations. The recommended technique allowed us to produce some dynamical wave patterns of kink, single soliton, compacton, periodic shape, multiple periodic waves, anti-kink, and other structures are developed, which are shown using 3D plots and contour plots to illustrate the physical layout clearly. The traveling waveform responses can be defined in terms of functions based on trigonometry, hyperbolic operations, and rational functions and that are quick, flexible, and simple to reproduce. Furthermore, the obtained closed-form solutions for nonlinear fractional evolution equations make stability analysis and accuracy comparison amongst numerical solvers easier, which asserts that the new generalized \((G^\prime / G)\)-expansion technique is one of the most proficient and effective approaches.Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertaintyhttps://zbmath.org/1544.351692024-11-01T15:51:55.949586Z"Alexanderian, Alen"https://zbmath.org/authors/?q=ai:alexanderian.alen"Nicholson, Ruanui"https://zbmath.org/authors/?q=ai:nicholson.ruanui"Petra, Noemi"https://zbmath.org/authors/?q=ai:petra.noemiSummary: We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unified framework. We build on the Bayesian approximation error (BAE) approach, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum \textit{a posteriori} probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.
{{\copyright} 2024 IOP Publishing Ltd}Volterra-Prabhakar function of distributed order and some applicationshttps://zbmath.org/1544.351762024-11-01T15:51:55.949586Z"Górska, K."https://zbmath.org/authors/?q=ai:gorska.katarzyna"Pietrzak, T."https://zbmath.org/authors/?q=ai:pietrzak.t"Sandev, T."https://zbmath.org/authors/?q=ai:sandev.trifce"Tomovski, Ž."https://zbmath.org/authors/?q=ai:tomovski.zivoradSummary: The paper studies the exact solution of two kinds of generalized Fokker-Planck equations in which the integral kernels are given either by the distributed order function \(k_1 ( t ) = \int_0^1 t^{- \mu} / \varGamma (1 - \mu) \operatorname{d} \mu\) or the distributed order Prabhakar function \(k_2 ( \alpha , \gamma ; \lambda ; t ) = \int_0^1 e_{\alpha , 1 - \mu}^{- \gamma} (\lambda ; t ) \operatorname{d} \mu \), where the Prabhakar function is denoted as \(e_{\alpha , 1 - \mu}^{- \gamma} ( \lambda ; t )\). Both of these integral kernels can be called the fading memory functions and are the Stieltjes functions. It is also shown that their Stieltjes character is enough to ensure the non-negativity of the mean square values and higher even moments. The odd moments vanish. Thus, the solution of generalized Fokker-Planck equations can be called the probability density functions. We introduce also the Volterra-Prabhakar function and its generalization which are involved in the definition of \(k_2 ( \alpha , \gamma ; \lambda ; t )\) and generated by it the probability density function \(p_2 ( x , t )\).The cortical V1 transform as a heterogeneous Poisson problemhttps://zbmath.org/1544.351782024-11-01T15:51:55.949586Z"Sarti, Alessandro"https://zbmath.org/authors/?q=ai:sarti.alessandro"Galeotti, Mattia"https://zbmath.org/authors/?q=ai:galeotti.mattia"Citti, Giovanna"https://zbmath.org/authors/?q=ai:citti.giovannaThe authors prove that the distribution of cells in the primary visual cortex is sufficient to reconstruct the perceived image without additional constraints. They focus on the perceptual phenomena of lightness and color constancy. The main model of cortical transform as a heterogeneous Poisson is presented. By employing a steepest descent method, the authors prove the existence of a weak solution of this problem. Then the notion of the \(H\)-convergence is introduced, and the proof of convergence of the heterogeneous problem to the homogeneous one is also given. Numerical results are finally addressed and discussed.
Reviewer: Rodica Luca (Iaşi)Nonlinear nonlocal equations involving subcritical or power nonlinearities and measure datahttps://zbmath.org/1544.351812024-11-01T15:51:55.949586Z"Gkikas, Konstantinos T."https://zbmath.org/authors/?q=ai:gkikas.konstantinos-tSummary: Let \(s\in(0, 1)\), \(1 < p < \frac{N}{s}\) and \(\Omega\subset\mathbb{R}^N\) be an open bounded set. In this work we study the existence of solutions to problems \((E_\pm) Lu\pm g(u) = \mu\) and \(u = 0\) a.e. in \(\mathbb{R}^N\setminus \Omega,\) where \(g\in C(\mathbb{R})\) is a nondecreasing function, \(\mu\) is a bounded Radon measure on \(\Omega\) and \(L\) is an integro-differential operator with order of differentiability \(s\in(0, 1)\) and summability \(p\in(1, \frac{N}{s}).\) More precisely, \(L\) is a fractional \(p\)-Laplace type operator. We establish sufficient conditions for the solvability of problems \((E_\pm)\). In the particular case \(g(t) = |t|^{ \kappa-1}t\); \(\kappa > p-1\), these conditions are expressed in terms of Bessel capacities.Retraction notice to: ``Novel analysis of fuzzy fractional Klein-Gordon model via semianalytical method''https://zbmath.org/1544.351822024-11-01T15:51:55.949586ZRetraction notice to the paper [\textit{M. Alshammari} et al., J. Funct. Spaces 2022, Article ID 4020269, 9 p. (2022; Zbl 1491.35426)].
From the retraction note, ``This article has been retracted by Hindawi following an investigation undertaken by the publisher. This investigation has
uncovered evidence of one or more [\dots] indicators of
systematic manipulation of the publication process.
The presence of these indicators undermines our confidence
in the integrity of the article's content and we cannot, therefore,
vouch for its reliability.''On non-autonomous fractional evolution equations and applicationshttps://zbmath.org/1544.351832024-11-01T15:51:55.949586Z"Achache, Mahdi"https://zbmath.org/authors/?q=ai:achache.mahdiSummary: We consider the problem of maximal regularity for semilinear non-autonomous fractional equations
\[
\sum_{i=1}^n \lambda_i \partial^{\alpha_i} (u-u_0)(t)+{\mathscr{A}}(t)u(t)=F(t,u(t))\quad t {\text{-a.e.}}, \,\lambda_i\in{\mathbb{C}}, \,n\in{\mathbb{N}}.
\]
Here, \( \partial^{\alpha_i}\) denotes the Riemann-Liouville fractional derivative of order \(\alpha_i \in (0,1)\) w.r.t. time and each operator \({\mathscr{A}}(t)\) arises from a time depending sesquilinear form \(\mathfrak{a}(t)\) on a Hilbert space \({\mathscr{H}}\) with constant domain \({\mathscr{V}}\), such that \({\mathscr{V}}\) is continuously and densely embedded into \({\mathscr{H}}\). We prove non-autonomous maximal \(L^p\)-regularity results on \({\mathscr{V}}^\prime\) and other regularity properties for the solutions of the above equation under minimal regularity assumptions on the forms, the initial data \(u_0\) and the inhomogeneous term \(F\).Invariance analysis and some new exact analytic solutions of the time-fractional coupled Drinfeld-Sokolov-Wilson equationshttps://zbmath.org/1544.351842024-11-01T15:51:55.949586Z"Astha, Chauhan"https://zbmath.org/authors/?q=ai:astha.chauhan"Rajan, Arora"https://zbmath.org/authors/?q=ai:rajan.aroraSummary: In this work, the fractional Lie symmetry method is used to find the exact solutions of the time-fractional coupled Drinfeld-Sokolov-Wilson equations with the Riemann-Liouville fractional derivative. Time-fractional coupled Drinfeld-Sokolov-Wilson equations are obtained by replacing the first-order time derivative to the fractional derivatives (FD) of order \(\alpha\) in the classical Drinfeld-Sokolov-Wilson (DSW) model. Using the fractional Lie symmetry method, the Lie symmetry generators are obtained. With the help of symmetry generators, FCDSW equations are reduced into fractional ordinary differential equations (FODEs) with Erdélyi-Kober fractional differential operator. Also, we have obtained the exact solution of FCDSW equations and shown the effects of non-integer order derivative value on the solutions graphically. The effect of fractional order \(\alpha\) on the behavior of solutions is studied graphically. Finally, new conservation laws are constructed along with the formal Lagrangian and fractional generalization of Noether operators. It is quite interesting the exact analytic solutions are obtained in explicit form.General solution of the singular fractional Fornasini-Marchesini linear systemshttps://zbmath.org/1544.351852024-11-01T15:51:55.949586Z"Benyettou, Kamel"https://zbmath.org/authors/?q=ai:benyettou.kamel"Ghezzar, Mohammed Amine"https://zbmath.org/authors/?q=ai:ghezzar.mohammed-amine"Bouagada, Djillali"https://zbmath.org/authors/?q=ai:bouagada.djillaliSummary: The purpose of this research is to compute the solution of two dimensional singular systems expressed by Fornasini-Marchesini models. A new result using some 2D transforms is given. The goal of this study is to discuss the applicability of the fundamental matrix and delta Kronecker to solve this class of system. The derived results are then compared with the existing the solution formula for the standard models. All the obtained study results are expressed numerically to demonstrate the validity and effectiveness of the proposed method.The fractional porous medium equation on noncompact Riemannian manifoldshttps://zbmath.org/1544.351862024-11-01T15:51:55.949586Z"Berchio, Elvise"https://zbmath.org/authors/?q=ai:berchio.elvise"Bonforte, Matteo"https://zbmath.org/authors/?q=ai:bonforte.matteo"Grillo, Gabriele"https://zbmath.org/authors/?q=ai:grillo.gabriele"Muratori, Matteo"https://zbmath.org/authors/?q=ai:muratori.matteoSummary: We study nonnegative solutions to the fractional porous medium equation on a suitable class of connected, noncompact Riemannian manifolds. We provide existence and smoothing estimates for solutions, in an appropriate weak (dual) sense, for data belonging either to the usual \(L^{1}\) space or to a considerably larger weighted space determined in terms of the fractional Green function. The class of manifolds for which the results hold includes both the Euclidean and the hyperbolic spaces and even in the Euclidean situation involves a class of data which is larger than the previously known one.Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacianhttps://zbmath.org/1544.351872024-11-01T15:51:55.949586Z"Borthagaray, Juan Pablo"https://zbmath.org/authors/?q=ai:borthagaray.juan-pablo"Nochetto, Ricardo H."https://zbmath.org/authors/?q=ai:nochetto.ricardo-h"Salgado, Abner J."https://zbmath.org/authors/?q=ai:salgado.abner-jSummary: We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian \((-\Delta)^s\) in a Lipschitz bounded domain \(\Omega\subset\mathbb{R}^n\) satisfying the exterior ball condition. The weight is a power of the distance to the boundary \(\partial\Omega\) of \(\Omega\) that accounts for the singular boundary behavior of the solution for any \(0 < s < 1\). These bounds then serve us as a guide in the design and analysis of a finite element scheme over graded meshes for any dimension \(n\), which is optimal for \(n = 2\).Highest cusped waves for the fractional KdV equationshttps://zbmath.org/1544.351882024-11-01T15:51:55.949586Z"Dahne, Joel"https://zbmath.org/authors/?q=ai:dahne.joelSummary: In this paper we prove the existence of highest, cusped, traveling wave solutions for the fractional KdV equations \(f_t + ff_x = |D|^\alpha f_x\) for all \(\alpha \in (-1, 0)\) and give their exact leading asymptotic behavior at zero. The proof combines careful asymptotic analysis and a computer-assisted approach.On a non-local Kirchhoff type equation with random terminal observationhttps://zbmath.org/1544.351892024-11-01T15:51:55.949586Z"Duc, Phuong Nguyen"https://zbmath.org/authors/?q=ai:duc.phuong-nguyen"Van, Tien Nguyen"https://zbmath.org/authors/?q=ai:van.tien-nguyen"Anh, Tuan Nguyen"https://zbmath.org/authors/?q=ai:anh.tuan-nguyenSummary: In this work, we are concerned with the terminal value problem for the time fractional equation (in the sense of Conformable fractional derivative) with a nonlocal term of the Kirchhoff type
\[
\partial_t^\alpha u = K\Big(\|\nabla u\|_{L^2(\mathcal{D})}\Big)\Delta u + f(x,t), \quad (x,t) \in (0,T)\times \mathcal{D}
\]
subject to the final data which is blurred by random Gaussian white noise. The principal goal of this article is to recover the solution \(u \). This problem is severely ill-posed in the sense of Hadamard, because of the violation of the continuous dependence of the solution on the data (the solution's behavior does not change continuously with the final condition). By applying non-parametric estimates of the value data from observation data and the truncation method for the Fourier series, we obtain a regularized solution. Under some priori assumptions, we derive an error estimate between a mild solution and its regularized solution.Normalized solution for fractional Choquard equation with potential and general nonlinearityhttps://zbmath.org/1544.351902024-11-01T15:51:55.949586Z"Jin, Zhen-Feng"https://zbmath.org/authors/?q=ai:jin.zhenfeng"Sun, Hong-Rui"https://zbmath.org/authors/?q=ai:sun.hongrui"Zhang, Jianjun"https://zbmath.org/authors/?q=ai:zhang.jianjun"Zhang, Weimin"https://zbmath.org/authors/?q=ai:zhang.weiminSummary: In this paper, we consider the following fractional Choquard equation:
\[
\begin{cases}
(-\Delta)^s u+V(x) u+\lambda u=(I_\alpha \ast F(u)) f(u) \quad \text{in } \mathbb{R}^N, \\
\displaystyle\int_{\mathbb{R}^N} u^2\,\mathrm{d}x=\rho^2, \rho>0,
\end{cases}
\]
where \(s \in (0,1)\), \(\alpha \in (0, N)\), \(N>2s\) and \(V:\mathbb{R}^N \to \mathbb{R}\) is a continuous nonnegative potential and vanishes at infinity. Under some mild assumptions imposed on \(V\) and \(f\), we establish the existence of \(L^2\) -normalized solution \((u, \lambda) \in H^s (\mathbb{R}^N) \times \mathbb{R}^+\).Multiple solutions for Kirchhoff-Schrödinger problems of fractional \(p\)-Laplacian involving Sobolev-Hardy critical exponenthttps://zbmath.org/1544.351912024-11-01T15:51:55.949586Z"Lin, Xiaolu"https://zbmath.org/authors/?q=ai:lin.xiaolu"Zheng, Shenzhou"https://zbmath.org/authors/?q=ai:zheng.shenzhouSummary: This paper is devoted to multiple solutions to a Kirchhoff-Schrödinger type problem of fractional \(p\)-Laplacian involving the Sobolev-Hardy critical exponent and a parameter \(\lambda > 0\). With some suitable assumptions on the potential \(V(x)\) and the nonlinearity \(f(x, u)\), the Krasnoselskii's genus argument is exploited to show the existence of infinitely many solutions if \(\lambda\) is sufficiently large. Furthermore, we employ a fractional version of the concentration-compactness to prove that there are \(m\)-pairs solutions of the problem provided that \(\lambda\) is small enough and the nonlinear force \(f(x, \cdot)\) is odd.\(q\)-homotopy analysis method for time-fractional Newell-Whitehead equation and time-fractional generalized Hirota-Satsuma coupled KdV systemhttps://zbmath.org/1544.351922024-11-01T15:51:55.949586Z"Liu, Di"https://zbmath.org/authors/?q=ai:liu.di"Gu, Qiongya"https://zbmath.org/authors/?q=ai:gu.qiongya"Wang, Lizhen"https://zbmath.org/authors/?q=ai:wang.lizhen(no abstract)On a viscoelastic Kirchhoff equation with fractional Laplacianhttps://zbmath.org/1544.351932024-11-01T15:51:55.949586Z"Liu, Yang"https://zbmath.org/authors/?q=ai:liu.yang.5"Li, Zhang"https://zbmath.org/authors/?q=ai:li.zhangSummary: In this paper, we study a viscoelastic Kirchhoff equation with fractional Laplacian. We first prove local existence and uniqueness of solutions. By using the theory of potential wells, we further obtain the global existence of solutions. Moreover, in the case where the Kirchhoff function takes a common form, we obtain asymptotic behavior and blow-up of solutions.A volume constraint problem for the nonlocal doubly nonlinear parabolic equationhttps://zbmath.org/1544.351942024-11-01T15:51:55.949586Z"Misawa, Masashi"https://zbmath.org/authors/?q=ai:misawa.masashi"Nakamura, Kenta"https://zbmath.org/authors/?q=ai:nakamura.kenta"Yamaura, Yoshihiko"https://zbmath.org/authors/?q=ai:yamaura.yoshihikoSummary: We consider a volume constraint problem for the nonlocal doubly nonlinear parabolic equation, called the nonlocal \(p\)-Sobolev flow, and introduce a nonlinear intrinsic scaling, converting a prototype nonlocal doubly nonlinear parabolic equation into the nonlocal \(p\)-Sobolev flow. This paper is dedicated to Giuseppe Mingione on the occasion of his 50th birthday, who is a maestro in the regularity theory of PDEs.Initial value and terminal value problems for distributed order fractional diffusion equationshttps://zbmath.org/1544.351952024-11-01T15:51:55.949586Z"Peng, Li"https://zbmath.org/authors/?q=ai:peng.li"Zhou, Yong"https://zbmath.org/authors/?q=ai:zhou.yongSummary: In this work, we introduce and study two problems for diffusion equations with the distributed order fractional derivatives including the initial value problem and the terminal value problem. For the initial value problem, we establish some existence results and Hölder regularity for the mild solution. On the other hand, we also show the existence results and a decay estimate of the mild solution for the terminal value problems. Especially, the polynomial decay of the solutions to the terminal value problems is firstly included when the source function is equal to zero.D'Alembert formula for diffusion-wave equationhttps://zbmath.org/1544.351962024-11-01T15:51:55.949586Z"Pskhu, A. V."https://zbmath.org/authors/?q=ai:pskhu.arsen-vladimirovichSummary: We construct a representation of solutions for diffusion-wave equations as a sum of two solutions of the first order PDEs. Fractional differentiation is given by the Liouville fractional derivative. The representation is an analogue of the d'Alembert formula known for the wave equation. In the case of an infinite rectangular domain (half-strip), we give relations that connect the traces of the solutions involved in the representation on the boundary of the domain.Applications of homogenous balance principles combined with fractional calculus approach and separate variable method on investigating exact solutions to multidimensional fractional nonlinear PDEshttps://zbmath.org/1544.351972024-11-01T15:51:55.949586Z"Ren, Ruichao"https://zbmath.org/authors/?q=ai:ren.ruichao"Zhang, Shunli"https://zbmath.org/authors/?q=ai:zhang.shunli"Rui, Weiguo"https://zbmath.org/authors/?q=ai:rui.weiguoSummary: We investigate the exact solutions of multidimensional time-fractional nonlinear PDEs (fnPDEs) in this paper. In terms of the fractional calculus properties and the separate variable method, we present a new homogenous balance principle (HBP) on the basis of the (1 + 1)-dimensional time fnPDEs. Taking advantage of the new types of HBP together with fractional calculus formulas that subtly avoid the chain rule, the fnPDEs can be reduced to spatial PDEs, and then we solve these PDEs by the fractional calculus method and the separate variable approach. In this way, some new type exact solutions of the certain time-fractional (2 + 1)-dimensional KP equation, (3 + 1)-dimensional Zakharov-Kuznetsov (ZK) equation, and Jimbo-Miwa (JM) equation are explicitly obtained under both Riemann-Liouville derivatives and Caputo derivatives. The dynamical analysis of solutions is shown by numerical simulations of taking property parameters as well.A Hopf type lemma for nonlocal pseudo-relativistic equations and its applicationshttps://zbmath.org/1544.351982024-11-01T15:51:55.949586Z"Wang, Pengyan"https://zbmath.org/authors/?q=ai:wang.pengyanSummary: In this paper, we consider the nonlinear equation involving the nonlocal pseudo-relativistic operators
\[
(-\Delta +m^2)^s u(x) = f(x, u(x)),
\]
where \(0<s<1\) and mass \(m>0\). The nonlocal pseudo-relativistic operator includes the pseudo-relativistic Schrödinger operator \(\sqrt{-\Delta +m^2}\). When \(m \to 0^+\), the nonlocal pseudo-relativistic operator \((-\Delta + m^2)^s\) is also closely related to the fractional Laplacian operator \((-\Delta)^s\). But these two operators are quite different. We first establish a Hopf type lemma for anti-symmetric functions to nonlocal pseudo-relativistic operators, which play a key role in the method of moving planes. The main difficulty is to construct a suitable sub-solution to nonlocal pseudo-relativistic operators. Then we prove a pointwise estimate to nonlocal pseudo-relativistic operators. As an application, combined with the Hopf type lemma and the pointwise estimate, we obtain the radial symmetry and monotonicity of positive solutions to the above nonlinear nonlocal pseudo-relativistic equation in the whole space. We believe that the Hopf type lemma will become a powerful tool in applying the method of moving planes on nonlocal pseudo-relativistic equations to obtain qualitative properties of solutions.The well-posedness of semilinear fractional dissipative equations on \(\mathbb{R}^n\)https://zbmath.org/1544.351992024-11-01T15:51:55.949586Z"Yang, Yong Zhen"https://zbmath.org/authors/?q=ai:yang.yong-zhen"Zhou, Yong"https://zbmath.org/authors/?q=ai:zhou.yongSummary: In this paper, we study the well-posedness of time-space fractional dissipative equations. Combining the Hörmander-multipliers theory and asymptotic property of the Mittag-Leffler functions, we give a useful method to estimate the solution operator which is independent of the \(C_0\)-semigroup generated by the fractional Laplace operator \((-\Delta)^{\frac{\beta}{2}}\) and the Mainardi's Wright type function. Furthermore, we discuss the global/local well-posedness in some appropriate time-space Lebesgue space and Besov spaces. We also obtain several results about blow-up alternative and asymptotic behavior. The approaches are based on the Hörmander-multipliers theory, Gagliardo-Nirenberg inequalities, fixed point techniques, Sobolev embedding and some methods in harmonic analysis.Retraction notice to: ``Analysis of fuzzy Kuramoto-Sivashinsky equations under a generalized fuzzy fractional derivative operator''https://zbmath.org/1544.352002024-11-01T15:51:55.949586ZRetraction notice to paper [\textit{N. H. Aljahdaly} et al., J. Funct. Spaces 2022, Article ID 9517158, 11 p. (2022; Zbl 1495.35205)].
From the retraction note, ``This article has been retracted by Hindawi following an investigation undertaken by the publisher. This investigation has
uncovered evidence of one or more [\dots] indicators of
systematic manipulation of the publication process.
The presence of these indicators undermines our confidence
in the integrity of the article's content and we cannot, therefore,
vouch for its reliability.''On terminal value problem for fractional superdiffusive of Sobolev equation typehttps://zbmath.org/1544.352012024-11-01T15:51:55.949586Z"Nam, Bui Duc"https://zbmath.org/authors/?q=ai:nam.bui-duc"Huynh, Le Nhat"https://zbmath.org/authors/?q=ai:huynh.le-nhat"Long, Le Dinh"https://zbmath.org/authors/?q=ai:long.le-dinh"Gurefe, Yusuf"https://zbmath.org/authors/?q=ai:gurefe.yusufSummary: In this paper, we consider the terminal value problem for fractional super diffusive equation in case linear source function. This equation has many applications in physical phenomena. The results in this study are mainly provide the existence and regularity of the mild solution under the various assumptions of the input data.Level set-based shape optimization approach for the inverse optical tomography problemhttps://zbmath.org/1544.352022024-11-01T15:51:55.949586Z"Belhachmi, Zakaria"https://zbmath.org/authors/?q=ai:belhachmi.zakaria"Dhif, Rabeb"https://zbmath.org/authors/?q=ai:dhif.rabeb"Meftahi, Houcine"https://zbmath.org/authors/?q=ai:meftahi.houcine.1The paper deals with an inverse optical tomography problem. The forward problem is governed by a steady-state reaction-diffusion equation posed in the two dimensional space. The basic idea consists in reconstructing diffusion and absorbed coefficients from the Cauchy data as well as from domain measurement of the associated potential. The coefficients to be reconstructed are assumed to be given by piecewise constants written in terms of characteristic functions of their geometrical supports. The inverse problem is rewritten as a shape optimization problem with respect to the supports of the unknown coefficients. Two formulations are considered, one in terms of the Cauchy data by using a \(H^1\)-version of the Kohn-Vogelius functional, and another one in terms of the distributed potential by using standard last square functional. Existence of optimal shapes is ensured by Theorem 1, which represents the main theoretical contribution of the paper. The resulting shape optimization problems are solved on the basis of a level-set method driven by the associated distributed shape gradients, which are derived with the help of the velocity method applied within the Lagrangian formalism. Finally, a nice set of numerical experiments is presented, showing different features of the proposed approach, including its resilience with respect to noisy data and capability in reconstructing both diffusion and absorbed coefficients simultaneously.
Reviewer: Antonio André Novotny (Petrópolis)Shape and location recovery of laser excitation sources in photoacoustic imaging using topological gradient optimizationhttps://zbmath.org/1544.352032024-11-01T15:51:55.949586Z"BenSalah, Mohamed"https://zbmath.org/authors/?q=ai:salah.mohamed-ben|bensalah.mohamed-oudiSummary: This study focuses on source term inversion in fractional partial differential equations, specifically applied to photoacoustic imaging. This work contributes to advancing imaging techniques and provides practical insights for medical diagnostics and materials characterization. Our aim in this paper is to develop an accurate method for recovering the location and the shape of a laser excitation source from partial boundary data. To achieve this, we reformulate our inverse problem as an optimization challenge. We utilize topological sensitivity analysis to establish an asymptotic expansion of a relevant shape function. These theoretical findings form the basis for a rapid and precise detection algorithm. Additionally, we present several numerical experiments that demonstrate the effectiveness and accuracy of our proposed approach.Determination of the time-dependent effective ion collision frequency from an integral observationhttps://zbmath.org/1544.352042024-11-01T15:51:55.949586Z"Cao, Kai"https://zbmath.org/authors/?q=ai:cao.kai"Lesnic, Daniel"https://zbmath.org/authors/?q=ai:lesnic.danielSummary: Identification of physical properties of materials is very important because they are in general unknown. Furthermore, their direct experimental measurement could be costly and inaccurate. In such a situation, a cheap and efficient alternative is to mathematically formulate an inverse, but difficult, problem that can be solved, in general, numerically; the challenge being that the problem is, in general, nonlinear and ill-posed. In this paper, the reconstruction of a lower-order unknown time-dependent coefficient in a Cahn-Hilliard-type fourth-order equation from an additional integral observation, which has application to characterizing the nonlinear saturation of the collisional trapped-ion mode in a tokamak, is investigated. The local existence and uniqueness of the solution to such inverse problem is established by utilizing the Rothe method. Moreover, the continuous dependence of the unknown coefficient upon the measured data is derived. Next, the Tikhonov regularization method is applied to recover the unknown coefficient from noisy measurements. The stability estimate of the minimizer is derived by investigating an auxiliary linear fourth-order inverse source problem. Henceforth, the variational source condition can be verified. Then the convergence rate is obtained under such source condition.Logarithmic type stability for the simultaneous identification of Robin coefficient and heat flux in an elliptic equationhttps://zbmath.org/1544.352052024-11-01T15:51:55.949586Z"Chen, De-Han"https://zbmath.org/authors/?q=ai:chen.dehan"Cheng, Ting"https://zbmath.org/authors/?q=ai:cheng.ting"Jiang, Daijun"https://zbmath.org/authors/?q=ai:jiang.daijunSummary: In this paper, we aim to reconstruct the unknown Robin coefficient and unknown heat flux simultaneously in an elliptic system from Cauchy data on arbitrarily small portion of accessible boundary. A novel logarithmic type stability in terms of the negative Sobolev norms is established.A high contrast and resolution reconstruction algorithm in quantitative photoacoustic tomographyhttps://zbmath.org/1544.352062024-11-01T15:51:55.949586Z"Dey, Anwesa"https://zbmath.org/authors/?q=ai:dey.anwesa"Borzì, Alfio"https://zbmath.org/authors/?q=ai:borzi.alfio"Roy, Souvik"https://zbmath.org/authors/?q=ai:roy.souvikSummary: A framework for reconstruction of optical diffusion and absorption coefficients in quantitative photoacoustic tomography is presented. This framework is based on a Tikhonov-type functional with a regularization term promoting sparsity of the absorption coefficient and a prior involving a Kubelka-Munk absorption-diffusion relation that allows to obtain superior reconstructions. The reconstruction problem is formulated as the minimization of this functional subject to the differential constraint given by a photon-propagation model. The solution of this problem is obtained by a fast and robust sequential quadratic hamiltonian algorithm based on the Pontryagin maximum principle. Results of several numerical experiments demonstrate that the proposed computational strategy is able to obtain reconstructions of the optical coefficients with high contrast and resolution for a wide variety of objects.The free boundary for a semilinear non-homogeneous Bernoulli problemhttps://zbmath.org/1544.352072024-11-01T15:51:55.949586Z"Du, Lili"https://zbmath.org/authors/?q=ai:du.lili"Yang, Chunlei"https://zbmath.org/authors/?q=ai:yang.chunleiSummary: In the classical homogeneous one-phase Bernoulli-type problem, the free boundary consists of a ``regular'' part and a ``singular'' part, as \textit{H. W. Alt} and \textit{L. A. Caffarelli} have shown in their pioneer work
[J. Reine Angew. Math. 325, 105--144 (1981; Zbl 0449.35105)]
that regular points are \(C^{1, \gamma}\) in two-dimensions. Later,
\textit{G. S. Weiss} [J. Geom. Anal. 9, No. 2, 317--326 (1999; Zbl 0960.49026)]
first realized that in higher dimensions a critical dimension \(d^\ast\) exists so that the singularities of the free boundary can only occur when \(d \geqslant d^\ast\).
In this paper, we consider a non-homogeneous semilinear one-phase Bernoulli-type problem, and we show that the free boundary is a disjoint union of a regular and a singular set. Moreover, the regular set is locally the graph of a \(C^{1, \gamma}\) function for some \(\gamma \in (0, 1)\). In addition, there exists a critical dimension \(d^\ast\) so that the singular set is empty if \(d < d^\ast\), discrete if \(d = d^\ast\) and of locally finite \(\mathcal{H}^{d - d^\ast}\) Hausdorff measure if \(d > d^\ast\). As a byproduct, we relate the existence of viscosity solutions of a non-homogeneous problem to the Weiss-boundary adjusted energy, which provides an alternative proof to existence of viscosity solutions for non-homogeneous problems.Determination of unknown time-dependent heat source in inverse problems under nonlocal boundary conditions by finite integration methodhttps://zbmath.org/1544.352082024-11-01T15:51:55.949586Z"Hazanee, Areena"https://zbmath.org/authors/?q=ai:hazanee.areena"Makaje, Nifatamah"https://zbmath.org/authors/?q=ai:makaje.nifatamahSummary: In this study, we investigate the unknown time-dependent heat source function in inverse problems. We consider three general nonlocal conditions; two classical boundary conditions and one nonlocal over-determination, condition, these genereate six different cases. The finite integration method (FIM), based on numerical integration, has been adapted to solve PDEs, and we use it to discretize the spatial domain; we use backward differences for the time variable. Since the inverse problem is ill-posed with instability, we apply regularization to reduce the instability. We use the first-order Tikhonov's regularization together with the minimization process to solve the inverse source problem. Test examples in all six cases are presented in order to illustrate the accuracy and stability of the numerical solutions.Simultaneous uniqueness for the diffusion coefficient and initial value identification in a time-fractional diffusion equationhttps://zbmath.org/1544.352092024-11-01T15:51:55.949586Z"Jing, Xiaohua"https://zbmath.org/authors/?q=ai:jing.xiaohua"Jia, Junxiong"https://zbmath.org/authors/?q=ai:jia.junxiong"Song, Xueli"https://zbmath.org/authors/?q=ai:song.xueliSummary: This article investigates the uniqueness of simultaneously determining the diffusion coefficient and initial value in a time-fractional diffusion equation with derivative order \(\alpha\in(0, 1)\). By additional boundary measurements and a priori assumption on the diffusion coefficient, the uniqueness of the eigenvalues and an associated integral equation for the diffusion coefficient are firstly established. The proof is based on the Laplace transform and the expansion of eigenfunctions for the solution to the initial value/boundary value problem. Furthermore, by using these two results, the simultaneous uniqueness in determining the diffusion coefficient and initial value is demonstrated from the Liouville transform and Gelfand-Levitan theory. The result shows that the uniqueness in simultaneous identification can be achieved, provided the initial values non-orthogonality to the eigenfunction of differential operators, which incorporates only one diffusion coefficient rather than scenarios involving two diffusion coefficients.Inverse problem of determining diffusion matrix between different structures for time fractional diffusion equationhttps://zbmath.org/1544.352102024-11-01T15:51:55.949586Z"Peng, Feiyang"https://zbmath.org/authors/?q=ai:peng.feiyang"Tang, Yanbin"https://zbmath.org/authors/?q=ai:tang.yanbinSummary: In this paper we consider some inverse problems of determining the diffusion matrix between different structures for the time fractional diffusion equation featuring a Caputo derivative. We first study an inverse problem of determining the diffusion matrix in the period structure using data from the corresponding homogenized equation, then we investigate an inverse problem of determining the diffusion matrix in the homogenized equation using data from the corresponding period structure of the oscillating equation. Finally, we establish the stability and uniqueness for the first inverse problem, and the asymptotic stability for the second inverse problem.Iterated fractional Tikhonov method for recovering the source term and initial data simultaneously in a two-dimensional diffusion equationhttps://zbmath.org/1544.352112024-11-01T15:51:55.949586Z"Qiao, Yu"https://zbmath.org/authors/?q=ai:qiao.yu.2|qiao.yu|qiao.yu.1"Xiong, Xiangtuan"https://zbmath.org/authors/?q=ai:xiong.xiangtuanSummary: In this paper, an inverse problem of a two-dimensional diffusion equation is considered. The purpose here includes recovering not only the source term but also the initial value from given observations at two fixed times \(t = T_1\) and \(t = T_2\). The uniqueness of the solutions for the inverse problem is given. Instead of the classical Tikhonov regularization strategy, we propose an iterated fractional Tikhonov regularization method to solve this problem, an \textit{a priori} and an \textit{a posterior} parameter selection rules and corresponding convergence rates are derived. For verification of the theoretical estimates, several numerical examples are constructed and compared with the standard iterated Tikhonov regularization method.An inverse problem for the wave equation with two nonlinear termshttps://zbmath.org/1544.352122024-11-01T15:51:55.949586Z"Romanov, V. G."https://zbmath.org/authors/?q=ai:romanov.vladimir-gSummary: An inverse problem for a second-order hyperbolic equation containing two nonlinear terms is studied. The problem is to reconstruct the coefficients of the nonlinearities. The Cauchy problem with a point source located at a point \(\mathbf{y}\) is considered. This point is a parameter of the problem and successively runs over a spherical surface \(S \). It is assumed that the desired coefficients are nonzero only in a domain lying inside \(S\). The trace of the solution of the Cauchy problem on \(S\) is specified for all possible values of \(\mathbf{y}\) and for times close to the arrival of the wave from the source to the points on the surface \(S \); this allows reducing the inverse problem under consideration to two successively solved problems of integral geometry. Solution stability estimates are found for these two problems.Regularity of flat free boundaries for two-phase \(p(x)\)-Laplacian problems with right hand sidehttps://zbmath.org/1544.352132024-11-01T15:51:55.949586Z"Ferrari, Fausto"https://zbmath.org/authors/?q=ai:ferrari.fausto"Lederman, Claudia"https://zbmath.org/authors/?q=ai:lederman.claudia-bThe paper studies viscosity solutions to two-phase free boundary problems involving the \(p(x)\)-Laplacian with a non-zero right-hand side. The authors establish that flat free boundaries are \(C^{1,\gamma}\), without assuming Lipschitz continuity of the solutions. These findings represent the first regularity results in the literature for such problems, even when \(p(x) \equiv\) constant, applicable to both singular and degenerate operators. By applying to viscosity solutions, the results offer wide applicability and mark a significant advancement in the understanding of free boundary problems associated with the \(p(x)\)-Laplacian.
Reviewer: Damião J. Araújo (João Pessoa)On nonminimizing solutions of elliptic free boundary problemshttps://zbmath.org/1544.352142024-11-01T15:51:55.949586Z"Perera, Kanishka"https://zbmath.org/authors/?q=ai:perera.kanishkaThe paper presents a variational framework for exploring nonminimizing solutions to elliptic free boundary problems, which are challenging due to their non-energy-minimizing nature. The authors employ this framework to derive mountain pass solutions for critical and subcritical superlinear problems. A significant achievement is the establishment of full regularity for the free boundary in two dimensions and partial regularity in higher dimensions.
Reviewer: Damião J. Araújo (João Pessoa)Generalized fractional Dirac type operatorshttps://zbmath.org/1544.470792024-11-01T15:51:55.949586Z"Restrepo, Joel E."https://zbmath.org/authors/?q=ai:restrepo.joel-esteban"Ruzhansky, Michael"https://zbmath.org/authors/?q=ai:ruzhansky.michael-v"Suragan, Durvudkhan"https://zbmath.org/authors/?q=ai:suragan.durvudkhanSummary: We introduce a class of fractional Dirac type operators with time variable coefficients by means of a Witt basis, the Djrbashian-Caputo fractional derivative and the fractional Laplacian, both operators defined with respect to some given functions. Direct and inverse fractional Cauchy type problems are studied for the introduced operators. We give explicit solutions of the considered fractional Cauchy type problems. We also use a recent method to recover a variable coefficient solution of some inverse fractional wave and heat type equations. Illustrative examples are provided.BV estimates on the transport density with Dirichlet region on the boundaryhttps://zbmath.org/1544.490092024-11-01T15:51:55.949586Z"Dweik, Samer"https://zbmath.org/authors/?q=ai:dweik.samerThe paper under review studies the regularity of the transport density \(\sigma\) in a PDE system of Monge-Kantorovich type, namely Equation (1.4) therein:
\[
\begin{cases}
-\nabla \cdot [\sigma \nabla u] = f \qquad & \text{ in } \mathring{\Omega},\\
u = g \qquad & \text{ on } \partial \Omega,\\
|\nabla u | \leq 1 \qquad & \text{ in } \Omega,\\
|\nabla u| = 1\qquad & \sigma\text{-a.e.}
\end{cases}
\]
Here \(\Omega\) is a compact domain \(\mathbb{R}^2\) with \(C^{2,1}\)-boundary. The Dirichlet boundary data \(g\) in in \(C^{2,1}(\partial \Omega)\) and \(\beta\)-Lipschitz in \(\Omega\) with \(\beta<1\).
It is proved that if \(f\) is in \(BV(\Omega)\cap L^\infty(\Omega)\) (or \(W^{1,1}(\Omega)\cap L^\infty(\Omega)\), resp.), then \(\sigma\) is of BV-regularity (or \(W^{1,1}\)-regularity, resp.). A counterexample is constructed that \(f \in C^\infty\) does \textit{not} imply \(\sigma \in W^{1,5}\).
The techniques used in this paper are specific for two dimensions. The BV-regularity in higher dimensions remains open.
Reviewer: Siran Li (Shanghai)Optimal control of stochastic delay differential equations and applications to path-dependent financial and economic modelshttps://zbmath.org/1544.490242024-11-01T15:51:55.949586Z"De Feo, Filippo"https://zbmath.org/authors/?q=ai:de-feo.filippo"Federico, Salvatore"https://zbmath.org/authors/?q=ai:federico.salvatore|federico.salvatore.1"Święch, Andrzej"https://zbmath.org/authors/?q=ai:swiech.andrzejA class of optimal control problems of stochastic differential delay equations is considered. First, the problem is rewritten in a suitable infinite-dimensional Hilbert space. Then, using the dynamic programming approach, the value function of the problem is characterized as the unique viscosity solution of the associated infinite-dimensional Hamilton-Jacobi-Bellman equation. Finally, the authors prove a \(C^{1, \alpha}\)-partial regularity of the value function. The optimal control problem at hand being not Markovian due to the delay, in order to regain Markovianity and approach the problem by dynamic programming, following a well-known procedure ([\textit{A. Bensoussan} et al., Representation and control of infinite dimensional systems. 2nd ed. Boston, MA: Birkhäuser (2007; Zbl 1117.93002)] for deterministic delay equations and [\textit{A. Chojnowska Michalik}, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 26, 635--642 (1978; Zbl 0415.60057); \textit{G. Da Prato} and \textit{J. Zabczyk}, Stochastic equations in infinite dimensions. 2nd ed. Cambridge: Cambridge University Press (2014; Zbl 1317.60077); \textit{S. Federico} et al., SIAM J. Control Optim. 48, No. 8, 4910--4937 (2010; Zbl 1208.49048)] for the stochastic case, the state equation is reformulated by lifting it to an infinite-dimensional space. The state equation is rewritten using an adequate maximal dissipative operator, the authors introducing an adequate associated operator which is shown to satisfy the weak \(B\)-condition.This is needed to be in the framework of the theory of viscosity solutions to the asscoiated HJB equation. Then estimates for solutions of the state equation, the cost functional and the value function are proven, as well as regularity properties. The value function \(V\) is characterized as the unique \(B\)-continuous viscosity solution to the associated HJB equation, thus providing existence and uniqueness results for fully nonlinear HJB equations in Hilbert spaces related to a general class of stochastic optimal control problems with delays involving controls in the diffusion coefficient. Note that besides the innovative results and so clear a writing, a very nice reliable synthetic introduction to important questions and paths to such issues is here provided. The results are applied to path dependent financial and economic problems, two examples are provided (one is a Merton-like problem with path dependent coefficients, and the other an optimal advertising with delays as in [\textit{F. Gozzi} and \textit{C. Marinelli}, Lect. Notes Pure Appl. Math. 245, 133--148 (2006; Zbl 1107.93035)]) and a section is dedicated to possible applications.
Reviewer: Lisa Morhaim (Paris)Solving inverse obstacle scattering problem with latent surface representationshttps://zbmath.org/1544.490322024-11-01T15:51:55.949586Z"Chen, Junqing"https://zbmath.org/authors/?q=ai:chen.junqing"Jin, Bangti"https://zbmath.org/authors/?q=ai:jin.bangti"Liu, Haibo"https://zbmath.org/authors/?q=ai:liu.haiboSummary: We propose a novel iterative numerical method to solve the three-dimensional inverse obstacle scattering problem of recovering the shape of an obstacle from far-field measurements. To address the inherent ill-posed nature of the inverse problem, we advocate the use of a trained latent representation of surfaces as the generative prior. This prior enjoys excellent expressivity within the given class of shapes, and meanwhile, the latent dimensionality is low, which greatly facilitates the computation. Thus, the admissible manifold of surfaces is realistic and the resulting optimization problem is less ill-posed. We employ the shape derivative to evolve the latent surface representation, by minimizing the loss, and we provide a local convergence analysis of a gradient descent type algorithm to a stationary point of the loss. We present several numerical examples, including also backscattered and phaseless data, to showcase the effectiveness of the proposed algorithm.
{{\copyright} 2024 IOP Publishing Ltd}A characterization of gauge balls in \(\mathbb{H}^n\) by horizontal curvaturehttps://zbmath.org/1544.530382024-11-01T15:51:55.949586Z"Guidi, Chiara"https://zbmath.org/authors/?q=ai:guidi.chiara"Martino, Vittorio"https://zbmath.org/authors/?q=ai:martino.vittorio"Tralli, Giulio"https://zbmath.org/authors/?q=ai:tralli.giulioSummary: In this paper, we aim at identifying the level sets of the gauge norm in the Heisenberg group \(\mathbb{H}^n\) via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in \(\mathbb{H}^1\) under an assumption on the location of the singular set, and in \(\mathbb{H}^n\) for \(n\geq 2\) in the proper class of horizontally umbilical hypersurfaces.Some gradient estimates for nonlinear heat-type equations on smooth metric measure spaces with compact boundaryhttps://zbmath.org/1544.580072024-11-01T15:51:55.949586Z"Abolarinwa, Abimbola"https://zbmath.org/authors/?q=ai:abolarinwa.abimbolaSummary: In this paper we prove some Hamilton type and Li-Yau type gradient estimates on positive solutions to generalized nonlinear parabolic equations on smooth metric measure space with compact boundary. The geometry of the space in terms of lower bounds on the weighted Bakry-Émery Ricci curvature tensor and weighted mean curvature of the boundary are key in proving generalized local and global gradient estimates. Various applications of these gradient estimates in terms of parabolic Harnack inequalities and Liouville type results are discussed. Further consequences to some specific models informed by the nature of the nonlinearities are highlighted.Schauder estimates for nonlocal equations with singular Lévy measureshttps://zbmath.org/1544.600532024-11-01T15:51:55.949586Z"Hao, Zimo"https://zbmath.org/authors/?q=ai:hao.zimo"Wang, Zhen"https://zbmath.org/authors/?q=ai:wang.zhen.7|wang.zhen.20|wang.zhen.5|wang.zhen.10|wang.zhen.2|wang.zhen.1|wang.zhen.12|wang.zhen.14|wang.zhen.3|wang.zhen|wang.zhen.17|wang.zhen.13|wang.zhen.8|wang.zhen.9|wang.zhen.23"Wu, Mingyan"https://zbmath.org/authors/?q=ai:wu.mingyanThe authors ``establish Schauder's estimates for the following non-local equations in \(\mathbb{R}^d\):
\[
\partial_t u=\mathcal{L}_{k,\sigma}^{(\alpha)}u +b \cdot \nabla u +f , u(0)=0
\]
where \(\alpha\in(1/2,2)\) and \(b:\mathbb{R}_+ \times \mathbb{R}^d\) is an unbounded local \(\beta\)-order Hölder function in \(x\) uniformly in \(t\)', and \(\mathcal{L}_{k,\sigma}^{(\alpha)}\) is a certain non-local \(\alpha\)-stable-like operator.'' ``It is well-known that Schauder's estimates play a basic role in constructing the classical solution for quasilinear partial differential equations (...), and also give an approach to show the well-posedness of stochastic differential equations (...).''
Reviewer: Alexander Schnurr (Siegen)Elastic drifted Brownian motions and non-local boundary conditionshttps://zbmath.org/1544.600832024-11-01T15:51:55.949586Z"D'Ovidio, Mirko"https://zbmath.org/authors/?q=ai:dovidio.mirko"Iafrate, Francesco"https://zbmath.org/authors/?q=ai:iafrate.francescoSummary: We provide a deep connection between elastic drifted Brownian motions and inverses to tempered subordinators. Based on this connection, we establish a link between multiplicative functionals and dynamical boundary conditions given in terms of non-local equations in time. Indeed, we show that the multiplicative functional associated to the elastic Brownian motion with drift is equivalent to a functional associated with non-local boundary conditions of tempered type. By exploiting such connections we write some functionals of the drifted Brownian motion in terms of a simple (positive and non-decreasing) process, the inverse of a tempered subordinator. In our view, such a representation is useful in many applications and brings new light on dynamic boundary value problems.Weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic SPDEs driven by additive noisehttps://zbmath.org/1544.650252024-11-01T15:51:55.949586Z"Mukam, Jean Daniel"https://zbmath.org/authors/?q=ai:mukam.jean-daniel"Tambue, Antoine"https://zbmath.org/authors/?q=ai:tambue.antoineSummary: This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.Ill-posedness of time-dependent inverse problems in Lebesgue-Bochner spaceshttps://zbmath.org/1544.650932024-11-01T15:51:55.949586Z"Burger, Martin"https://zbmath.org/authors/?q=ai:burger.martin"Schuster, Thomas"https://zbmath.org/authors/?q=ai:schuster.thomas"Wald, Anne"https://zbmath.org/authors/?q=ai:wald.anneSummary: We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover parameters from given observations where the parameters or the data depend on time. There are various important applications being subject of current research that belong to this class of problems. Typically inverse problems are ill-posed in the sense that already small noise in the data causes tremendous errors in the solution. In this article we present two different concepts of ill-posedness: temporally (pointwise) ill-posedness and uniform ill-posedness with respect to the Lebesgue-Bochner setting. We investigate the two concepts by means of a typical setting consisting of a time-depending observation operator composed by a compact operator. Furthermore we develop regularization methods that are adapted to the respective class of ill-posedness.
{{\copyright} 2024 The Author(s). Published by IOP Publishing Ltd}On the convergence of locally one-dimensional schemes for the differential equation in partial derivatives of fractional orders in a multidimensional domainhttps://zbmath.org/1544.651262024-11-01T15:51:55.949586Z"Bazzaev, Alexander K."https://zbmath.org/authors/?q=ai:bazzaev.aleksandr-kazbekovich(no abstract)Time two-grid technique combined with temporal second order difference method for semilinear fractional diffusion-wave equationshttps://zbmath.org/1544.651282024-11-01T15:51:55.949586Z"Cen, Dakang"https://zbmath.org/authors/?q=ai:cen.dakang"Ou, Caixia"https://zbmath.org/authors/?q=ai:ou.caixia"Wang, Zhibo"https://zbmath.org/authors/?q=ai:wang.zhibo"Vong, Seakweng"https://zbmath.org/authors/?q=ai:vong.seakwengSummary: In this paper, we construct two high order difference schemes for semilinear fractional diffusion-wave equations by a symmetric order reduction method and standard order reduction method at first. To improve the computational efficiency, an efficient time two-grid algorithm is then proposed. The global convergence order of the two-grid schemes reaches \(O(\tau_F^2 +\tau_C^4 +h^2)\), where \(\tau_F\) and \(\tau_C\) represent the time-step sizes on the fine and coarse grids respectively, while \(h\) is the space-step size. Furthermore, stability and convergence analysis of the derived schemes are carefully verified by the energy method. Finally, a numerical experiment is carried out to show the effectiveness of theoretical statements.Construction and analysis of structure-preserving numerical algorithm for two-dimensional damped nonlinear space fractional Schrödinger equationhttps://zbmath.org/1544.651292024-11-01T15:51:55.949586Z"Ding, Hengfei"https://zbmath.org/authors/?q=ai:ding.hengfei"Qu, Haidong"https://zbmath.org/authors/?q=ai:qu.haidong"Yi, Qian"https://zbmath.org/authors/?q=ai:yi.qianSummary: In this paper, we present a novel high-order structure-preserving numerical scheme for solving the damped nonlinear space fractional Schrödinger equation (DNSFSE) in two spatial dimensions. The main idea of constructing new algorithm consists of two parts. Firstly, we introduce an auxiliary exponential variable to transform the original DNSFSE into a modified one. The modified DNSFSE subjects to the conservation of mass and energy, which is crucial to develop structure-preserving numerical schemes. Secondly, we construct a high-order numerical differential formula to approximate the Riesz derivative in space, which contributes to a semi-discrete difference scheme for the modified DNSFSE. Combining the semi-discrete scheme with the variant Crank-Nicolson method in time, we can obtain the fully-discrete difference scheme for solving the modified DNSFSE. The advantage of the proposed scheme is that a fourth-order convergent accuracy can be achieved in space while maintaining the conservation of mass and energy. Subsequently, we conduct a detailed study on the boundedness, uniqueness, and convergence of solution for fully-discrete scheme. Furthermore, an improved efficient iterative algorithm is proposed for the fully-discrete scheme, which has the advantage of maintaining the same convergence order as the original difference scheme. Finally, extensive numerical results are reported to further verify the correctness of theoretical analysis and the effectiveness of the proposed numerical algorithm.A fast compact finite difference method for fractional Cattaneo equation based on Caputo-Fabrizio derivativehttps://zbmath.org/1544.651392024-11-01T15:51:55.949586Z"Qiao, Haili"https://zbmath.org/authors/?q=ai:qiao.haili"Liu, Zhengguang"https://zbmath.org/authors/?q=ai:liu.zhengguang"Cheng, Aijie"https://zbmath.org/authors/?q=ai:cheng.aijieSummary: The Cattaneo equations with Caputo-Fabrizio fractional derivative are investigated. A compact finite difference scheme of Crank-Nicolson type is presented and analyzed, which is proved to have temporal accuracy of second order and spatial accuracy of fourth order. Since this derivative is defined with an integral over the whole passed time, conventional direct solvers generally take computational complexity of \(O(M N^2)\) and require memory of \(O(M N)\), with \(M\) and \(N\) the number of space steps and time steps, respectively. We develop a fast evaluation procedure for the Caputo-Fabrizio fractional derivative, by which the computational cost is reduced to \(O(M N)\) operations; meanwhile, only \(O(M)\) memory is required. In the end, several numerical experiments are carried out to verify the theoretical results and show the applicability of the fast compact difference procedure.Second-order error analysis of the averaged L1 scheme \(\overline{\text{L1}}\) for time-fractional initial-value and subdiffusion problemshttps://zbmath.org/1544.651412024-11-01T15:51:55.949586Z"Shen, Jinye"https://zbmath.org/authors/?q=ai:shen.jinye"Zeng, Fanhai"https://zbmath.org/authors/?q=ai:zeng.fanhai"Stynes, Martin"https://zbmath.org/authors/?q=ai:stynes.martinSummary: Fractional initial-value problems (IVPs) and time-fractional initial-boundary value problems (IBVPs), each with a Caputo temporal derivative of order \(\alpha \in (0, 1)\), are considered. An averaged variant of the well-known L1 scheme is proved to be \(O(N^{-2})\) convergent for IVPs on suitably graded meshes with \(N\) points, thereby improving the \(O(N^{-(2- \alpha)})\) convergence rate of the standard L1 scheme. The analysis relies on a delicate decomposition of the temporal truncation error that yields a sharp dependence of the order of convergence on the degree of mesh grading used. This averaged L1 scheme can be combined with a finite difference or piecewise linear finite element discretization in space for IBVPs, and under a restriction on the temporal mesh width, one gets again \(O(N^{-2})\) convergence in time, together with \(O(h^2)\) convergence in space where \(h\) is the spatial mesh width. Numerical experiments support our results.Fast high-order compact finite difference methods based on the averaged \(L1\) formula for a time-fractional mobile-immobile diffusion problemhttps://zbmath.org/1544.651522024-11-01T15:51:55.949586Z"Zheng, Zi-Yun"https://zbmath.org/authors/?q=ai:zheng.zi-yun"Wang, Yuan-Ming"https://zbmath.org/authors/?q=ai:wang.yuanming.1|wang.yuanmingSummary: A two-dimensional time-fractional mobile-immobile diffusion problem with the Caputo time-fractional derivative of order \(\alpha \in (0,1)\) is considered. We show that the solution of the problem has a weak singularity at the initial time. Using the averaged \(L1\) formula to approximate the Caputo time-fractional derivative and using a compact finite difference approximation to discretize the space derivatives, we propose a high-order averaged \(L1\)-type compact finite difference method on the uniform space-time mesh for the problem. We then base on this method to develop an averaged \(L1\)-type compact alternating direction implicit (ADI) finite difference method and a fast sum-of-exponentials compact ADI finite difference method, both of which significantly reduce the storage requirements and the computational costs while maintaining the same global convergence rate. By using the discrete energy analysis technique, we rigorously prove that all methods are unconditionally stable and convergent, and they have the spatial global fourth-order convergence rate and the temporal global convergence rate of order \(\min \{2, 3-2\alpha\}\). For the case of \(\alpha >1/2\), we use the discrete minimum-maximum principle to prove that the temporal second-order convergence rate can also be achieved in positive time. Numerical results confirm the theoretical analysis results and demonstrate the computational efficiency of the methods.When data driven reduced order modeling meets full waveform inversionhttps://zbmath.org/1544.651592024-11-01T15:51:55.949586Z"Borcea, Liliana"https://zbmath.org/authors/?q=ai:borcea.liliana"Garnier, Josselin"https://zbmath.org/authors/?q=ai:garnier.josselin"Mamonov, Alexander V."https://zbmath.org/authors/?q=ai:mamonov.alexander-v"Zimmerling, Jörn"https://zbmath.org/authors/?q=ai:zimmerling.jorn-tSummary: Waveform inversion is concerned with estimating a heterogeneous medium, modeled by variable coefficients of wave equations, using sources that emit probing signals and receivers that record the generated waves. It is an old and intensively studied inverse problem with a wide range of applications, but the existing inversion methodologies are still far from satisfactory. The typical mathematical formulation is a nonlinear least squares data fit optimization and the difficulty stems from the nonconvexity of the objective function that displays numerous local minima at which local optimization approaches stagnate. This pathological behavior has at least three unavoidable causes: (1) The mapping from the unknown coefficients to the wave field is nonlinear and complicated. (2) The sources and receivers typically lie on a single side of the medium, so only backscattered waves are measured. (3) The probing signals are band limited and with high frequency content. There is a lot of activity in the computational science and engineering communities that seeks to mitigate the difficulty of estimating the medium by data fitting. In this paper we present a different point of view, based on reduced order models (ROMs) of two operators that control the wave propagation. The ROMs are called data driven because they are computed directly from the measurements, without any knowledge of the wave field inside the inaccessible medium. This computation is noniterative and uses standard numerical linear algebra methods. The resulting ROMs capture features of the physics of wave propagation in a complementary way and have surprisingly good approximation properties that facilitate waveform inversion.Uniqueness and numerical inversion in bioluminescence tomography with time-dependent boundary measurementhttps://zbmath.org/1544.651602024-11-01T15:51:55.949586Z"Gong, Rongfang"https://zbmath.org/authors/?q=ai:gong.rongfang"Liu, Xinran"https://zbmath.org/authors/?q=ai:liu.xinran"Shen, Jun"https://zbmath.org/authors/?q=ai:shen.jun.3"Huang, Qin"https://zbmath.org/authors/?q=ai:huang.qin"Sun, Chunlong"https://zbmath.org/authors/?q=ai:sun.chunlong"Zhang, Ye"https://zbmath.org/authors/?q=ai:zhang.ye.1Summary: In the paper, an inverse source problem in bioluminescence tomography (BLT) is investigated. BLT is a method of light imaging and offers many advantages such as sensitivity, cost-effectiveness, high signal-to-noise ratio and non-destructivity. It thus has promising prospects for many applications such as cancer diagnosis, drug discovery and development as well as gene therapies. In the literature, BLT is extensively studied based on the (stationary) diffusion approximation (DA) equation, where the distribution of peak sources is reconstructed and no solution uniqueness is guaranteed without proper \textit{a priori} information. In this work, motivated by solution uniqueness, a novel dynamic coupled DA model is proposed. Theoretical analysis including the well-posedness of the forward problem and the solution uniqueness of the inverse problem are given. Based on the new model, iterative inversion algorithms under the framework of regularizing schemes are introduced and applied to reconstruct the smooth and non-smooth sources. We discretize the regularization functional with the finite element method and give the convergence rate of numerical solutions. Several numerical examples are implemented to validate the effectiveness of the new model and the proposed algorithms.
{{\copyright} 2024 IOP Publishing Ltd}An inverse problem for a two-dimensional time-fractional sideways heat equationhttps://zbmath.org/1544.651612024-11-01T15:51:55.949586Z"Liu, Songshu"https://zbmath.org/authors/?q=ai:liu.songshu"Feng, Lixin"https://zbmath.org/authors/?q=ai:feng.lixinSummary: In this paper, we consider a two-dimensional (2D) time-fractional inverse diffusion problem which is severely ill-posed; i.e., the solution (if it exists) does not depend continuously on the data. A modified kernel method is presented for approximating the solution of this problem, and the convergence estimates are obtained based on both a priori choice and a posteriori choice of regularization parameters. The numerical examples illustrate the behavior of the proposed method.High-order splitting finite element methods for the subdiffusion equation with limited smoothing propertyhttps://zbmath.org/1544.651712024-11-01T15:51:55.949586Z"Li, Buyang"https://zbmath.org/authors/?q=ai:li.buyang"Yang, Zongze"https://zbmath.org/authors/?q=ai:yang.zongze"Zhou, Zhi"https://zbmath.org/authors/?q=ai:zhou.zhiSummary: In contrast with the diffusion equation which smoothens the initial data to \(C^\infty\) for \(t>0\) (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac-Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods.Preconditioning techniques of all-at-once systems for multi-term time-fractional diffusion equationshttps://zbmath.org/1544.651832024-11-01T15:51:55.949586Z"Gan, Di"https://zbmath.org/authors/?q=ai:gan.di"Zhang, Guo-Feng"https://zbmath.org/authors/?q=ai:zhang.guofeng"Liang, Zhao-Zheng"https://zbmath.org/authors/?q=ai:liang.zhaozhengSummary: In this paper, we consider solutions for discrete systems arising from multi-term time-fractional diffusion equations. Using discrete sine transform techniques, we find that all-at-once systems of such equations have a structure similar to that of diagonal-plus-Toeplitz matrices. We establish a generalized circulant approximate inverse preconditioner for the all-at-once systems. Through a detailed analysis of the preconditioned matrices, we show that the spectrum of the obtained preconditioned matrices is clustered around one. We give some numerical examples to demonstrate the effectiveness of the proposed preconditioner.A new analysis of fractional Drinfeld-Sokolov-Wilson model with exponential memoryhttps://zbmath.org/1544.651902024-11-01T15:51:55.949586Z"Bhatter, Sanjay"https://zbmath.org/authors/?q=ai:bhatter.sanjay"Mathur, Amit"https://zbmath.org/authors/?q=ai:mathur.amit"Kumar, Devendra"https://zbmath.org/authors/?q=ai:kumar.devendra.3"Singh, Jagdev"https://zbmath.org/authors/?q=ai:singh.jagdevSummary: The key purpose of this study is to suggest a new fractional extension of nonlinear Drinfeld-Sokolov-Wilson (DSW) equation with exponential memory. The nonlinear DSW equation plays a great role in describing dispersive water waves. The stability analysis is executed with the aid of fixed point theory. The advantage of FHATM over the other existing techniques is that its solution contains an auxiliary parameter \(\hbar\), which plays a big role in controlling the convergence of the solution. The outcomes of the study are presented in the form of graphs and tables. The results achieved by the use of the suggested scheme unfold that the used computational algorithm is very accurate, flexible, effective and simple to perform to examine the fractional order mathematical models.A novel Newton method for inverse elastic scattering problemshttps://zbmath.org/1544.651982024-11-01T15:51:55.949586Z"Chang, Yan"https://zbmath.org/authors/?q=ai:chang.yan"Guo, Yukun"https://zbmath.org/authors/?q=ai:guo.yukun"Liu, Hongyu"https://zbmath.org/authors/?q=ai:liu.hongyu"Zhang, Deyue"https://zbmath.org/authors/?q=ai:zhang.deyueSummary: This work is concerned with an inverse elastic scattering problem of identifying the unknown rigid obstacle embedded in an open space filled with a homogeneous and isotropic elastic medium. A Newton-type iteration method relying on the boundary condition is designed to identify the boundary curve of the obstacle. Based on the Helmholtz decomposition and the Fourier-Bessel expansion, we explicitly derive the approximate scattered field and its derivative on each iterative curve. Rigorous mathematical justifications for the proposed method are provided. Numerical examples are presented to verify the effectiveness of the proposed method.
{{\copyright} 2024 IOP Publishing Ltd}Copious closed forms of solutions for the fractional nonlinear longitudinal strain wave equation in microstructured solidshttps://zbmath.org/1544.740452024-11-01T15:51:55.949586Z"Qin, Haiyong"https://zbmath.org/authors/?q=ai:qin.haiyong"Khater, Mostafa M. A."https://zbmath.org/authors/?q=ai:khater.mostafa-m-a"Attia, Raghda A. M."https://zbmath.org/authors/?q=ai:attia.raghda-a-mSummary: A computational scheme is employed to investigate various types of the solution of the fractional nonlinear longitudinal strain wave equation. The novelty and advantage of the proposed method are illustrated by applying this model. A new fractional definition is used to convert the fractional formula of these equations into integer-order ordinary differential equations. Soliton, rational functions, the trigonometric function, the hyperbolic function, and many other explicit wave solutions are obtained.Incompressible jet flow past an obstaclehttps://zbmath.org/1544.760172024-11-01T15:51:55.949586Z"Cheng, Jianfeng"https://zbmath.org/authors/?q=ai:cheng.jianfeng"Huang, Jinli"https://zbmath.org/authors/?q=ai:huang.jinliThe authors consider the well-posedness of the incompressible jet flow issuing from a semi-infinitely long nozzle and moving around an obstacle. They obtain that one can find a differential pressure, such that there exists a unique solution to the incompressible jet flow, with the upper free boundary initiating smoothly from the endpoint of the nozzle wall, and the lower free boundary initiating smoothly from the surface of the obstacle. Moreover, the nonexistence of a finite cavity between the cavity boundary and the obstacle is established. Additionally, the authors show the optimal regularity at the detachment of the lower free boundary.
Reviewer: Fatma Gamze Düzgün (Ankara)An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasmahttps://zbmath.org/1544.760902024-11-01T15:51:55.949586Z"Goswami, Amit"https://zbmath.org/authors/?q=ai:goswami.amit"Singh, Jagdev"https://zbmath.org/authors/?q=ai:singh.jagdev"Kumar, Devendra"https://zbmath.org/authors/?q=ai:kumar.devendra.3"Sushila"https://zbmath.org/authors/?q=ai:sushila.Summary: In this paper, we present a coupling of homotopy perturbation technique and sumudu transform known as homotopy perturbation sumudu transform method (HPSTM). We show applicability of this method by solving fractional equal width (EW) equation, fractional modified equal width (MEW) equation and variant of fractional modified equal width (VMEW) equation. The fractional equal width equations play a key role in describing hydro-magnetic waves in cold plasma. Our aim is to study the nonlinear behavior of plasma system and highlight the important points. We examine the ability of HPSTM to study the fractional nonlinear systems and show its supremacy over other available numerical techniques. The other key point of this investigation is to examine two important fractional equations with different nonlinearity. The HPSTM gives excellent accuracy in analogous with the numerical solution. The numerical solutions indicate that the HPSTM is a powerful technique for studying the nonlinear behavior of plasma system very precisely and accurately.A reliable numerical algorithm for fractional advection-dispersion equation arising in contaminant transport through porous mediahttps://zbmath.org/1544.760922024-11-01T15:51:55.949586Z"Singh, Harendra"https://zbmath.org/authors/?q=ai:singh.harendra-prasad"Pandey, Rajesh K."https://zbmath.org/authors/?q=ai:pandey.rajesh-kumar"Singh, Jagdev"https://zbmath.org/authors/?q=ai:singh.jagdev"Tripathi, M. P."https://zbmath.org/authors/?q=ai:tripathi.manoj-pSummary: This article presents a reliable numerical approach for the fractional advection-dispersion equation by making use of Legendre scaling functions as a basis. The fractional advection-dispersion equation describes the anomalous transport in surface and subsurface water. Using two dimensional basis formed by Legendre scaling functions, we get operational matrix for fractional integrations and differentiations. Substituting these operational matrices in the equation leads linear algebraic equations whose solutions can be derived with the aid of Sylvester's approach; this in turn yields approximate solutions for advection-dispersion equation. Convergence analysis of the proposed scheme is presented. The potency and accuracy of the proposed numerical algorithm are shown by plotting error figures.A consensus-based alternating direction method for mixed-integer and PDE-constrained gas transport problemshttps://zbmath.org/1544.900482024-11-01T15:51:55.949586Z"Krug, Richard"https://zbmath.org/authors/?q=ai:krug.richard"Leugering, Günter"https://zbmath.org/authors/?q=ai:leugering.gunter"Martin, Alexander"https://zbmath.org/authors/?q=ai:martin.alexander"Schmidt, Martin"https://zbmath.org/authors/?q=ai:schmidt.martin"Weninger, Dieter"https://zbmath.org/authors/?q=ai:weninger.dieterSummary: We consider dynamic gas transport optimization problems, which lead to large-scale and nonconvex mixed-integer nonlinear optimization problems (MINLPs) on graphs. Usually, the resulting instances are too challenging to be solved by state-of-the-art MINLP solvers. In this paper, we use graph decompositions to obtain multiple optimization problems on smaller blocks, which can be solved in parallel and may result in simpler classes of optimization problems because not every block necessarily contains mixed-integer or nonlinear aspects. For achieving feasibility at the interfaces of the several blocks, we employ a tailored consensus-based penalty alternating direction method. Our numerical results show that such decomposition techniques can outperform the baseline approach of just solving the overall MINLP from scratch. However, a complete answer to the question of how to decompose MINLPs on graphs in dependence of the given model is still an open topic for future research.A wavelet collocation method for fractional Black-Scholes equations by subdiffusive modelhttps://zbmath.org/1544.913562024-11-01T15:51:55.949586Z"Damircheli, Davood"https://zbmath.org/authors/?q=ai:damircheli.davood"Razzaghi, Mohsen"https://zbmath.org/authors/?q=ai:razzaghi.mohsenIn this article, a spectral method based on the fractional-order generalized Taylor wavelets (FGTW) is proposed to tackle the problem of option pricing under the fractional Black-Scholes (B-S) model. The exact expression of the Riemann-Liouville fractional integral operator (RLFIO) of the FGTW is calculated by utilizing the regularized beta function. The results of computational experiments showed that this method can effectively, and very accurately estimate the target problems. Error analysis of the proposed numerical method is studied in Section 5.
Reviewer: Nikolay Kyurkchiev (Plovdiv)Time-fractional order biological systems with uncertain parametershttps://zbmath.org/1544.920012024-11-01T15:51:55.949586Z"Chakraverty, Snehashish"https://zbmath.org/authors/?q=ai:chakraverty.snehashish"Jena, Rajarama Mohan"https://zbmath.org/authors/?q=ai:jena.rajarama-mohan"Jena, Subrat Kumar"https://zbmath.org/authors/?q=ai:jena.subrat-kumarSee the review of the original edition in [Zbl 1465.92002].A reaction-diffusion fractional model for cancer virotherapy with immune response and Hattaf time-fractional derivativehttps://zbmath.org/1544.920772024-11-01T15:51:55.949586Z"El Younoussi, Majda"https://zbmath.org/authors/?q=ai:el-younoussi.majda"Hajhouji, Zakaria"https://zbmath.org/authors/?q=ai:hajhouji.zakaria"Hattaf, Khalid"https://zbmath.org/authors/?q=ai:hattaf.khalid"Yousfi, Noura"https://zbmath.org/authors/?q=ai:yousfi.nouraSummary: Recently, fractional partial differential equations (FPDEs) play a crucial role in the modeling of the dynamics of many systems arising from biology and other fields of science and engineering. The aim of this work is to model the interaction between nutrient, normal cells, tumor cells, M1 virus, and cytotoxic T lymphocyte (CTL) cells by using the new generalized Hattaf fractional (GHF) derivative. The mathematical model describing such type of interaction is rigorously formulated. Furthermore, the dynamical behaviors of the model are determined by three threshold parameters, namely, the ability of absorbing nutrient by normal cells, the ability of absorbing nutrient by tumor cells, and the reproduction number for cellular immunity.
For the entire collection see [Zbl 1531.92006].Implementation of the functional response in marine ecosystem: a state-of-the-art plankton modelhttps://zbmath.org/1544.922142024-11-01T15:51:55.949586Z"Chatterjee, Anal"https://zbmath.org/authors/?q=ai:chatterjee.anal"Pal, Samares"https://zbmath.org/authors/?q=ai:pal.samareshSummary: In this article, we apply different functional responses to introduce new mathematical feature in marine ecosystem. Strength of phytoplankton refuge and zooplankton refuge play big impacts in our system. We examine the different bifurcation scenarios when one or two different parameters vary together at the same time. A comparison of deterministic and stochastic approaches for analyzing the system dynamics are adopted. Analytical as well as numerical simulations are carried out to establish our findings.
For the entire collection see [Zbl 1515.92004].Control of the Cauchy system for an elliptic operator: the controllability methodhttps://zbmath.org/1544.933042024-11-01T15:51:55.949586Z"Guel, Bylli André B."https://zbmath.org/authors/?q=ai:guel.bylli-andre-b(no abstract)Accuracy estimation of the fractional, discrete-continuous model of the one-dimensional heat transfer processhttps://zbmath.org/1544.933092024-11-01T15:51:55.949586Z"Oprzędkiewicz, Krzysztof"https://zbmath.org/authors/?q=ai:oprzedkiewicz.krzysztof"Dziedzic, Klaudia"https://zbmath.org/authors/?q=ai:dziedzic.klaudiaSummary: In the paper a new, state space, finite dimensional, non integer order model of a one-dimensional heat transfer process is considered. The proposed model uses a well known finite difference method and fractional Caputo operator to express the time derivative. The second order backward difference describes the derivative along the length. The analytical formula of the step response is given. Accuracy and convergence of the proposed model are numerically analyzed and compared to previously proposed state space model using semigroup approach. Results of simulations point that the good accuracy of the proposed model can be achieved for its relatively low order.
For the entire collection see [Zbl 1522.93010].Discrete-time state observations control to synchronization of hybrid-impulses complex-valued multi-links coupled systemshttps://zbmath.org/1544.933162024-11-01T15:51:55.949586Z"Dai, Guang"https://zbmath.org/authors/?q=ai:dai.guang"Gao, Ruijie"https://zbmath.org/authors/?q=ai:gao.ruijie"Zhang, Chunmei"https://zbmath.org/authors/?q=ai:zhang.chunmei"Liu, Yan"https://zbmath.org/authors/?q=ai:liu.yan.31(no abstract)A solution of the complex fuzzy heat equation in terms of complex Dirichlet conditions using a modified Crank-Nicolson methodhttps://zbmath.org/1544.934452024-11-01T15:51:55.949586Z"Zureigat, Hamzeh"https://zbmath.org/authors/?q=ai:zureigat.hamzeh"Tashtoush, Mohammad A."https://zbmath.org/authors/?q=ai:tashtoush.mohammad-a"Jassar, Ali F. Al"https://zbmath.org/authors/?q=ai:jassar.ali-f-al"Az-Zo'bi, Emad A."https://zbmath.org/authors/?q=ai:az-zobi.emad-a"Alomari, Mohammad W."https://zbmath.org/authors/?q=ai:alomari.mohammad-wajeeh(no abstract)Adaptive synchronization of quaternion-valued neural networks with reaction-diffusion and fractional orderhttps://zbmath.org/1544.936602024-11-01T15:51:55.949586Z"Zhang, Weiwei"https://zbmath.org/authors/?q=ai:zhang.weiwei.2"Zhao, Hongyong"https://zbmath.org/authors/?q=ai:zhao.hongyong"Sha, Chunlin"https://zbmath.org/authors/?q=ai:sha.chunlinSummary: This paper is dedicated to the study of adaptive finite-time synchronization (FTS) for generalized delayed fractional-order reaction-diffusion quaternion-valued neural networks (GDFORDQVNN). Utilizing the suitable Lyapunov functional, Green's formula, and inequalities skills, testable algebraic criteria for ensuring the FTS of GDFORDQVNN are established on the basis of two adaptive controllers. Moreover, the numerical examples validate that the obtained results are feasible. Furthermore, they are also verified in image encryption as the application.Exponential stability of large-scale stochastic reaction-diffusion equationshttps://zbmath.org/1544.936722024-11-01T15:51:55.949586Z"Wang, Yuan"https://zbmath.org/authors/?q=ai:wang.yuan.4|wang.yuan|wang.yuan.2|wang.yuan.3|wang.yuan.1"Ren, Yong"https://zbmath.org/authors/?q=ai:ren.yong.2|ren.yong.4|ren.yong|ren.yong.1Summary: In this paper, we consider a class of large-scale stochastic reaction-diffusion systems. To prove the exponential stability of the system, we introduce the corresponding isolated subsystems. We show that the exponential stability of the isolated systems implies the exponential stability of the large-scale stochastic reaction-diffusion system under some conditions. Furthermore, we discuss a special case where the large-scale stochastic reaction-diffusion system is described in a hierarchical form. In this case, we prove that the original system is exponentially stable if and only if the corresponding subsystems are exponentially stable.Optimal stopping of conditional McKean-Vlasov jump diffusionshttps://zbmath.org/1544.937612024-11-01T15:51:55.949586Z"Agram, Nacira"https://zbmath.org/authors/?q=ai:agram.nacira"Øksendal, Bernt"https://zbmath.org/authors/?q=ai:oksendal.bernt-karstenSummary: The purpose of this paper is to study the optimal stopping problem of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal.
The key is that we combine the conditional McKean-Vlasov equation with the associated stochastic Fokker-Planck partial integro-differential equation for the conditional law of the state. This leads to a Markovian system which can be handled by using a version of a Dynkin formula.
Our verification result is illustrated by finding the optimal time to sell in a market with common noise and jumps.Improved image denoising through fractional anisotropic diffusion and resolution-tailored differentiation in the Fourier domainhttps://zbmath.org/1544.940922024-11-01T15:51:55.949586Z"Paskaš, Milorad P."https://zbmath.org/authors/?q=ai:paskas.milorad-pSummary: Fractional-order anisotropic diffusion-based denoising of images has been implemented in the literature through fractional differentiation in the Fourier domain. Furthermore, Fourier transform of the schemes on half-integer or integer mesh points has been used for the differentiation of images. In this paper, differentiation in the Fourier domain is proposed using schemes on fractional mesh points, aiming to enhance the performance of the algorithm. This can be regarded as employing fractional schemes at various resolutions, governed by a parameter of resolution within the range of \((0.5, 1)\). Variations in resolution affect the pixel neighborhood by incorporating additional information from interpolated pixels. Experiments conducted on a reference image dataset, using three quantitative measures, demonstrate that the proposed method surpasses the method from the literature for higher values of the parameter of resolution. The improvement is particularly noticeable at higher noise levels, where the proposed method consistently outperforms the method from the literature across all values of the parameter of resolution. As a side effect, the proposed method is less time-consuming than the original method, as it requires higher time steps within the numerical scheme. Experiments and stability analysis show that the proposed method reduces the number of iterations by three to four times compared to the original method.