Recent zbMATH articles in MSC 45https://zbmath.org/atom/cc/452024-04-15T15:10:58.286558ZWerkzeugOn traces in Hardy type analytic spaces in bounded strictly pseudoconvex domains and in tubular domains over symmetric coneshttps://zbmath.org/1530.320052024-04-15T15:10:58.286558Z"Shamoyan, Romi F."https://zbmath.org/authors/?q=ai:shamoyan.romi-f"Kurilenko, Sergey M."https://zbmath.org/authors/?q=ai:kurilenko.sergey-mSummary: We provide some new estimates on traces in new mixed norm Hardy type spaces and related new results on Bergman type intergal operators in Hardy type spaces in tubular domains over symmetric cones and bounded striclty pseudoconvex domains with smooth boundary. We generalize a well-known one dimensional result concerning traces of Hardy spaces obtained previously in the unit disk by various authors.Extended incomplete Riemann-Liouville fractional integral operators and related special functionshttps://zbmath.org/1530.330122024-04-15T15:10:58.286558Z"Özarslan, Mehmet Ali"https://zbmath.org/authors/?q=ai:ozarslan.mehmet-ali"Ustaoğlu, Ceren"https://zbmath.org/authors/?q=ai:ustaoglu.cerenSummary: In this study, we introduce the extended incomplete versions of the Riemann-Liouville (R-L) fractional integral operators and investigate their analytical properties rigorously. More precisely, we investigate their transformation properties in \(L_1\) and \(L_\infty\) spaces, and we observe that the extended incomplete fractional calculus operators can be used in the analysis of a wider class of functions than the extended fractional calculus operator. Moreover, by considering the concept of analytical continuation, definitions for extended incomplete R-L fractional derivatives are given and therefore the full fractional calculus model has been completed for each complex order. Then the extended incomplete \(\tau\)-Gauss, confluent and Appell's hypergeometric functions are introduced by means of the extended incomplete beta functions and some of their properties such as integral representations and their relations with the extended R-L fractional calculus has been given. As a particular advantage of the new fractional integral operators, some generating relations of linear and bilinear type for extended incomplete \(\tau\)-hypergeometric functions have been derived.On the Ulam type stabilities of a general iterative integro-differential equation including a variable delayhttps://zbmath.org/1530.340132024-04-15T15:10:58.286558Z"Tunç, Osman"https://zbmath.org/authors/?q=ai:tunc.osman"Sahu, D. R."https://zbmath.org/authors/?q=ai:sahu.daya-ram"Tunç, Cemil"https://zbmath.org/authors/?q=ai:tunc.cemilSummary: This work is interested in the existence and uniqueness of solutions, the Hyers-Ulam stability (HUS) and the Hyers-Ulam-Rassias stability (HURS) of a new and general iterative integro-delay differential equation (IIDDE) of first order including a variable delay. We prove the results regarding the HUS and the HURS of the considered problem on a bounded interval, half infinite intervals and an infinite interval using the Banach fixed point theorem (BFPT). The outcomes of this research work are new and they allow new enhancements to the existence theory, the HUS and HURS of the nonlinear IIDDEs of first order including a variable delay.A general decay result for the Cauchy problem of a fractional Laplace viscoelastic equationhttps://zbmath.org/1530.350552024-04-15T15:10:58.286558Z"Messaoudi, Salim A."https://zbmath.org/authors/?q=ai:messaoudi.salim-a"Lacheheb, Ilyes"https://zbmath.org/authors/?q=ai:lacheheb.ilyes(no abstract)Regularity theory for fractional reaction-subdiffusion equation and application to inverse problemhttps://zbmath.org/1530.350992024-04-15T15:10:58.286558Z"Tran Van Bang"https://zbmath.org/authors/?q=ai:tran-van-bang."Tran Van Tuan"https://zbmath.org/authors/?q=ai:tran-van-tuan.(no abstract)Using the decomposition method to solve the fractional order temperature distribution equation: a new approachhttps://zbmath.org/1530.351082024-04-15T15:10:58.286558Z"Rawashdeh, Mahmoud S."https://zbmath.org/authors/?q=ai:rawashdeh.mahmoud-saleh"Obeidat, Nazek A."https://zbmath.org/authors/?q=ai:obeidat.nazek-ahmad"Ababneh, Omar M."https://zbmath.org/authors/?q=ai:ababneh.omar-m(no abstract)Mean first-passage time of cell migration in confined domainshttps://zbmath.org/1530.351212024-04-15T15:10:58.286558Z"Serrano, Hélia"https://zbmath.org/authors/?q=ai:serrano.helia"Álvarez-Estrada, Ramón F."https://zbmath.org/authors/?q=ai:alvarez-estrada.ramon-f"Calvo, Gabriel F."https://zbmath.org/authors/?q=ai:calvo.gabriel-f(no abstract)Initial-boundary value problem for flows of a fluid with memory in a 3D network-like domainhttps://zbmath.org/1530.351782024-04-15T15:10:58.286558Z"Baranovskii, E. S."https://zbmath.org/authors/?q=ai:baranovskii.evgeni-sergeevich|baranovskii.evgenii-sergeevichSummary: We consider an initial-boundary value problem for an integro-differential system that describes 3D flows of a non-Newtonian fluid with memory in a network-like domain. The problem statement uses the Dirichlet boundary conditions for the velocity and pressure fields as well as Kirchhoff-type transmission conditions at the internal nodes of the network. A theorem on the existence and uniqueness of a time-continuous weak solution is proved. In addition, an energy equality for this solution is derived.On uniqueness results for solutions of the Benjamin equationhttps://zbmath.org/1530.352112024-04-15T15:10:58.286558Z"Cunha, Alysson"https://zbmath.org/authors/?q=ai:cunha.alyssonSummary: We prove that the uniqueness results obtained in [\textit{J. Jiménez Urrea}, J. Differ. Equations 254, No. 4, 1863--1892 (2013; Zbl 1259.35217)] for the Benjamin equation, cannot be extended for any pair of non-vanishing solutions. On the other hand, we study uniqueness results of solutions of the Benjamin equation. With this purpose, we showed that for any solutions \(u\) and \(v\) defined in \(\mathbb{R}\times[0,T]\), if there exists an open set \(I\subset\mathbb{R}\) such that \(u(\cdot, 0)\) and \(v(\cdot, 0)\) agree in \(I,\partial_t u(\cdot,0)\) and \(\partial_tv(\cdot,0)\) agree in \(I\), then \(u\equiv v\). A better version of this uniqueness result is also established. To finish, this type of uniqueness results were also proved for the nonlocal perturbation of the Benjamin-Ono equation (npBO) and for the regularized Benjamin-Ono equation (rBO).``Gradient-free'' diffuse approximations of the Willmore functional and Willmore flowhttps://zbmath.org/1530.352122024-04-15T15:10:58.286558Z"Dabrock, Nils"https://zbmath.org/authors/?q=ai:dabrock.nils"Knüttel, Sascha"https://zbmath.org/authors/?q=ai:knuttel.sascha"Röger, Matthias"https://zbmath.org/authors/?q=ai:roger.matthiasSummary: We introduce new diffuse approximations of the Willmore functional and the Willmore flow. They are based on a corresponding approximation of the perimeter that has been studied by \textit{S. Amstutz} and \textit{N. Van Goethem} [Interfaces Free Bound. 14, No. 3, 401--430 (2012; Zbl 1255.49070)]. We identify the candidate for the \(\Gamma\)-convergence, prove the \(\Gamma\)-limsup statement and justify the convergence to the Willmore flow by an asymptotic expansion. Furthermore, we present numerical simulations that are based on the new approximation.Asymptotic profiles of solutions for the generalized Fornberg-Whitham equation with dissipationhttps://zbmath.org/1530.352512024-04-15T15:10:58.286558Z"Fukuda, Ikki"https://zbmath.org/authors/?q=ai:fukuda.ikkiSummary: We consider the Cauchy problem for the generalized Fornberg-Whitham equation with dissipation. This is one of the nonlinear, nonlocal and dispersive-dissipative equations. The main topic of this paper is an asymptotic analysis for the solutions to this problem. We prove that the solution to this problem converges to the modified heat kernel. Moreover, we construct the second term of asymptotics for the solutions depending on the degree of the nonlinearity. In view of those second asymptotic profiles, we investigate the effects of the dispersion, dissipation and nonlinear terms on the asymptotic behavior of the solutions.Optimal distributed control for a viscous non-local tumour growth modelhttps://zbmath.org/1530.353222024-04-15T15:10:58.286558Z"Fornoni, Matteo"https://zbmath.org/authors/?q=ai:fornoni.matteoSummary: In this paper, we address an optimal distributed control problem for a non-local model of phase-field type, describing the evolution of tumour cells in presence of a nutrient. The model couples a non-local and viscous Cahn-Hilliard equation for the phase parameter with a reaction-diffusion equation for the nutrient. The optimal control problem aims at finding a therapy, encoded as a source term in the system, both in the form of radiotherapy and chemotherapy, which could lead to the evolution of the phase variable towards a desired final target. First, we prove strong well-posedness for the system of non-linear partial differential equations. In particular, due to the presence of a viscous regularisation, we can also consider double-well potentials of singular type and cross-diffusion terms related to the effects of chemotaxis. Moreover, the particular structure of the reaction terms allows us to prove new regularity results for this kind of system. Then, turning to the optimal control problem, we prove the existence of an optimal therapy and, by studying Fréchet-differentiability properties of the control-to-state operator and the corresponding adjoint system, we obtain the first-order necessary optimality conditions.Global dynamics and traveling waves for a diffusive SEIVS epidemic model with distributed delayshttps://zbmath.org/1530.353272024-04-15T15:10:58.286558Z"Wang, Lianwen"https://zbmath.org/authors/?q=ai:wang.lianwen.1"Wang, Xingyu"https://zbmath.org/authors/?q=ai:wang.xingyu"Liu, Zhijun"https://zbmath.org/authors/?q=ai:liu.zhijun.1"Wang, Yating"https://zbmath.org/authors/?q=ai:wang.yatingSummary: This contribution develops a delayed diffusive SEIVS epidemic model for predicting and quantifying transmission dynamics for some slowly progressive diseases with long-term latent stage, governed by reaction-diffusion integro-differential equations taking distributed delays of latency and waning immunity, spatial mobility, vaccination strategies, temporary immunity into account. Necessary and sufficient conditions not only for global asymptotic stability of the disease-free and endemic equilibria are just determined by the basic reproduction number, but also for the existence and nonexistence of traveling wave solution connecting the two equilibria fully depend on the minimal wave velocity and the basic reproduction number. The targeted model with exponential distributions is applied to fit the pulmonary tuberculosis (TB) case data in China, predict its spread trend and provide us with an improving understanding of the effectiveness of a few interventions. Furthermore, our analytical findings are numerically corroborated to characterize the spatiotemporal evolution of pulmonary TB.Viscosity solutions for nonlocal equations with space-dependent operatorshttps://zbmath.org/1530.353392024-04-15T15:10:58.286558Z"Buccheri, Stefano"https://zbmath.org/authors/?q=ai:buccheri.stefano"Stefanelli, Ulisse"https://zbmath.org/authors/?q=ai:stefanelli.ulisseSummary: We consider a class of elliptic and parabolic problems featuring a specific nonlocal operator of fractional Laplacian type, where integration is taken on variable domains. Both elliptic and parabolic problems are proved to be uniquely solvable in the viscosity sense. Moreover, some spectral properties of the elliptic operator are investigated, proving existence and simplicity of the first eigenvalue. Eventually, parabolic solutions are proven to converge to the corresponding limiting elliptic solution in the long-time limit.Existence and bifurcation of positive solutions for fractional \(p\)-Kirchhoff problemshttps://zbmath.org/1530.353542024-04-15T15:10:58.286558Z"Wang, Linlin"https://zbmath.org/authors/?q=ai:wang.linlin"Xing, Yuming"https://zbmath.org/authors/?q=ai:xing.yuming"Zhang, Binlin"https://zbmath.org/authors/?q=ai:zhang.binlin(no abstract)Volterra integral equation with power nonlinearityhttps://zbmath.org/1530.450012024-04-15T15:10:58.286558Z"Askhabov, Sultan Nazhmudinovich"https://zbmath.org/authors/?q=ai:askhabov.sultan-nazhmudinovichThe paper is devoted to the following nonlinear integral equations with power nonlinearities: \[ u^{\alpha}(x)=\int_{0}^{x}K(x,t)u(t)dt, \qquad \alpha>0, \qquad x\geq 0, \] with respect to the unknown function \(u(x)\) in the cone consisting of all non-negative and continuous functions on the positive half-axis \[Q_0=\left\{ u(x):\; u(x)\in C[0,\infty),\; u(x)>0 \text{ for } x>0\right\}.\]
A global theorem on the existence and uniqueness of solutions is proved by the method of weight metrics. The exact two-sided estimates of the solution are obtained using an integral inequality generalizing, in particular, Chebyshev's inequality. It is shown that this solution can be found by the method of successive approximations of Picard type. An estimate of the rate of convergence of successive approximations to an exact solution in terms of a weight metric is given. It is shown that, unlike the linear case, the nonlinear homogeneous Volterra integral equation, in addition to a trivial solution, can also have a non-trivial solution. Examples illustrating the results are discussed.
Reviewer: Alexander N. Tynda (Penza)Spike variations for stochastic Volterra integral equationshttps://zbmath.org/1530.450022024-04-15T15:10:58.286558Z"Wang, Tianxiao"https://zbmath.org/authors/?q=ai:wang.tianxiao"Yong, Jiongmin"https://zbmath.org/authors/?q=ai:yong.jiongminSummary: The spike variation technique plays a crucial role in deriving Pontryagin's type maximum principle of optimal controls for ordinary differential equations (ODEs), partial differential equations (PDEs), stochastic differential equations (SDEs), and (deterministic forward) Volterra integral equations (FVIEs), when the control domains are not assumed to be convex. It is natural to expect that such a technique could be extended to the case of (forward) stochastic Volterra integral equations (FSVIEs). However, by mimicking the case of SDEs, one encounters an essential difficulty of handling an involved quadratic term. To overcome this difficulty, we introduce an auxiliary process for which one can use Itô's formula, and develop new technologies inspired by stochastic linear-quadratic optimal control problems. Then the suitable representation of the above-mentioned quadratic form is obtained, and the second-order adjoint equations are derived. Consequently, the maximum principle of Pontryagin type is established. Some relevant extensions are investigated as well.Closed-form solutions for several classes of singular integral equations with convolution and Cauchy operatorhttps://zbmath.org/1530.450032024-04-15T15:10:58.286558Z"Bai, Songwei"https://zbmath.org/authors/?q=ai:bai.songwei"Li, Pingrun"https://zbmath.org/authors/?q=ai:li.pingrun"Sun, Meng"https://zbmath.org/authors/?q=ai:sun.mengThe authors discuss the existence of solutions to certain types of singular integral equations on the real line. The equations are assumed to contain a combination of integrals with Cauchy kernels and integrals with certain convolution kernels. Solutions are sought in the class \(L_2(\mathbb R) \cap \hat H\), where \(\hat H\) is the class of functions \(f\) that satisfy a Hölder condition of arbitrary order \(\nu \in (0,1]\) on some finite interval \([-A, A]\) and a corresponding condition \(|f(x) - f(y)| \le C |x^{-1} - y^{-1}|^\nu\) for \(x, y \notin [-A,A]\). The proofs are based on constructing certain Riemann boundary value problems related to the integral equations in question and on analyzing these boundary value problems.
Reviewer: Kai Diethelm (Schweinfurt)The attractivity of functional hereditary integral equationshttps://zbmath.org/1530.450042024-04-15T15:10:58.286558Z"Fang, Qingxiang"https://zbmath.org/authors/?q=ai:fang.qingxiang"Liu, Xiaoping"https://zbmath.org/authors/?q=ai:liu.xiaoping"Peng, Jigen"https://zbmath.org/authors/?q=ai:peng.jigenSummary: In this paper, the problem of attractivity of solutions for functional hereditary integral equations is initiated. A hereditary integral equation is converted into a second kind Volterra integral equation of convolution type by use of the property of convolution. Sufficient conditions for the existence of attractive solutions are established according to the fixed point theorem. The conclusions presented in this paper extend some published results. The effectiveness of the proposed results is illustrated by three numerical examples.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}An existence result with numerical solution of nonlinear fractional integral equationshttps://zbmath.org/1530.450052024-04-15T15:10:58.286558Z"Kazemi, Manochehr"https://zbmath.org/authors/?q=ai:kazemi.manochehr"Deep, Amar"https://zbmath.org/authors/?q=ai:deep.amar"Nieto, Juan"https://zbmath.org/authors/?q=ai:nieto.juanjo|gonzalez-nieto.juan-manuel|nieto.juan-antonio|nieto.juan-jose|nieto.juan-miguel|nieto.juan-iSummary: By utilizing the technique of Petryshyn's fixed point theorem in Banach algebra, we examine the existence of solutions for fractional integral equations, which include as special cases of many fractional integral equations that arise in various branches of mathematical analysis and their applications. Also, the numerical iterative method is employed successfully to find the solutions to fractional integral equations. Lastly, we recall some different cases and examples to verify the applicability of our study.
{{\copyright} 2023 John Wiley \& Sons, Ltd.}Overdetermined problems for negative power integral equations on bounded domainhttps://zbmath.org/1530.450062024-04-15T15:10:58.286558Z"Liu, Zhao"https://zbmath.org/authors/?q=ai:liu.zhao.1|liu.zhao"Hu, Yunyun"https://zbmath.org/authors/?q=ai:hu.yunyunSummary: The aim of this paper is to study symmetry and monotonicity of positive solutions for the following overdetermined problem
\[
\begin{cases}
u(x) = A \displaystyle\int_\Omega |x-y|^{\alpha-n} u^{-p} (y) \mathrm{d}y + B, \quad &\text{in } \Omega,\\
u > 0, & \text{in } \Omega,
\end{cases}
\]
where \(\alpha>n\), \(p>0\), \(A>0\), \(B \geq 0\), \(\Omega \subset \mathbb{R}^n\) is a bounded domain. We first prove that \(u=\text{const.}\) on \(\partial\Omega\) if and only if \(\Omega\) is a ball. Next we consider the partially overdetermined problem. If \(\Gamma\) is a proper open set of \(\partial\Omega\) and \(u = C\) in \(\Gamma \subseteq \partial\Omega\), we show that under some assumptions on the geometry of \(\Gamma\), \(\Omega\) is a ball. Furthermore, we derive that all positive solutions of above equations are radially symmetric and monotone increasing with respect to the radius by using the method of moving planes.On the solvability of a class of nonlinear functional integral equations involving Erdélyi-Kober fractional operatorhttps://zbmath.org/1530.450072024-04-15T15:10:58.286558Z"Pathak, Vijai Kumar"https://zbmath.org/authors/?q=ai:pathak.vijai-kumar"Mishra, Lakshmi Narayan"https://zbmath.org/authors/?q=ai:mishra.lakshmi-narayan"Mishra, Vishnu Narayan"https://zbmath.org/authors/?q=ai:mishra.vishnu-narayanSummary: In this paper, utilizing the technique of generalized Darbo's fixed-point theorem associated with measure of noncompactness in Banach space, we analyze the existence of solution for a class of nonlinear functional integral equations involving Erdélyi-Kober fractional operator. The existing result was obtained to strengthen the ones mentioned previously in the literature. An example for a class of nonlinear functional integral equations is also presented to validate our main result. Finally, we propose the numerical method formed by the modified homotopy perturbation approach to resolving the problem with acceptable accuracy.
{{\copyright} 2023 John Wiley \& Sons, Ltd.}On controllability for fractional Volterra-Fredholm systemhttps://zbmath.org/1530.450082024-04-15T15:10:58.286558Z"Hamoud, Ahmed A."https://zbmath.org/authors/?q=ai:hamoud.ahmed-abdullah"Jameel, Saif Aldeen M."https://zbmath.org/authors/?q=ai:jameel.saif-aldeen-m"Mohammed, Nedal M."https://zbmath.org/authors/?q=ai:mohammed.nedal-m"Emadifar, Homan"https://zbmath.org/authors/?q=ai:emadifar.homan"Parvaneh, Foroud"https://zbmath.org/authors/?q=ai:parvaneh.foroud"Khademi, Masoumeh"https://zbmath.org/authors/?q=ai:khademi.masoumehSummary: In this manuscript, we study the sufficient conditions for controllability of Volterra-Fredholm type fractional integro-differential systems in a Banach space. Fractional calculus and the fixed point theorem are used to derive the findings. Some examples are provided to illustrate the obtained results.The existence and compactness of the set of solutions for a 2-order nonlinear integrodifferential equation in \(N\) variables in a Banach spacehttps://zbmath.org/1530.450092024-04-15T15:10:58.286558Z"Le Thi Phuong Ngoc"https://zbmath.org/authors/?q=ai:le-thi-phuong-ngoc."Nguyen Thanh Long"https://zbmath.org/authors/?q=ai:nguyen-thanh-long.Summary: In this paper, by applying the fixed point theorem of Krasnosel'skii, we prove the existence and compactness of the set of solutions for a 2-order nonlinear integrodifferential equation in \(N\) variables in an arbitrary Banach space \(E\). Here, an appropriate Banach space \(X_1\) for the above equation is defined and a sufficient condition for relatively compact subsets in \(X_1\) is proved. An example is given to verify the efficiency of the used method.A well-posed parameter identification for nonlocal diffusion problemshttps://zbmath.org/1530.450102024-04-15T15:10:58.286558Z"Zhang, Shangyuan"https://zbmath.org/authors/?q=ai:zhang.shangyuan"Nie, Yufeng"https://zbmath.org/authors/?q=ai:nie.yufengSummary: This paper concerns the problem of identifying the diffusion parameter in the nonlocal diffusion model. The parameter identification problem is formulated as an optimal control problem, and the objective of the control is a cost functional formulated by the energy functional. By using the nonlocal vector calculus, in a manner analogous to the local partial differential equations counterpart, we proved that the cost functional is a strictly convex functional with a unique global minimizer. Moreover, the existence and uniqueness of parameter identification are further demonstrated. Finally, one-dimensional numerical experiments are given to illustrate our theoretical results and show that continuous and discontinuous parameters can be estimated.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}Different stabilities for oscillatory Volterra integral equationshttps://zbmath.org/1530.450112024-04-15T15:10:58.286558Z"Simões, Alberto Manuel"https://zbmath.org/authors/?q=ai:simoes.alberto-manuelSummary: Inspired by the increasing development of theories subordinate to the topic of stability in the sense of Ulam-Hyers and Ulam-Hyers-Rassias, we present in this paper new sufficient conditions for concluding the stability of classes of integral equations with kernels depending on sine and cosine functions. This will be done by taking the profit of fixed-point arguments in the framework of spaces of continuous functions endowed with a generalization of the Bielecki metric. After presenting the main theorems, some examples are provided to verify the effectiveness of the proposed theoretical results.
{{\copyright} 2023 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley \& Sons Ltd.}The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functionshttps://zbmath.org/1530.470602024-04-15T15:10:58.286558Z"Carlone, Raffaele"https://zbmath.org/authors/?q=ai:carlone.raffaele"Fiorenza, Alberto"https://zbmath.org/authors/?q=ai:fiorenza.alberto"Tentarelli, Lorenzo"https://zbmath.org/authors/?q=ai:tentarelli.lorenzoSummary: For kernels \(\nu\) which are positive and integrable we show that the operator \(g \mapsto J_\nu g = \int_0^x \nu(x - s) g(s) ds\) on a finite time interval enjoys a regularizing effect when applied to Hölder continuous and Lebesgue functions and a ``contractive'' effect when applied to Sobolev functions. For Hölder continuous functions, we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel, namely by the factor \(N(x) = \int_0^x \nu(s) d s\). For functions in Lebesgue spaces, we prove that an improvement always exists, and it can be expressed in terms of Orlicz integrability. Finally, for functions in Sobolev spaces, we show that the operator \(J_\nu\) ``shrinks'' the norm of the argument by a factor that, as in the Hölder case, depends on the function \(N\) (whereas no regularization result can be obtained).
These results can be applied, for instance, to Abel kernels and to the Volterra function \(\mathcal{I}(x) = \mu(x, 0, - 1) = \int_0^\infty x^{s - 1} /\Gamma(s) d s\), the latter being relevant for instance in the analysis of the Schrödinger equation with concentrated nonlinearities in \(\mathbb{R}^2\).Existence results for a system of nonlinear operator equations and block operator matrices in locally convex spaceshttps://zbmath.org/1530.470632024-04-15T15:10:58.286558Z"Bahidi, Fatima"https://zbmath.org/authors/?q=ai:bahidi.fatima"Krichen, Bilel"https://zbmath.org/authors/?q=ai:krichen.bilel"Mefteh, Bilel"https://zbmath.org/authors/?q=ai:mefteh.bilelSummary: The purpose of this paper is to prove some fixed point results dealing with a system of nonlinear equations defined in an angelic Hausdorff locally convex space \((X,\{|\,{\cdot}\,|_p\}_{p\in\Lambda})\) having the \(\tau \)-Krein-Šmulian property, where \(\tau\) is a weaker Hausdorff locally convex topology of \(X\). The method applied in our study is connected with a family \(\Phi_{\Lambda}^{\tau}\)-MNC of measures of weak noncompactness and with the concept of \(\tau \)-sequential continuity. As a special case, we discuss the existence of solutions for a \(2\times 2\) block operator matrix with nonlinear inputs. Furthermore, we give an illustrative example for a system of nonlinear integral equations in the space \(C(\mathbb{R}^+)\times C(\mathbb{R}^+)\) to verify the effectiveness and applicability of our main result.Pontryagin's maximum principle for a fractional integro-differential Lagrange problemhttps://zbmath.org/1530.490202024-04-15T15:10:58.286558Z"Kamocki, Rafał"https://zbmath.org/authors/?q=ai:kamocki.rafalThe author considers the optimal control problem: minimize \( \int_{a}^{b}g_{0}(t,y(t),u(t),v(t))dt\), subject to \((^{C}D_{a+}^{\alpha }y)(t)=g(t,y(t),u(t))+\int_{a}^{t}\frac{\Psi (t,s,y(s),v(s))}{ (t-s)^{1-\alpha }}ds\), \(u(t)\in M\subset \mathbb{R}^{k}\), \(v(t)\in N\subset \mathbb{R}^{l}\), a.e. \(t\in \lbrack a,b]\), \(y(a)=y_{0}\in \mathbb{R}\), where \(\alpha \in (0,1)\), \(g_{0}:[a,b]\times \mathbb{R}^{n}\times \mathbb{R} ^{k}\times \mathbb{R}^{l}\rightarrow \mathbb{R}\), \(g:[a,b]\times \mathbb{R} ^{n}\times \mathbb{R}^{k}\rightarrow \mathbb{R}^{n}\), \(\Psi :\Delta \times \mathbb{R}^{n}\times \mathbb{R}^{l}\rightarrow \mathbb{R}^{n}\), with \(\Delta =\{(t,s)\in \lbrack a,b]\times \lbrack a,b]:s<t\}\). Here \((D_{a+}^{\alpha }z)(\cdot )\) is the left-sided Riemann-Liouville derivative of order \(\alpha \) of the function \(z\in L_{n}^{1}\), the space of all summable functions \( z(\cdot ):[a,b]\times \mathbb{R}^{n}\), defined through: \((D_{a+}^{\alpha }z)(t)=\frac{d}{dt}(I_{a+}^{1-\alpha }z)(t)\), a.e. \(t\in \lbrack a,b]\), with \((I_{a+}^{\alpha }z)(t)=\int_{a}^{t}\frac{z(\tau )}{(t-\tau )^{1-\alpha }} d\tau \), the function \((I_{a+}^{1-\alpha }z)\) being absolutely continuous on \([a,b]\). \((^{C}D_{a+}^{\alpha }z)\) is the left-sided Caputo derivative of order \(\alpha \) of the function \(z\in C_{n}\), the space of all continuous functions from \([a,b]\) to \(\mathbb{R}^{n}\), such that the function \(z(\cdot )-z(a)\) has a Riemann-Liouville derivative, defined through \( (^{C}D_{a+}^{\alpha }z)(t)=D_{a+}^{\alpha }(z(t)-z(a))\), a.e. \(t\in \lbrack a,b]\).
The main result of the paper is that, under appropriate hypotheses on the data, if \((y_{\ast }(\cdot ),u_{\ast }(\cdot ),v_{\ast }(\cdot ))\in _{C}AC_{a+}^{\alpha ,p}\times L_{k}^{\infty }\times L_{k}^{\infty }\) is a local minimum to the above problem, there exists \(\gamma (\cdot )\in I_{b-}^{\alpha }(L_{n}^{p/(p-1)})\), such that \((D_{b-}^{\alpha }\gamma )(s)=\int_{s}^{b}\frac{[\Psi _{y}(t,s,y_{\ast }(s),v_{\ast }(s))]^{T}\gamma (t)}{(t-\tau )^{1-\alpha }}dt+[g_{y}(s,y_{\ast }(s),u_{\ast }(s))]^{T}\gamma (s)+(g_{0})_{y}(s,y_{\ast }(s),u_{\ast }(s),v_{\ast }(s))\), for a.e. \(s\in \lbrack a,b]\), and \(I_{b-}^{1-\alpha }(b)=0\). Here \((D_{b-}^{\alpha }z)(\cdot )\) is the right-sided Riemann-Liouville derivative of order \(\alpha \) of the function \(z\) and \(_{C}AC_{a+}^{\alpha ,p}([a,b]\times \mathbb{R}^{n})\) is the set of all functions \(z(\cdot ):[a,b]\rightarrow \mathbb{R}^{n}\) that have the representation \(z(t)=c_{a}+(I_{a+}^{\alpha }\varphi )(t)\), a.e. \( t\in \lbrack a,b]\), for some \(c_{a}\in \mathbb{R}^{n}\) and \(\varphi (\cdot )\in L_{n}^{p}\). For the proof, the author recalls properties of the left- and right-sided Riemann-Liouville derivatives, and he first proves the optimality conditions in the case of a zero initial condition. The paper ends with the presentation of an example.
Reviewer: Alain Brillard (Riedisheim)Maximal regularity for fractional Cauchy equation in Hölder space and its approximationhttps://zbmath.org/1530.650562024-04-15T15:10:58.286558Z"Liu, Li"https://zbmath.org/authors/?q=ai:liu.li.7"Fan, Zhenbin"https://zbmath.org/authors/?q=ai:fan.zhenbin"Li, Gang"https://zbmath.org/authors/?q=ai:li.gang.8"Piskarev, Sergey"https://zbmath.org/authors/?q=ai:piskarev.s-iThe well-posedness and maximal regularity of space-time discrete approximations are investigated for the nonhomogeneous fractional differential equation in Hölder space. The stability and regularity of the new implicit difference schemes are based on \(\beta\)-resolvent theory, using the concept of Riemann-Liouville fractional difference derivative with order \(\beta\) in [\textit{A. Ashyralyev}, J. Math. Anal. Appl. 357, No. 1, 232--236 (2009; Zbl 1175.26004)]. The conditions for the existence and stability of implicit difference schemes are derived using tools from functional analysis and numerical analysis. Numerical results for the new schemes are not provided.
Reviewer: Bülent Karasözen (Ankara)A second-order finite difference scheme for nonlinear tempered fractional integrodifferential equations in three dimensionshttps://zbmath.org/1530.650962024-04-15T15:10:58.286558Z"Wang, R."https://zbmath.org/authors/?q=ai:wang.ruru"Qiao, L."https://zbmath.org/authors/?q=ai:qiao.leijie"Zaky, M. A."https://zbmath.org/authors/?q=ai:zaky.mahmoud-a"Hendy, A. S."https://zbmath.org/authors/?q=ai:hendy.ahmed-sSummary: In this paper, we provide a numerical solution for the nonlinear tempered fractional integrodifferential equation in three dimensions. We use the trapezoidal convolution rule with backward differences (BDF2) for temporal discretization, and develop an alternating direction implicit difference scheme for spatial discretization. A novel fast approximation is applied to tackle the nonlinear term. The stability and convergence analysis of the numerical scheme are analyzed. Furthermore, some numerical experiments are provided to confirm the theoretical results.Unconditionally stable and convergent difference scheme for superdiffusion with extrapolationhttps://zbmath.org/1530.650972024-04-15T15:10:58.286558Z"Yang, Jinping"https://zbmath.org/authors/?q=ai:yang.jinping"Green, Charles Wing Ho"https://zbmath.org/authors/?q=ai:green.charles-wing-ho"Pani, Amiya K."https://zbmath.org/authors/?q=ai:pani.amiya-kumar"Yan, Yubin"https://zbmath.org/authors/?q=ai:yan.yubinSummary: Approximating the Hadamard finite-part integral by the quadratic interpolation polynomials, we obtain a scheme for approximating the Riemann-Liouville fractional derivative of order \(\alpha \in (1, 2)\) and the error is shown to have the asymptotic expansion \(\big( d_3 \tau^{3-\alpha} + d_4 \tau^{4-\alpha} + d_5 \tau^{5-\alpha} + \cdots \big) + \big( d_2^* \tau^4 +d_3^* \tau^6 +d_4^* \tau^8 +\cdots \big)\) at any fixed time, where \(\tau\) denotes the step size and \(d_l\), \(l = 3, 4, \dots\) and \(d_l^*\), \(l = 2, 3, \dots\) are some suitable constants. Applying the proposed scheme in temporal direction and the central difference scheme in spatial direction, a new finite difference method is developed for approximating the time fractional wave equation. The proposed method is unconditionally stable, convergent with order \(O (\tau^{3- \alpha})\), \(\alpha \in (1, 2)\) and the error has the asymptotic expansion. Richardson extrapolation is applied to improve the accuracy of the numerical method. The convergence orders are \(O(\tau^{4-\alpha})\) and \(O(\tau^{2(3-\alpha)})\), \(\alpha \in (1, 2)\), respectively, after first two extrapolations. Numerical examples are presented to show that the numerical results are consistent with the theoretical findings.High-order schemes based on extrapolation for semilinear fractional differential equationhttps://zbmath.org/1530.650982024-04-15T15:10:58.286558Z"Yang, Yuhui"https://zbmath.org/authors/?q=ai:yang.yuhui"Green, Charles Wing Ho"https://zbmath.org/authors/?q=ai:green.charles-wing-ho"Pani, Amiya K."https://zbmath.org/authors/?q=ai:pani.amiya-kumar"Yan, Yubin"https://zbmath.org/authors/?q=ai:yan.yubinSummary: By rewriting the Riemann-Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann-Liouville fractional derivative of order \(\alpha \in (1,2)\). The error has the asymptotic expansion \(\big( d_3 \tau^{3- \alpha} + d_4 \tau^{4-\alpha} + d_5 \tau^{5-\alpha} + \cdots \big) + \big( d_2^* \tau^4 + d_3^* \tau^6 + d_4^* \tau^8 + \cdots \big)\) at any fixed time \(t_N = T\), \(N \in\mathbb{Z}^+\), where \(d_i\), \(i=3, 4,\dots\) and \(d_i^*\), \(i=2, 3,\dots\) denote some suitable constants and \(\tau = T/N\) denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order \(\alpha \in (1,2)\) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.The compact difference scheme for the fourth-order nonlocal evolution equation with a weakly singular kernelhttps://zbmath.org/1530.651002024-04-15T15:10:58.286558Z"Zhou, Ziyi"https://zbmath.org/authors/?q=ai:zhou.ziyi"Zhang, Haixiang"https://zbmath.org/authors/?q=ai:zhang.haixiang"Yang, Xuehua"https://zbmath.org/authors/?q=ai:yang.xuehuaSummary: In this paper, we main discuss an efficient numerical algorithm for the fourth-order nonlocal evolution equation with a weakly singular kernel. The second-order fractional convolution quadrature rule and L1 method are proposed to approximate the Riemann-Liouville (R-L) fractional integral term and the temporal Caputo derivative, respectively. In order to obtain a fully discrete method, the compact difference scheme is used to discretize the second-order and fourth-order spatial derivative. Further, two new approaches are adopted for stability analysis, and then the optimal error estimates in the discrete \({L}^{\infty }\)-norm and \({L}^2\)-norm are obtained. At last, we give three test problems to illustrate the validity of the methods.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}Numerical solution of fractional model of Atangana-Baleanu-Caputo integrodifferential equations with integral boundary conditionshttps://zbmath.org/1530.651122024-04-15T15:10:58.286558Z"Alneimat, Mohammad"https://zbmath.org/authors/?q=ai:alneimat.mohammad"Moakher, Maher"https://zbmath.org/authors/?q=ai:moakher.maher"Djeddi, Nadir"https://zbmath.org/authors/?q=ai:djeddi.nadir"Al-Omari, Shrideh"https://zbmath.org/authors/?q=ai:al-omari.shrideh-khalaf-qasemSummary: In this analysis, we propose an advanced numerical technique, reproducing kernel discretization method (RKDM), to investigate numerical solutions for a class of systems of fractional integro-differential equations (SFIDE) with integral boundary conditions. The Atangana-Baleanu fractional derivative is used to formulate the fractional integro-differential equations. The solution methodology is mainly based on constructing a reproducing kernel function, that satisfies the integral boundary conditions, in order to construct an orthonormal basis to formulate the solution in form of Fourier series that is uniformly convergent in the specified space \(W^2_2[a, b]\). Numerical applications are investigated to represent the hypothesis and to confirm the design steps of the proposed advanced technique. The numerical viewpoint indicates that the RKDM is an important tool for dealing with such issues arising in physics and engineering fields.Correction to: ``Robust bivariate polynomials scheme with convergence analysis for two-dimensional nonlinear optimal control problem''https://zbmath.org/1530.651302024-04-15T15:10:58.286558Z"Ebrahimzadeh, Asiyeh"https://zbmath.org/authors/?q=ai:ebrahimzadeh.asiyeh"Panjeh Ali Beik, Samaneh"https://zbmath.org/authors/?q=ai:beik.samaneh-panjeh-aliFrom the text: Unfortunately, the corresponding author is misspelled in the original article [ibid. 17, No. 3, 325--335 (2023; Zbl 1522.65182)] and the correct name is Asiyeh Ebrahimzadeh.
The original article is now corrected.A fully discrete high-order fast multiscale Galerkin method for solving boundary integral equations in a domain with cornershttps://zbmath.org/1530.651632024-04-15T15:10:58.286558Z"Fang, Yiying"https://zbmath.org/authors/?q=ai:fang.yiying"Jiang, Ying"https://zbmath.org/authors/?q=ai:jiang.yingSummary: In this paper, we develop a fully discrete fast multiscale Galerkin method for solving the boundary integral equation derived from the interior Dirichlet problem in a domain with corners. The integral operator in the equation can be split into two operators: a noncompact operator with a singular kernel and a compact operator with a piecewise smooth kernel. As shown in the paper, we develop two fast schemes to evaluate the entries of representation matrices for noncompact and compact operators. One is designed to compute integrals with algebraic singularities, and the other is a row-column scheme for evaluating the entries of the representing matrix of compact operators. We prove the proposed fully discrete method can achieve high-order convergence rates and only use the total number \(\mathscr{O}(n2^n)\) of arithmetic operators to generate the representing matrices where \(2^n\) is the number of the basis functions used in the method. Numerical examples are presented to confirm the theoretical results for the computational complexity, stability, and accuracy of the proposed method.On existence and numerical solution of higher order nonlinear integro-differential equations involving variable coefficientshttps://zbmath.org/1530.651842024-04-15T15:10:58.286558Z"Amin, Rohul"https://zbmath.org/authors/?q=ai:amin.rohul"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamal"Gao, Liping"https://zbmath.org/authors/?q=ai:gao.liping"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabetSummary: We investigate the approximate solution to a class of fourth-order Volterra-Fredholm integro differential equations (VFIDEs). Additionally, we are able to get some adequate results for the existence of a solution with the use of nonlinear analysis techniques. The basis for the required numerical computation is provided by the Haar wavelet collocations (HWCs) technique, which converts the problems into a system of algebraic equations. Then, in order to obtain the required numerical solution, we solve the resulting system of algebraic equations using Gauss elimination and Broyden's techniques. These methods can be used to show convergence, and the predicted rate of convergence as well. Along with relevant examples, we have included a graphical presentation to demonstrate our suggested approach.Solving nonlinear Volterra integral equations of the first kind with discontinuous kernels by using the operational matrix methodhttps://zbmath.org/1530.651852024-04-15T15:10:58.286558Z"Amirkhizi, Simin Aghaei"https://zbmath.org/authors/?q=ai:amirkhizi.simin-aghaei"Mahmoudi, Yaghoub"https://zbmath.org/authors/?q=ai:mahmoudi.yaghoub"Shamloo, Ali Salimi"https://zbmath.org/authors/?q=ai:shamloo.ali-salimiSummary: A numerical method to solve the nonlinear Volterra integral equations of the first kind with discontinuous kernels is proposed. Usage of operational matrices for this kind of equation is a cost-efficient scheme. Shifted Legendre polynomials are applied for solving Volterra integral equations with discontinuous kernels by converting the equation to a system of nonlinear algebraic equations. The convergence analysis is given for the approximated solution and numerical examples are demonstrated to denote the precision of the proposed method.Chebyshev spectral method for solving fuzzy fractional Fredholm-Volterra integro-differential equationhttps://zbmath.org/1530.651862024-04-15T15:10:58.286558Z"Kumar, Sachin"https://zbmath.org/authors/?q=ai:kumar.sachin"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-jose"Ahmad, Bashir"https://zbmath.org/authors/?q=ai:ahmad.bashir.2Summary: The fuzzy integral equation is used to model many physical phenomena which arise in many fields like chemistry, physics, and biology, etc. In this article, we emphasize on mathematical modeling of the fuzzy fractional Fredholm-Volterra integral equation. The numerical solution of the fuzzy fractional Fredholm-Volterra equation is determined in which model contains fuzzy coefficients and fuzzy initial condition. First, an operational matrix of Chebyshev polynomial of Caputo type fractional fuzzy derivative is derived in fuzzy environment. The integral term is approximated by the Chebyshev spectral method and the differential term is approximated by the operational matrix. This method converted the given fuzzy fractional integral equation into algebraic equations which are fuzzy in nature. The desired numerical solution is to find out by solving these algebraic equations. The different particular cases of our model have been solved which depict the feasibility of our method. The error tables show the accuracy of the method. We also can see the accuracy of our method by 3D figures of exact and obtained numerical solutions. Hence, our method is suitable to deal with the fuzzy fractional Fredholm-Volterra equation.Superconvergence and postprocessing of collocation methods for fractional differential equationshttps://zbmath.org/1530.651872024-04-15T15:10:58.286558Z"Wang, Lu"https://zbmath.org/authors/?q=ai:wang.lu.7|wang.lu.5|wang.lu|wang.lu.15|wang.lu.3|wang.lu.1"Liang, Hui"https://zbmath.org/authors/?q=ai:liang.huiSummary: This paper aims to propose a complete superconvergence analysis for a postprocessing technique based on collocation methods for fractional differential equations (FDEs). We start with the simple linear FDEs with Caputo derivative of order \(0<\alpha <1\). The problem is reformulated as a weakly singular Volterra integral equation (VIE), and based on the resolvent theory of VIEs, the existence, uniqueness and regularity for the exact solution for the original FDE are obtained. Then the piecewise polynomial collocation method is adopted to solve the reformulated VIE, and based on the regularity of the original FDE, the convergence for the collocation method and the superconvergence for the iterated collocation method are investigated in detail, respectively. Further, based on the obtained collocation solution, the interpolation postprocessing approximation of higher accuracy is constructed on graded mesh, and the superconvergence is obtained. Compared with classical iterated collocation method, the cost on computation of interpolation postprocessing technique is less. Numerical experiments are given to illustrate the theoretical results, and it is also shown that the proposed postprocessing technique can be extended to certain nonlinear and systems of FDEs.Singular expansions and collocation methods for generalized Abel integral equationshttps://zbmath.org/1530.651882024-04-15T15:10:58.286558Z"Wang, Tongke"https://zbmath.org/authors/?q=ai:wang.tongke"Liu, Sijing"https://zbmath.org/authors/?q=ai:liu.sijing"Zhang, Zhiyue"https://zbmath.org/authors/?q=ai:zhang.zhiyueSummary: In this paper, the generalized Abel integral equation with singular solution is studied. First, the finite-term psi-series expansion of the solution about the origin is derived using the method of undetermined coefficients, which gives the complete singular information of the solution. Second, this singular expansion is used to separate the singularity such that the Abel integral equation is converted into a perturbed one with smooth solution on a regular interval. Third, a product trapezoidal method is designed to solve the transformed equation on the regular interval and the convergence analysis is conducted to show that the scheme has second order accuracy combining with a controllable perturbation term. Fourth, a Chebyshev collocation method is further constructed on the regular interval to show that the global orthogonal polynomial interpolation is also suitable for solving the transformed Abel integral equation with high accuracy. Finally, two numerical examples confirm the correctness of the truncated psi-series solution and the effectiveness of the piecewise and global collocation methods with singularity separation for solving this kind of Abel integral equation.Asymptotically exact method for calculation of density of states in HTSChttps://zbmath.org/1530.820152024-04-15T15:10:58.286558Z"Kashurnikov, Vladimir A."https://zbmath.org/authors/?q=ai:kashurnikov.vladimir-a"Krasavin, Andrey V."https://zbmath.org/authors/?q=ai:krasavin.andrey-v"Zhumagulov, Yaroslav V."https://zbmath.org/authors/?q=ai:zhumagulov.yaroslav-vSummary: The work presents the method of restoring a spectral density from known Green's function. The method is based on a combination of Monte Carlo and gradient descent algorithms, which avoids the problem of distortion of the equation by nonlinear terms and, therefore, analyzes the most representative range of small deviations. Furthermore, the method does not contain sources of systematic errors and, in principle, any spectral function can be parameterized with any desired accuracy. With the use of the method, the spectral density of states was restored for FeAs-based superconductors. The method works well also for metal nanoclusters and many other systems.A density description of a bounded-confidence model of opinion dynamics on hypergraphshttps://zbmath.org/1530.914802024-04-15T15:10:58.286558Z"Chu, Weiqi"https://zbmath.org/authors/?q=ai:chu.weiqi"Porter, Mason A."https://zbmath.org/authors/?q=ai:porter.mason-aSummary: Social interactions often occur between three or more agents simultaneously. Examining opinion dynamics on hypergraphs allows one to study the effect of such polyadic interactions on the opinions of agents. In this paper, we consider a bounded-confidence model (BCM), in which opinions take continuous values and interacting agents comprise their opinions if they are close enough to each other. We study a density description of a Deffuant-Weisbuch BCM on hypergraphs. We derive a rate equation for the mean-field opinion density as the number of agents becomes infinite, and we prove that this rate equation yields a probability density that converges to noninteracting opinion clusters. Using numerical simulations, we examine bifurcations of the density-based BCM's steady-state opinion clusters and demonstrate that the agent-based BCM converges to the density description of the BCM as the number of agents becomes infinite.The perturbed compound Poisson risk model with proportional investmenthttps://zbmath.org/1530.915972024-04-15T15:10:58.286558Z"Deng, Nai-dan"https://zbmath.org/authors/?q=ai:deng.nai-dan"Wang, Chun-wei"https://zbmath.org/authors/?q=ai:wang.chunwei"Xu, Jia-en"https://zbmath.org/authors/?q=ai:xu.jia-enSummary: In this paper, the insurance company considers venture capital and risk-free investment in a constant proportion. The surplus process is perturbed by diffusion. At first, the integro-differential equations satisfied by the expected discounted dividend payments and the Gerber-Shiu function are derived. Then, the approximate solutions of the integro-differential equations are obtained through the sinc method. Finally, the numerical examples are given when the claim sizes follow different distributions. Furthermore, the errors between the explicit solution and the numerical solution are discussed in a special case.State feedback control law design for an age-dependent SIR modelhttps://zbmath.org/1530.921302024-04-15T15:10:58.286558Z"Sonveaux, Candy"https://zbmath.org/authors/?q=ai:sonveaux.candy"Winkin, Joseph J."https://zbmath.org/authors/?q=ai:winkin.joseph-jSummary: An age-dependent SIR model is considered with the aim to develop a state-feedback vaccination law in order to eradicate a disease. A dynamical analysis of the system is performed using the principle of linearized stability and shows that, if the basic reproduction number is larger than 1, the disease free equilibrium is unstable. This result justifies the development of a vaccination law. Two approaches are used. The first one is based on a discretization of the partial integro-differential equations (PIDE) model according to the age. In this case a linearizing feedback law is found using Isidori's theory. Conditions guaranteeing stability and positivity are established. The second approach yields a linearizing feedback law developed for the PIDE model. This law is deduced from the one obtained for the ODE case. Using semigroup theory, stability conditions are also obtained. Finally, numerical simulations are presented to reinforce the theoretical arguments.Superinfection and the hypnozoite reservoir for \textit{Plasmodium vivax}: a general frameworkhttps://zbmath.org/1530.922572024-04-15T15:10:58.286558Z"Mehra, Somya"https://zbmath.org/authors/?q=ai:mehra.somya"McCaw, James M."https://zbmath.org/authors/?q=ai:mccaw.james-m"Taylor, Peter G."https://zbmath.org/authors/?q=ai:taylor.peter-gSummary: A characteristic of malaria in all its forms is the potential for superinfection (that is, multiple concurrent blood-stage infections). An additional characteristic of \textit{Plasmodium vivax} malaria is a reservoir of latent parasites (hypnozoites) within the host liver, which activate to cause (blood-stage) relapses. Here, we present a model of hypnozoite accrual and superinfection for \textit{P. vivax}. To couple host and vector dynamics for a homogeneously-mixing population, we construct a density-dependent Markov population process with countably many types, for which disease extinction is shown to occur almost surely. We also establish a functional law of large numbers, taking the form of an infinite-dimensional system of ordinary differential equations that can also be recovered by coupling expected host and vector dynamics (i.e. a hybrid approximation) or through a standard compartment modelling approach. Recognising that the subset of these equations that model the infection status of the human hosts has precisely the same form as the Kolmogorov forward equations for a Markovian network of infinite server queues with an inhomogeneous batch arrival process, we use physical insight into the evolution of the latter process to write down a time-dependent multivariate generating function for the solution. We use this characterisation to collapse the infinite-compartment model into a single integrodifferential equation (IDE) governing the intensity of mosquito-to-human transmission. Through a steady state analysis, we recover a threshold phenomenon for this IDE in terms of a parameter \(R_0\) expressible in terms of the primitives of the model, with the disease-free equilibrium shown to be uniformly asymptotically stable if \(R_0<1\) and an endemic equilibrium solution emerging if \(R_0>1\). Our work provides a theoretical basis to explore the epidemiology of \textit{P. vivax}, and introduces a strategy for constructing tractable population-level models of malarial superinfection that can be generalised to allow for greater biological realism in a number of directions.Symmetry in a multi-strain epidemiological model with distributed delay as a general cross-protection period and disease enhancement factorhttps://zbmath.org/1530.922742024-04-15T15:10:58.286558Z"Steindorf, Vanessa"https://zbmath.org/authors/?q=ai:steindorf.vanessa"Oliva, Sergio"https://zbmath.org/authors/?q=ai:oliva.sergio-m"Stollenwerk, Nico"https://zbmath.org/authors/?q=ai:stollenwerk.nico"Aguiar, Maíra"https://zbmath.org/authors/?q=ai:aguiar.mairaSummary: Important biological features of viral infectious diseases caused by multiple agents with interacting strain dynamics continue to pose challenges for mathematical modelling development. Motivated by dengue fever epidemiology, we study a system of integro-differential equations (IDE) considering strain structure of pathogens. Knowing that complex dynamics observed in dengue models are driven by the combination of two biological features, the temporary cross-immunity (TCI) and disease enhancement via the antibody-dependent enhancement process (ADE), our IDE system incorporates the TCI with a general time delay term, and the ADE effect by a constant factor to differentiate the susceptibility of individuals experiencing a primary or a secondary infection. Aiming at analysing the effect of the symmetry on dengue serotypes in the IDE framework, a detailed qualitative analysis of the model is performed and the instability of the coexistence steady state is shown using the perturbation theory approach. Numerical simulations identify the bifurcation structures and confirm the stability analysis. Results for the symmetric and asymmetric models are discussed.Pattern formation in mesic savannashttps://zbmath.org/1530.923082024-04-15T15:10:58.286558Z"Patterson, Denis"https://zbmath.org/authors/?q=ai:patterson.denis-d"Levin, Simon"https://zbmath.org/authors/?q=ai:levin.simon-a"Staver, Ann Carla"https://zbmath.org/authors/?q=ai:staver.ann-carla"Touboul, Jonathan"https://zbmath.org/authors/?q=ai:touboul.jonathan-davidSummary: We analyze a spatially extended version of a well-known model of forest-savanna dynamics, which presents as a system of nonlinear partial integro-differential equations, and study necessary conditions for pattern-forming bifurcations. Homogeneous solutions dominate the dynamics of the standard forest-savanna model, regardless of the length scales of the various spatial processes considered. However, several different pattern-forming scenarios are possible upon including spatial resource limitation, such as competition for water, soil nutrients, or herbivory effects. Using numerical simulations and continuation, we study the nature of the resulting patterns as a function of system parameters and length scales, uncovering subcritical pattern-forming bifurcations and observing significant regions of multistability for realistic parameter regimes. Finally, we discuss our results in the context of extant savanna-forest modeling efforts and highlight ongoing challenges in building a unifying mathematical model for savannas across different rainfall levels.On a general degenerate/singular parabolic equation with a nonlocal space termhttps://zbmath.org/1530.930232024-04-15T15:10:58.286558Z"Allal, Brahim"https://zbmath.org/authors/?q=ai:allal.brahim"Fragnelli, Genni"https://zbmath.org/authors/?q=ai:fragnelli.genni"Salhi, Jawad"https://zbmath.org/authors/?q=ai:salhi.jawad(no abstract)Results on approximate controllability of fractional stochastic Sobolev-type Volterra-Fredholm integro-differential equation of order \(1 < r < 2\)https://zbmath.org/1530.930262024-04-15T15:10:58.286558Z"Dineshkumar, Chendrayan"https://zbmath.org/authors/?q=ai:dineshkumar.chendrayan"Udhayakumar, Ramalingam"https://zbmath.org/authors/?q=ai:udhayakumar.r(no abstract)Existence and controllability of higher-order nonlinear fractional integrodifferential systems via fractional resolventhttps://zbmath.org/1530.930282024-04-15T15:10:58.286558Z"Haq, Abdul"https://zbmath.org/authors/?q=ai:haq.abdul"Sukavanam, Nagarajan"https://zbmath.org/authors/?q=ai:sukavanam.nagarajan(no abstract)Fast-time complete controllability of nonlinear fractional delay integrodifferential evolution equations with nonlocal conditions and a parameterhttps://zbmath.org/1530.930382024-04-15T15:10:58.286558Z"Zhao, Daliang"https://zbmath.org/authors/?q=ai:zhao.daliang"Liu, Yansheng"https://zbmath.org/authors/?q=ai:liu.yansheng"Li, Haitao"https://zbmath.org/authors/?q=ai:li.haitao(no abstract)