Recent zbMATH articles in MSC 34Ahttps://zbmath.org/atom/cc/34A2024-09-27T17:47:02.548271ZUnknown authorWerkzeugOn the extreme eigenvalues and asymptotic conditioning of a class of Toeplitz matrix-sequences arising from fractional problemshttps://zbmath.org/1541.150392024-09-27T17:47:02.548271Z"Bogoya, Manuel"https://zbmath.org/authors/?q=ai:bogoya.manuel"Grudsky, Sergei"https://zbmath.org/authors/?q=ai:grudsky.sergei-m"Mazza, Mariarosa"https://zbmath.org/authors/?q=ai:mazza.mariarosa"Serra-Capizzano, Stefano"https://zbmath.org/authors/?q=ai:serra-capizzano.stefanoSummary: The analysis of the spectral features of a Toeplitz matrix-sequence \(\{T_n(f)\}_{n\in\mathbb{N}}\), generated by the function \(f\in L^1([-\pi,\pi])\), real-valued almost everywhere (a.e.), has been provided in great detail in the last century, as well as the study of the conditioning, when \(f\) is nonnegative a.e. Here we consider a novel type of problem arising in the numerical approximation of distributed-order fractional differential equations (FDEs), where the matrices under consideration take the form
\[
\begin{aligned}
\mathcal{T}_n = c_0\mathcal{T}_n(f_0) + c_1h^h T_n(f_1) + c_2h^{2h} T_n(f_2) +\cdots \\
+ c_{n-1}h^{(n-1)h} T_n(f_n-1),
\end{aligned}
\]
\(c_0,c_1,\dots,c_{n-1}\) belong to the interval \([c_*,c^*]\) with \(c^*\geqslant c_*>0\) independent of \(n\), \(h=\frac{1}{n}\), \(f_j\sim g_j\), and \(g_j(\theta)=|\theta |^{2-jh}\) for every \(j=0,\dots,n-1\). For nonnegative functions or sequences, the notation \(s(x)\sim t(x)\) means that there exist positive constants \(c\), \(d\), independent of the variable \(x\) in the definition domain such that \(cs(x)\leqslant t(x)\leqslant ds(x)\) for any \(x\). Since the resulting generating function depends on \(n\), the standard theory cannot be applied and the analysis has to be performed using new ideas. Few selected numerical experiments are presented, also in connection with matrices that come from distributed-order FDE problems, and the adherence with the theoretical analysis is discussed, together with open questions and future investigations.Stieltjes analytic functions and higher order linear differential equationshttps://zbmath.org/1541.260132024-09-27T17:47:02.548271Z"Cora, Víctor"https://zbmath.org/authors/?q=ai:cora.victor"Fernández, F. Javier"https://zbmath.org/authors/?q=ai:fernandez.francisco-javier"Tojo, F. Adrián F."https://zbmath.org/authors/?q=ai:tojo.f-adrian-fSummary: In this work we develop a theory of Stieltjes-analytic functions. We first define the Stieltjes monomials and polynomials and we study them exhaustively. Then, we introduce the \(g\)-analytic functions locally, as an infinite series of these Stieltjes monomials and we study their properties in depth and how they relate to higher order Stieltjes differentiation. We define the exponential series and prove that it solves the first order linear problem. Finally, we apply the theory to solve higher order linear homogeneous Stieltjes differential equations with constant coefficients.Omega derivativehttps://zbmath.org/1541.260192024-09-27T17:47:02.548271Z"Castillo, Rene Erín"https://zbmath.org/authors/?q=ai:castillo.rene-erin"Valdés, Juan E. Nápoles"https://zbmath.org/authors/?q=ai:valdes.juan-eduardo-napoles|napoles-valdes.juan-e"Chaparro, Héctor"https://zbmath.org/authors/?q=ai:chaparro.hector-camiloSummary: In this paper, we introduce the \(\Omega\)-derivative, which generalizes the classical concept of derivative. Main properties of this new derivative are revised. We also study \(\Omega\)-differential equations and some of its applications.Properties of vector-valued \(\tau \)-discrete fractional calculus and its connection with Caputo fractional derivativeshttps://zbmath.org/1541.260202024-09-27T17:47:02.548271Z"Chang, Yong-Kui"https://zbmath.org/authors/?q=ai:chang.yong-kui"Ponce, Rodrigo"https://zbmath.org/authors/?q=ai:ponce.rodrigo-fSummary: In this paper, for a given vector-valued sequence \((v^n)_{n\in \mathbb{N}_0}\), we study its discrete fractional derivative in the sense of Caputo for \(0<\alpha <1\) and its connection with the Caputo fractional derivative. Moreover, we study the convergence of this Caputo fractional difference operator to the Caputo fractional derivative.New fractional integral formulas and kinetic model associated with the hypergeometric superhyperbolic sine functionhttps://zbmath.org/1541.260232024-09-27T17:47:02.548271Z"Geng, Lu-Lu"https://zbmath.org/authors/?q=ai:geng.lu-lu"Yang, Xiao-Jun"https://zbmath.org/authors/?q=ai:yang.xiao-jun"Alsolami, Abdulrahman Ali"https://zbmath.org/authors/?q=ai:alsolami.abdulrahman-ali(no abstract)A new weighted fractional operator with respect to another function via a new modified generalized Mittag-Leffler lawhttps://zbmath.org/1541.260302024-09-27T17:47:02.548271Z"Thabet, Sabri T. M."https://zbmath.org/authors/?q=ai:thabet.sabri-t-m"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet"Kedim, Imed"https://zbmath.org/authors/?q=ai:kedim.imed"Ayari, M. Iadh"https://zbmath.org/authors/?q=ai:ayari.mohamed-iadhSummary: In this paper, new generalized weighted fractional derivatives with respect to another function are derived in the sense of Caputo and Riemann-Liouville involving a new modified version of a generalized Mittag-Leffler function with three parameters, as well as their corresponding fractional integrals. In addition, several new and existing operators of nonsingular kernels are obtained as special cases of our operator. Many important properties related to our new operator are introduced, such as a series version involving Riemann-Liouville fractional integrals, weighted Laplace transforms with respect to another function, etc. Finally, an example is given to illustrate the effectiveness of the new results.On simple and interlaced property of the zeros of two entire functionshttps://zbmath.org/1541.260522024-09-27T17:47:02.548271Z"Liu, Tao"https://zbmath.org/authors/?q=ai:liu.tao.6|liu.tao"Wei, Guangsheng"https://zbmath.org/authors/?q=ai:wei.guangshengSummary: This paper is concerned with two pairs \((A_j,B_j)\), \(j=0,1\) of entire functions of \(m\)-type. We give conditions under which the zeros of \(A_0+tA_1\) and \(B_0+tB_1\) are real and interlaced for each \(t\in \mathbb{R}\). This result is used to deal with the inverse problems for the special transmission eigenvalue problem: \(-u^{\prime \prime}+qu=\lambda u\) with
\[
u(0)=0= u^{\prime}(1)\frac{\sin \sqrt{\lambda}}{\sqrt{\lambda}}-u(1)\cos \sqrt{\lambda}.
\]
We prove that, if its characteristic function \(\Delta (\lambda ,q)\) has only nonreal zeros, then for each \(t\in \mathbb{R}\), there exists a unique real-valued function \(q(\cdot ,t)\in L^2(0,1)\) such that the corresponding characteristic function of \(q(\cdot, t)\) is \(t\Delta (\lambda ,q)\).
{{\copyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}Integral representations for products of two solutions of the Airy equation with shifted arguments and their applications in physicshttps://zbmath.org/1541.330062024-09-27T17:47:02.548271Z"Bazarov, Kirill V."https://zbmath.org/authors/?q=ai:bazarov.kirill-v"Tolstikhin, Oleg I."https://zbmath.org/authors/?q=ai:tolstikhin.oleg-iSummary: Integral representations for a complete set of linearly independent products of two solutions of the Airy equation whose arguments differ by \(z_0\) are obtained using the Laplace contour integral method. This generalizes similar integral representations for the case \(z_0 =0\) obtained by \textit{W. H. Reid} [Z. Angew. Math. Phys. 46, No. 2, 159--170 (1995; Zbl 0824.33002)]. The relation to other previous results is discussed. The results are used to obtain the outgoing-wave Green's function for an electron in a static electric field in a closed analytic form.A computational approach to polynomial conservation lawshttps://zbmath.org/1541.340032024-09-27T17:47:02.548271Z"Desoeuvres, Aurélien"https://zbmath.org/authors/?q=ai:desoeuvres.aurelien"Iosif, Alexandru"https://zbmath.org/authors/?q=ai:iosif.alexandru"Lüders, Christoph"https://zbmath.org/authors/?q=ai:luders.christoph"Radulescu, Ovidiu"https://zbmath.org/authors/?q=ai:radulescu.ovidiu"Rahkooy, Hamid"https://zbmath.org/authors/?q=ai:rahkooy.hamid"Seiß, Matthias"https://zbmath.org/authors/?q=ai:seiss.matthias"Sturm, Thomas"https://zbmath.org/authors/?q=ai:sturm.thomasSummary: For polynomial ODE models, we introduce and discuss the concepts of exact and approximate conservation laws, which are the first integrals of the full and truncated sets of ODEs. For fast-slow systems, truncated ODEs describe the fast dynamics. We define compatibility classes as subsets of the state space, obtained by equating the conservation laws to constants. A set of conservation laws is complete when the corresponding compatibility classes contain a finite number of steady states. Complete sets of conservation laws can be used for model order reduction and for studying the multistationarity of the model. We provide algorithmic methods for computing linear, monomial, and polynomial conservation laws of polynomial ODE models and for testing their completeness. The resulting conservation laws and their completeness are either independent or dependent on the parameters. In the latter case, we provide parametric case distinctions. In particular, we propose a new method to compute polynomial conservation laws by comprehensive Gröbner systems and syzygies.Solving nonlinear ordinary differential equations using the invariant manifolds and Koopman eigenfunctionshttps://zbmath.org/1541.340042024-09-27T17:47:02.548271Z"Morrison, Megan"https://zbmath.org/authors/?q=ai:morrison.megan"Kutz, J. Nathan"https://zbmath.org/authors/?q=ai:kutz.j-nathanSummary: Nonlinear ODEs can rarely be solved analytically. Koopman operator theory provides a way to solve two-dimensional nonlinear systems, under suitable restrictions, by mapping nonlinear dynamics to a linear space using Koopman eigenfunctions. Unfortunately, finding such eigenfunctions is difficult. We introduce a method for constructing Koopman eigenfunctions from a two-dimensional nonlinear ODE's one-dimensional invariant manifolds. This method, when successful, allows us to find analytical solutions for autonomous, nonlinear systems. Previous data-driven methods have used Koopman theory to construct local Koopman eigenfunction approximations valid in different regions of phase space; our method finds analytic Koopman eigenfunctions that are exact and globally valid. We demonstrate our Koopman method of solving nonlinear systems on one-dimensional and two-dimensional ODEs. The nonlinear examples considered have simple expressions for their codimension-1 invariant manifolds which produce tractable analytical solutions. Thus our method allows for the construction of analytical solutions for previously unsolved ODEs. It also highlights the connection between invariant manifolds and eigenfunctions in nonlinear ODEs and presents avenues for extending this method to solve more nonlinear systems.Stability of set differential equations in Fréchet spaceshttps://zbmath.org/1541.340052024-09-27T17:47:02.548271Z"Bao, Junyan"https://zbmath.org/authors/?q=ai:bao.junyan"Chen, Wei"https://zbmath.org/authors/?q=ai:chen.wei.67"Wang, Peiguang"https://zbmath.org/authors/?q=ai:wang.peiguang(no abstract)On the solutions of fractional differential equations via Geraghty type hybrid contractionshttps://zbmath.org/1541.340062024-09-27T17:47:02.548271Z"Adıgüzel, Rezan Sevinik"https://zbmath.org/authors/?q=ai:adiguzel.rezan-sevinik"Aksoy, Ümit"https://zbmath.org/authors/?q=ai:aksoy.umit"Karapınar, Erdal"https://zbmath.org/authors/?q=ai:karapinar.erdal"Erhan, İnci M."https://zbmath.org/authors/?q=ai:erhan.inci-mSummary: The aim of this article is twofold. Firstly, to study fixed points of mappings on \(b\)-metric spaces satisfying a general contractive condition called Geraghty type hybrid contraction. Secondly, to apply the theoretical results to the problem of existence and uniqueness of solutions of boundary value problems with integral boundary conditions associated with a certain type of nonlinear fractional differential equations. The conditions for the existence of fixed points for Geraghty type hybrid contractions are determined and several consequences of the main results are deduced. Some examples on boundary value problems for nonlinear fractional differential equations of order \(3 < \alpha \leq 4\) are provided, where the existence and uniqueness of solutions are shown by using Geraghty type contractions.On a type of differential calculus in the frame of generalized Hilfer integro-differential equationhttps://zbmath.org/1541.340072024-09-27T17:47:02.548271Z"Alkord, Mohammed N."https://zbmath.org/authors/?q=ai:alkord.mohammed-n-a"Shaikh, Sadikali L."https://zbmath.org/authors/?q=ai:shaikh.sadikali-l"Altalla, Mohammed B. M."https://zbmath.org/authors/?q=ai:altalla.mohammed-b-mSummary: In this paper, we investigate the existence and uniqueness of solutions to a new class of integro-differential equation boundary value problems (BVPs) with \(\top\)-Hilfer operator. Our problem is converted into an equivalent fixed-point problem by introducingan operator whose fixed points coincide with the solutions to the given problem. Using Banach's and Schauder's fixed point techniques, the uniqueness and existence result for thegiven problem are demonstrated. The stability results for solutions of the given problem arealso discussed. In the end. One example is provided to demonstrate the obtained results.A novel approach on the sequential type \(\psi\)-Hilfer pantograph fractional differential equation with boundary conditionshttps://zbmath.org/1541.340082024-09-27T17:47:02.548271Z"Aly, Elkhateeb S."https://zbmath.org/authors/?q=ai:aly.elkhateeb-s"Maheswari, M. Latha"https://zbmath.org/authors/?q=ai:maheswari.muthukrishnan-latha"Shri, K. S. Keerthana"https://zbmath.org/authors/?q=ai:shri.kolathur-srinivasan-keerthana"Hamali, Waleed"https://zbmath.org/authors/?q=ai:hamali.waleedSummary: This article investigates sufficient conditions for the existence and uniqueness of solutions to the \(\psi\)-Hilfer sequential type pantograph fractional boundary value problem. Considering the system depends on a lower-order fractional derivative of an unknown function, the study is carried out in a special working space. Standard fixed point theorems such as the Banach contraction principle and Krasnosel'skii's fixed point theorem are applied to prove the uniqueness and the existence of a solution, respectively. Finally, an example demonstrating our results with numerical simulations is presented.The sequential conformable Langevin-type differential equations and their applications to the RLC electric circuit problemshttps://zbmath.org/1541.340092024-09-27T17:47:02.548271Z"Aydin, M."https://zbmath.org/authors/?q=ai:aydin.mustafa"Mahmudov, N. I."https://zbmath.org/authors/?q=ai:mahmudov.nazim-idrisogluSummary: In this paper, the sequential conformable Langevin-type differential equation is studied. A representation of a solution consisting of the newly defined conformable bivariate Mittag-Leffler function to its nonhomogeneous and linear version is obtained via the conformable Laplace transforms' technique. Also, existence and uniqueness of a global solution to its nonlinear version are obtained. The existence and uniqueness of solutions are shown with respect to the weighted norm defined in compliance with (conformable) exponential function. The concept of the Ulam-Hyers stability of solutions is debated based on the fixed-point approach. The LRC electrical circuits are presented as an application to the described system. Simulated and numerical instances are offered to instantiate our abstract findings.Dynamics analysis of fractional-order Hopfield neural networkshttps://zbmath.org/1541.340102024-09-27T17:47:02.548271Z"Batiha, Iqbal M."https://zbmath.org/authors/?q=ai:batiha.iqbal-m"Albadarneh, Ramzi B."https://zbmath.org/authors/?q=ai:albadarneh.ramzi-b"Momani, Shaher"https://zbmath.org/authors/?q=ai:momani.shaher-m"Jebril, Iqbal H."https://zbmath.org/authors/?q=ai:jebril.iqbal-hamzhSummary: This paper proposes fractional-order systems for Hopfield Neural Network (HNN). The so-called Predictor-Corrector Adams-Bashforth-Moulton Method (PCABMM) has been implemented for solving such systems. Graphical comparisons between the PCABMM and the Runge-Kutta Method (RKM) solutions for the classical HNN reveal that the proposed technique is one of the powerful tools for handling these systems. To determine all Lyapunov exponents for them, the Benettin-Wolf algorithm has been involved in the PCABMM. Based on such algorithm, the Lyapunov exponents as a function of a given parameter and as another function of the fractional-order have been described, the intermittent chaos for these systems has been explored. A new result related to the Mittag-Leffler stability of some nonlinear Fractional-order Hopfield Neural Network (FoHNN) systems has been shown. Besides, the description and the dynamic analysis of those phenomena have been discussed and verified theoretically and numerically via illustrating the phase portraits and the Lyapunov exponents' diagrams.Weakly perturbed linear boundary-value problem for system of fractional differential equations with Caputo derivativehttps://zbmath.org/1541.340112024-09-27T17:47:02.548271Z"Boichuk, Oleksandr"https://zbmath.org/authors/?q=ai:boichuk.oleksandr-andriiovych"Feruk, Viktor"https://zbmath.org/authors/?q=ai:feruk.victorSummary: We consider a perturbed linear boundary-value problem for a system of fractional differential equations with Caputo derivative. The boundary-value problem is specified by a linear vector functional, the number of components of which does not coincide with the dimension of the system of differential equations. This formulation of the problem is being considered for the first time and includes both underdetermined and overdetermined boundary-value problems. Under the condition that the solution of the homogeneous generating boundary-value problem is not unique and that the inhomogeneous generating boundary-value problem is unsolvable, the conditions for the bifurcation of solutions of this problem are determined. An iterative procedure for constructing a family of solutions of the perturbed linear boundary-value problem in the form of Laurent series in powers of a small parameter \(\varepsilon\) with singularity at the point \(\varepsilon = 0\) is proposed. The results obtained by us generalize the known results of perturbation theory for boundary-value problems for ordinary differential equations.On the \(\rho\)-Caputo impulsive \(p\)-Laplacian boundary problem: an existence analysishttps://zbmath.org/1541.340122024-09-27T17:47:02.548271Z"Chabane, Farid"https://zbmath.org/authors/?q=ai:chabane.farid"Benbachir, Maamar"https://zbmath.org/authors/?q=ai:benbachir.maamar"Etemad, Sina"https://zbmath.org/authors/?q=ai:etemad.sina"Rezapour, Shahram"https://zbmath.org/authors/?q=ai:rezapour.shahram"Avcı, İbrahim"https://zbmath.org/authors/?q=ai:avci.ibrahimIn this paper the authors study a boundary value problem of impulsive differential equations with a nonlinear non-symmetric \(\rho\)-Caputo fractional derivative and an operator of \(p\)-Laplacian type and integral boundary conditions. Existence and uniqueness of solutions are established via Schauder's and Schaefer's fixed point theorems, together with the Banach contraction mapping principle. Examples illustrating the obtained results are also presented.
Reviewer: Sotiris K. Ntouyas (Ioannina)New results for impulsive fractional differential equations through variational methodshttps://zbmath.org/1541.340132024-09-27T17:47:02.548271Z"Gao, Dongdong"https://zbmath.org/authors/?q=ai:gao.dongdong"Li, Jianli"https://zbmath.org/authors/?q=ai:li.jianliSummary: In this paper, we mainly discuss the existence of solutions for impulsive fractional differential equations. By applying variational methods and critical point theory, some new criteria to guarantee that the impulsive fractional differential equation has infinitely many solutions are established. Moreover, we improve and extend some previous results.
{{\copyright} 2021 Wiley-VCH GmbH}Analysis of existence and stability results for fractional impulsive \(\mathfrak{J}\)-Hilfer Fredholm-Volterra modelshttps://zbmath.org/1541.340142024-09-27T17:47:02.548271Z"Ismaael, Fawzi Muttar"https://zbmath.org/authors/?q=ai:ismaael.fawzi-muttarSummary: In this paper, we investigate the suitable conditions for the existence results for a class of \(\mathfrak{J}\)-Hilfer fractional nonlinear Fredholm-Volterra models with new conditions. The findings are based on Banach contraction principle and Schauder's fixed point theorem. Also, the generalized Hyers-Ulam stability and generalized Hyers-Ulam-Rassias stability for solutions of the given problem are provided.Differential equations with fractional derivatives with fixed memory lengthhttps://zbmath.org/1541.340152024-09-27T17:47:02.548271Z"Ledesma, César T."https://zbmath.org/authors/?q=ai:torres-ledesma.cesar-e"Rodríguez, Jesús A."https://zbmath.org/authors/?q=ai:rodriguez.jesus-a.1"da C. Sousa, J. Vanterler"https://zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.joseSummary: In the present paper, we have as main purpose to contribute significantly with deep analysis of the properties of the fractional operators with fixed memory length, in particular, involving the Laplace transform of the Riemann-Liouville fractional integral and derivative with fixed memory length. In this sense, we tackle a problem of the existence and uniqueness of a fractional differential equation with fixed memory length, by means of two fundamental lemmas in the discussion of the integral equation and the Banach fixed point.Alternative solution to the fractional differential equation with recurrence relationshiphttps://zbmath.org/1541.340162024-09-27T17:47:02.548271Z"Luque, Luciano L."https://zbmath.org/authors/?q=ai:luque.luciano-leonardo"Dorrego, Gustavo Abel"https://zbmath.org/authors/?q=ai:dorrego.gustavo-abelSummary: A different solution from the one already known for sequential fractional differential equations with recurrence relation is proposed. This solution involves a Mittag-Leffler type function, which satisfies a recurrence property compatible with the behavior of sequential fractional differential equations with recurrence relation.Homogeneity-based exponential stability analysis for conformable fractional-order systemshttps://zbmath.org/1541.340172024-09-27T17:47:02.548271Z"Mabrouk, Fehmi"https://zbmath.org/authors/?q=ai:mabrouk.fehmiSummary: We study the exponential stability of homogeneous fractional time-varying systems and the existence of Lyapunov homogeneous function for the conformable fractional homogeneous systems. We also prove that the local and global behaviors are similar. A numerical example is given to illustrate the efficiency of the obtained results.Stability of some generalized fractional differential equations in the sense of Ulam-Hyers-Rassiashttps://zbmath.org/1541.340182024-09-27T17:47:02.548271Z"Makhlouf, Abdellatif Ben"https://zbmath.org/authors/?q=ai:ben-makhlouf.abdellatif"El-hady, El-sayed"https://zbmath.org/authors/?q=ai:el-hady.el-sayed"Arfaoui, Hassen"https://zbmath.org/authors/?q=ai:arfaoui.hassen"Boulaaras, Salah"https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud"Mchiri, Lassaad"https://zbmath.org/authors/?q=ai:mchiri.lassaadSummary: In this paper, we investigate the existence and uniqueness of fractional differential equations (FDEs) by using the fixed-point theory (FPT). We discuss also the Ulam-Hyers-Rassias (UHR) stability of some generalized FDEs according to some classical mathematical techniques and the FPT. Finally, two illustrative examples are presented to show the validity of our results.\((\omega,c)\)-asymptotically periodic mild solutions to semilinear two terms fractional differential equationshttps://zbmath.org/1541.340192024-09-27T17:47:02.548271Z"Ouena, Pihire Vincent"https://zbmath.org/authors/?q=ai:ouena.pihire-vincent"Moumini, Kéré"https://zbmath.org/authors/?q=ai:moumini.kereSummary: In this article, we first explore new properties of \((\omega,c)\)-asymptotically periodic functions. Then using the Banach fixed point principle and the Leray-Schauder alternative theorem, we prove the existence and uniqueness of \((\omega,c)\)-asymptotically periodic mild solutions to the abstract semilinear fractional differential equation of the form:
\begin{align*}
D_t^\alpha u(t) & = Au(t)+D_t^{\alpha-1}f(t,u(t)),\quad 1<\alpha<2,\, t\geq 0, \\
u(0) & = u_0,
\end{align*}
where \(A:D(A)\subset\mathbb{X}\to\mathbb{X}\) is a linear densely defined operator of sectorial type on a complex Banach space \(\mathbb{X}\), \(u_0\in \mathbb{X}\), \(f:\mathbb{R}_+\times\mathbb{X}\to\mathbb{X}\) is \((\omega,c)\)-asymptotically periodic in \(t\in\mathbb{R}_+\), and \(D_t^\alpha(\cdot)(1<\alpha<2)\) is the Riemann-Liouville fractional derivative.A study on \(k\)-generalized \(\psi\)-Hilfer derivative operatorhttps://zbmath.org/1541.340202024-09-27T17:47:02.548271Z"Salim, Abdelkrim"https://zbmath.org/authors/?q=ai:salim.abdelkrim"Lazreg, Jamal Eddine"https://zbmath.org/authors/?q=ai:lazreg.jamal-eddine"Ahmad, Bashir"https://zbmath.org/authors/?q=ai:ahmad.bashir.2"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffak"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseSummary: In this paper, we generalize the \(\psi\)-Hilfer fractional derivative and discuss some of its properties. We prove existence, uniqueness and stability results for a class of initial value problems for implicit nonlinear fractional differential equations involving generalized \(\psi\)-Hilfer fractional derivative. The uniqueness result for the given problem is obtained via the Banach contraction mapping principle. In addition, two examples are given to illustrate our results.Positive solutions for the Riemann-Liouville-type fractional differential equation system with infinite-point boundary conditions on infinite intervalshttps://zbmath.org/1541.340212024-09-27T17:47:02.548271Z"Yu, Yang"https://zbmath.org/authors/?q=ai:yu.yang.6|yu.yang|yu.yang.1"Ge, Qi"https://zbmath.org/authors/?q=ai:ge.qiSummary: In this paper, we study the existence and uniqueness of positive solutions for a class of a fractional differential equation system of Riemann-Liouville type on infinite intervals with infinite-point boundary conditions. First, the higher-order equation is reduced to the lower-order equation, and then it is transformed into the equivalent integral equation. Secondly, we obtain the existence and uniqueness of positive solutions for each fixed parameter \(\lambda >0\) by using the mixed monotone operators fixed-point theorem. The results obtained in this paper show that the unique positive solution has good properties: continuity, monotonicity, iteration, and approximation. Finally, an example is given to demonstrate the application of our main results.Periodic boundary-value problem for a Rayleigh-type equation unsolved with respect to the derivativehttps://zbmath.org/1541.340222024-09-27T17:47:02.548271Z"Chuiko, Serhii"https://zbmath.org/authors/?q=ai:chuiko.serhii"Nesmelova, Olha"https://zbmath.org/authors/?q=ai:nesmelova.olhaSummary: We establish constructive necessary and sufficient conditions of solvability and propose a scheme for the construction of solutions to a nonautonomous nonlinear periodic boundary-value problem for a Rayleightype equation unsolved with respect to the derivative. The urgency of investigation of nonautonomous boundary-value problems unsolved with respect to the derivative is explained by the fact that the analysis of traditional problems solved with respect to the derivative is sometimes significantly complicated, e.g., in the presence of nonlinearities that are not integrable in elementary functions. We consider the critical case in which the equation for generating amplitudes of a weakly nonlinear periodic boundary-value problem for a Rayleigh-type equation does not turn into the identity. The least-squares method is used to establish constructive conditions for the solvability and propose convergent iterative schemes for the construction of approximate solutions to a nonautonomous nonlinear boundary-value problem unsolved with respect to the derivative. As an example of application of the proposed iterative scheme, we find approximations to the solutions of periodic boundary-value problems unsolved with respect to the derivative in the case of periodic problem for the equation that describes the motion of a satellite on the elliptic orbit. We obtain an estimate for the range of values of a small parameter in which the iterative procedure used for the construction of solutions to a weakly nonlinear periodic boundary-value problem for a Rayleigh-type equation unsolved with respect to the derivative is convergent. To check the accuracy of the proposed approximations, we estimate the discrepancies appearing in the equation used to simulate the motion of satellites along the elliptic orbits.Comparison theorem for systems of differential equations and its application to estimate the average time benefit from resource collectionhttps://zbmath.org/1541.340232024-09-27T17:47:02.548271Z"Bazulkina, Anastasiya Andreevna"https://zbmath.org/authors/?q=ai:bazulkina.anastasiya-andreevna"Rodina, Lyudmila Ivanovna"https://zbmath.org/authors/?q=ai:rodina.lyudmila-ivanovnaSummary: One of the versions of the comparison theorem for systems of ordinary differential equations is proved, the consequence of which is the property of monotonicity of solutions with respect to initial data. We consider the problem of estimating the average time profit from resource extraction for a structured population consisting of individual species \(x_1,\ldots,x_n\), or divided into \(n\) age groups. We assume that the dynamics of the population in the absence of exploitation is given by a system of differential equations \(\dot x =f(x)\), and at times \(\tau(k)=kd\), \(d>0\), a certain share of the biological resource is extracted from the population \(u(k)=(u_1(k),\ldots,u_n(k))\in [0,1]^n\), \(k=1,2,\ldots.\) It is shown that using the comparison theorem it is possible to find estimates of the average time benefit in cases when analytical solutions for relevant systems are not known. The results obtained are illustrated for models of interaction between two species, such as symbiosis and competition. It is shown that for models of symbiosis, commensalism and neutralism, the greatest value of the average time profit is achieved with the simultaneous exploitation of two types of resources. For populations between which an interaction of the ``competition'' type is observed, cases in which it is advisable to extract only one type of resource are highlighted.To a problem on distinction of a centre, a focus and a saddle-focus for Darboux systemhttps://zbmath.org/1541.340242024-09-27T17:47:02.548271Z"Blashkyevitch, V. V."https://zbmath.org/authors/?q=ai:blashkyevitch.v-v"Dyeniskovets, A. A."https://zbmath.org/authors/?q=ai:dyeniskovets.a-a"Tyshchenko, V. Y."https://zbmath.org/authors/?q=ai:tyshchenko.valentin-yurevichSummary: The problem of definition of topological type (a centre, a focus or a saddle-focus) for the isolated equilibrium state of Darboux system having to steam purely imaginary and one real characteristic roots is solved. The class of systems for which the received algorithm of a solution of the given problem is applicable is reduced.Invariant differential polynomialshttps://zbmath.org/1541.340252024-09-27T17:47:02.548271Z"Malyshev, Fëdor Mikhaĭlovich"https://zbmath.org/authors/?q=ai:malyshev.fedor-mSummary: Based on the method proposed in the article for solving the so-called \((r,s)\)-systems of linear equations proven that the orders of homogeneous invariant differential operators \(n\) of smooth real functions of one variable take values from \(n\) to \(\frac{n(n+1)}{2} \), and the dimension of the space of all such operators does not exceed \(n!\). A~classification of invariant differential operators of order \(n+s\) is obtained for \(s=1,2,3,4\), and for \(n=4\) for all orders from 4 to 10. The only, up to factors, homogeneous invariant differential operators of the smallest order \(n\) and the largest order \(\frac{n(n+1)}{2}\) are given, respectively, by the product of the \(n\) first differentials \((s=0\) ) and the Wronskian \((s=(n-1)n/2)\). The existence of nonzero homogeneous invariant differential operators of order \(n+s\) for \(s<\frac{1+\sqrt{5}}{2}(n-1)\) is proved.On the Hill discriminant of Lamé's differential equationhttps://zbmath.org/1541.340262024-09-27T17:47:02.548271Z"Volkmer, Hans"https://zbmath.org/authors/?q=ai:volkmer.hans-wSummary: Lamé's differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function \(\mathrm{sn}\) depending on the modulus \(k\), and two additional parameters \(h\) and \(\nu\). This differential equation appears in several applications, for example, the motion of coupled particles in a periodic potential. Stability and existence of periodic solutions of Lamé's equations is determined by the value of its Hill discriminant \(D(h,\nu, k)\). The Hill discriminant is compared to an explicitly known quantity including explicit error bounds. This result is derived from the observation that Lamé's equation with \(k=1\) can be solved by hypergeometric functions because then the elliptic function \(\mathrm{sn}\) reduces to the hyperbolic tangent function. A connection relation between hypergeometric functions then allows the approximation of the Hill discriminant by a simple expression. In particular, one obtains an asymptotic approximation of \(D(h,\nu,k)\) when the modulus \(k\) tends to \(1\).Spreading speeds for time heterogeneous prey-predator systems with nonlocal diffusion on a latticehttps://zbmath.org/1541.340272024-09-27T17:47:02.548271Z"Ducrot, Arnaud"https://zbmath.org/authors/?q=ai:ducrot.arnaud"Jin, Zhucheng"https://zbmath.org/authors/?q=ai:jin.zhuchengSummary: We investigate the spreading behaviour for the solutions of a non-autonomous prey-predator system on a discrete lattice. These time variations are assumed to enjoy an averaging property. This includes periodicity, almost periodicity and unique ergodicity as special cases. The spatial motion of individuals from one site to another is modelled by a discrete convolution operator. In order to take into account external fluctuations such as seasonality, daily variations and so on, the convolution kernels and reaction terms may vary with time. Our analysis of the spreading speeds of invasion of the species is based on the careful and detailed study of the hair-trigger effect and spreading speed for a non-autonomous scalar Fisher-KPP equation on a lattice. Then, we are able to compare the solutions of the prey-predator system with those of a suitable scalar Fisher-KPP equation and derive the invasion speeds of the prey and of the predator.Unbounded asymmetric stationary solutions of lattice Nagumo equationshttps://zbmath.org/1541.340282024-09-27T17:47:02.548271Z"Hesoun, Jakub"https://zbmath.org/authors/?q=ai:hesoun.jakub"Stehlík, Petr"https://zbmath.org/authors/?q=ai:stehlik.petr"Volek, Jonáš"https://zbmath.org/authors/?q=ai:volek.jonasA complete characterization of a class of unbounded asymmetric stationary solutions of the lattice Nagumo equations is analyzed in this article. The authors use the tool of iterative mirroring technique to show that for any bistable cubic nonlinearity and arbitrary diffusion rate there exists a two-parametric set of equivalence classes of generally asymmetric stationary solutions which diverge to infinity. They also generalize the result for a broad class of reaction functions. The idea and approach of this article could be applicable to other problems related to lattice equations.
Reviewer: Caidi Zhao (Wenzhou)Assessing the impact of host predation with Holling II response on the transmission of Chagas diseasehttps://zbmath.org/1541.340292024-09-27T17:47:02.548271Z"Jiang, Jiahao"https://zbmath.org/authors/?q=ai:jiang.jiahao.3|jiang.jiahao.2"Gao, Daozhou"https://zbmath.org/authors/?q=ai:gao.daozhou"Jiang, Jiao"https://zbmath.org/authors/?q=ai:jiang.jiao"Wu, Xiaotian"https://zbmath.org/authors/?q=ai:wu.xiaotianSummary: Chagas disease is a zoonosis caused by the protozoan parasite \textit{Trypanosoma cruzi} and transmitted by a broad range of blood-sucking triatomine species. Recently, it is recognized that the parasite can also be transmitted by host ingestion. In this paper, we propose a Chagas disease model incorporating two transmission routes of biting-defecation and host predation between vectors and hosts with Holling II functional response. The basic reproduction number \(\mathcal{R}_v\) of triatomine population and basic reproduction numbers \(\mathcal{R}_0\) of disease population are derived analytically, and it is shown that they are insufficient to serve as threshold quantities to determine dynamics of the model. Our results have revealed the phenomenon of bistability, with backward and forward bifurcations. Specifically, if \(\mathcal{R}_v>1\), the dynamic is rather simple, namely, the disease-free equilibrium is globally asymptotically stable as \(\mathcal{R}_0<1\) and a unique endemic equilibrium is globally asymptotically stable as \(\mathcal{R}_0>1\). However, if \(\mathcal{R}_v<1\), there exists a backward bifurcation with one unstable and one stable positive vector equilibria, and bistability phenomenon occurs, revealing that different initial conditions may lead to disease extinction or persistence even if the corresponding \(\mathcal{R}_0>1\). In conclusion, predation transmission in general reduces the risk of Chagas disease, whilst it makes the complexity of Chagas disease transmission, requiring an integrated strategy for the prevention and control of Chagas disease.Numerical evidence of hyperbolic dynamics and coding of solutions for Duffing-type equations with periodic coefficientshttps://zbmath.org/1541.340302024-09-27T17:47:02.548271Z"Lebedev, Mikhail E."https://zbmath.org/authors/?q=ai:lebedev.mikhail-e"Alfimov, Georgy L."https://zbmath.org/authors/?q=ai:alfimov.georgii-leonidovichSummary: In this paper, we consider the equation \(u_{xx}+Q(x)u+P(x)u^3=0\) where \(Q(x)\) and \(P(x)\) are periodic functions. It is known that, if \(P(x)\) changes sign, a ``great part'' of the solutions for this equation are singular, i. e., they tend to infinity at a finite point of the real axis. Our aim is to describe as completely as possible solutions, which are regular (i. e., not singular) on \(\mathbb{R}\). For this purpose we consider the Poincaré map \(\mathcal{P}\) (i. e., the map-over-period) for this equation and analyse the areas of the plane \((u,u_x)\) where \(\mathcal{P}\) and \(\mathcal{P}^{-1}\) are defined. We give sufficient conditions for hyperbolic dynamics generated by \(\mathcal{P}\) in these areas and show that the regular solutions correspond to a Cantor set situated in these areas. We also present a numerical algorithm for verifying these sufficient conditions at the level of ``numerical evidence''. This allows us to describe regular solutions of this equation, completely or within some class, by means of symbolic dynamics. We show that regular solutions can be coded by bi-infinite sequences of symbols of some alphabet, completely or within some class. Examples of the application of this technique are given.Beyond the Bristol book: advances and perspectives in non-smooth dynamics and applicationshttps://zbmath.org/1541.340312024-09-27T17:47:02.548271ZSummary: Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic ``Bristol book'' [\textit{M. di Bernardo} et al., Piecewise-smooth dynamical systems. Theory and applications. New York, NY: Springer (2008; Zbl 1146.37003)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied.Nilpotent center conditions in cubic switching polynomial Liénard systems by higher-order analysishttps://zbmath.org/1541.340322024-09-27T17:47:02.548271Z"Chen, Ting"https://zbmath.org/authors/?q=ai:chen.ting.2"Li, Feng"https://zbmath.org/authors/?q=ai:li.feng"Yu, Pei"https://zbmath.org/authors/?q=ai:yu.peiIn this paper, the authors investigate nilpotent center conditions and bifurcation of limit cycles in cubic switching polynomial Liénard systems. Assume that the origin \((0, 0)\) is an isolated nilpotent singular point. Then under some generic conditions, the system considered by the authors can be reduced to the following form:
\begin{eqnarray*}
\left( \begin{array}{c} \dot{x} \\
\dot{y} \end{array} \right) = \left \{ \begin{array}{ll} \left( \begin{array}{c} y-(a_{2}^{+} x^2 +a_{3}^{+} x^3) \\
-(b_{2}^{+} x^2 +b_{3}^{+} x^3) \end{array} \right) & \mbox{if } x\geq 0, \\
\left( \begin{array}{c} y-(a_{2}^{-} x^2 +a_{3}^{-} x^3) \\
-(b_{2}^{-} x^2 +b_{3}^{-} x^3) \end{array} \right) & \mbox{if } x< 0, \end{array} \right.
\end{eqnarray*}
where \(a_{k}^{\pm}\in\mathbb{R}\), \(b_{k}^{\pm}\in\mathbb{R}\) for \(k=2, 3\) are system parameters, and the \(y\)-axis is the unique switching line. The authors develop a higher-order Poncaré-Lyapunov method and obtain conditions under which the origin is a nilpotent center and conditions under which the origin is a nilpotent global center. Then based on the Bogdanov-Takens bifurcation theory, they prove that at least five small amplitude limit cycles can bifurcate from the nilpotent center of a class of cubic switching Liénard systems, which is a new lower bound of cyclicity for such polynomial systems around nilpotent center.
Reviewer: Zhengdong Du (Chengdu)The particle paths of hyperbolic conservation lawshttps://zbmath.org/1541.340332024-09-27T17:47:02.548271Z"Fjordholm, Ulrik S."https://zbmath.org/authors/?q=ai:fjordholm.ulrik-skre"Mæhlen, Ola H."https://zbmath.org/authors/?q=ai:maehlen.ola-h"Ørke, Magnus C."https://zbmath.org/authors/?q=ai:orke.magnus-cSummary: Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy solution if and only if the ODE corresponding to its velocity field is well-posed. We also show that the flow of the ODE is 1/2-Hölder regular. Finally, we give several examples showing that our results are sharp, and we provide explicit computations in the case of a Riemann problem.Optimization spectral problem for the Sturm-Liouville operator in a vector function spacehttps://zbmath.org/1541.340342024-09-27T17:47:02.548271Z"Sadovnichii, V. A."https://zbmath.org/authors/?q=ai:sadovnichii.viktor-antonovich"Sultanaev, Ya. T."https://zbmath.org/authors/?q=ai:sultanaev.yaudat-talgatovich"Valeev, N. F."https://zbmath.org/authors/?q=ai:valeev.nur-f|valeev.nurmukhamet-fuatovichThe authors consider the matrix Sturm-Liouville operator defined by the differential expression
\[
L[Q](Y)=-\dfrac{d^2}{dx^2}Y(x)+Q(x)Y(x) \text{ for } x\in (0, 1)
\]
together with the boundary conditions
\[
Y(0)-hY'(0)=0 \text{ and } Y(1)+HY(1)=0.
\]
Note that \(L[Q]\) is acting in the Hilbert space of complex vector valued functions \(L^2_n(0,1)= L^2(0,1)\times\cdots \times L^2(0,1)\) and \(Q(x)\in M^2_n(0,1)\) is a Hermitian matrix, so the spectrum is real, discrete and bounded below.
The main result can be stated as follows: Given \(\lambda^*\in \mathbb{R}\), and \(Q_0 \in M^2_n(0,1)\), then there exists an optimal potential \(\hat{Q}\in M^2_n(0,1)\) such that \(\lambda_k(\hat{Q})= \lambda^*\) while it minimizes \(\Vert Q_0-Q\Vert\) for all Hermitian matrices \(Q \in M^2_n(0,1)\).
It is also shown that if \(\lambda_1(Q_0)\leq \lambda^*\) then \(\hat{Q}\) is unique.
Reviewer: Amin Boumenir (Carrollton)Sharp bounds for Dirichlet eigenvalue ratios of the Camassa-Holm equationshttps://zbmath.org/1541.340352024-09-27T17:47:02.548271Z"Chu, Jifeng"https://zbmath.org/authors/?q=ai:chu.jifeng.2|chu.jifeng"Meng, Gang"https://zbmath.org/authors/?q=ai:meng.gang.1|meng.gangThe authors give a proof of the maximization of Dirichlet eigenvalue ratios for the Camassa-Holm equation:
\[
y''=\frac{y}{4}+\lambda m(x),
\]
where \(m\) is some potential and \(\lambda\) a spectral parameter. For some \(m\), they solve an associated infinitely dimensional maximization problem and find an explicit solution. Then the distribution of all Dirichlet eigenvalues is obtained and the zeros of the associated eigenfunctions discussed.
Reviewer: Smail Djebali (Riyadh)An application of Sobolev's inequality to one-dimensional Kirchhoff equationshttps://zbmath.org/1541.340442024-09-27T17:47:02.548271Z"Goodrich, Christopher S."https://zbmath.org/authors/?q=ai:goodrich.christopher-sAuthor's abstract: We consider nonlocal differential equations with convolution coefficients of the form
\[
- M \left( \left(a \ast(g \circ | u^\prime |) \right)(1) \right) u''(t) = \lambda f(t, u(t))\text{ for } t \in(0, 1),
\]
in which the coefficient function \(M\) imparts a nonlocal structure to the problem. The function \(g\) satisfies \(p\)-\(q\) growth. A model case occurs when \(g(t) : = t^p\), where \(p > 1\), and \(a(t) \equiv 1\) - i.e., the problem
\[
- M \left( \| u^\prime \|_{L^p}^p \right) u''(t) = \lambda f(t, u(t)),\;\; t \in(0, 1).
\]
By an application of the one-dimensional Sobolev inequality, together with a specially constructed order cone, we are able to demonstrate existence of at least one positive solution to this equation when subjected to Dirichlet boundary conditions. Our methodology utilises a more recently developed topological fixed point theory, which allows for the use of sets that are unbounded in the ambient norm.
Reviewer: Haiyan Wang (Phoenix)Positive and decreasing solutions for higher order Caputo boundary value problems with sign-changing Green's functionhttps://zbmath.org/1541.340452024-09-27T17:47:02.548271Z"Yan, Rian"https://zbmath.org/authors/?q=ai:yan.rian"Zhao, Yige"https://zbmath.org/authors/?q=ai:zhao.yige"Leng, Xuan"https://zbmath.org/authors/?q=ai:leng.xuan"Li, Yabing"https://zbmath.org/authors/?q=ai:li.yabingSummary: In this paper, Caputo boundary value problems of order \(3 < \zeta \leq 4\) are investigated on the interval \([0, 1]\). By Guo-Krasnoselskii fixed point theorem, some criteria of existence and multiplicity of positive and decreasing solutionsare established. The main novelty of the paper lies in its capability to achieve positive solutions while the corresponding Green's function changes sign. Finally, two examples are provided to illustrate the application of these results.Ambarzumyan's theorem for the Dirac operator on equilateral tree graphshttps://zbmath.org/1541.340492024-09-27T17:47:02.548271Z"Wu, Dong-Jie"https://zbmath.org/authors/?q=ai:wu.dongjie"Xu, Xin-Jian"https://zbmath.org/authors/?q=ai:xu.xinjian"Yang, Chuan-Fu"https://zbmath.org/authors/?q=ai:yang.chuanfuThe paper deals with the system of Dirac equations
\[
\left\{ B_0 \frac{d}{dx} + V_j(x)\right\} y_j = \lambda y_j, \quad 0 < x < \pi,
\]
\[
B_0 = \begin{pmatrix} 0 & 1 \\
-1 & 0 \end{pmatrix}, \quad V_j(x) = \begin{pmatrix} p_j(x) & q_j(x) \\
q_j(x) & -p_j(x) \end{pmatrix}, \quad y_j(x) = \begin{pmatrix} y_{j,1}(x) \\
y_{j,2}(x) \end{pmatrix}
\]
on a tree graph (i.e. graph without cycles) with appropriate matching conditions in the vertices. The authors prove the Ambarzumyan-type theorem: If \(V(0) = V(\pi)\) and \(\{ 2n \colon n \in \mathbb Z \}\) is a subset of the eigenvalues of the considered system, then \(V(x) = 0\) on \([0,\pi]\), where \(V(x)\) is the diagonal matrix with the elements \(V_j(x)\) on the main diagonal.
Reviewer: Natalia Bondarenko (Saratov)The number of limit cycles of a kind of piecewise quadratic systems with switching curve \(y = x^m\)https://zbmath.org/1541.340552024-09-27T17:47:02.548271Z"Si, Zheng"https://zbmath.org/authors/?q=ai:si.zheng"Zhao, Liqin"https://zbmath.org/authors/?q=ai:zhao.liqin|zhao.liqin.1The authors consider the following real planar autonomous piecewise differential system:
\[
\begin{array}{lll} \left( \begin{array}{c} \dot{x} \\
\dot{y} \end{array} \right) & = & \left\{ \begin{array}{ll} \displaystyle \left( y + \epsilon f_1^{+}(x,y) + \epsilon^2 f_2^{+}(x,y), -x + \epsilon g_1^{+}(x,y) + \epsilon^2 g_2^{+}(x,y) \right) \text{ for }&y \geq x^m, \vspace{0.2cm} \\
\displaystyle \left( y + \epsilon f_1^{-}(x,y) + \epsilon^2 f_2^{-}(x,y), -x + \epsilon g_1^{-}(x,y) + \epsilon^2 g_2^{-}(x,y) \right)\text{ for } &y < x^m, \end{array} \right\} \end{array}
\]
where \(0< |\epsilon|<<1\), \(m \in \mathbb{N}\), \(m \geq 1\) and
\[
\begin{array}{ll} \displaystyle f_1^{\pm} (x,y) \, = \, \sum_{i+j=0}^{2} a_{ij}^{\pm} x^i y^j, & \quad \displaystyle g_1^{\pm} (x,y) \, = \, \sum_{i+j=0}^{2} b_{ij}^{\pm} x^i y^j, \vspace{0.2cm} \\
\displaystyle f_2^{\pm} (x,y) \, = \, \sum_{i+j=0}^{2} c_{ij}^{\pm} x^i y^j, & \quad \displaystyle g_2^{\pm} (x,y) \, = \, \sum_{i+j=0}^{2} d_{ij}^{\pm} x^i y^j \end{array}
\]
with \(a_{ij}\), \(b_{ij}\), \(c_{ij}\) and \(d_{ij}\) real parameters. They study the bifurcation of limit cycles from the period annulus corresponding to the value \(\epsilon=0\). \newline
The authors first give the cases where the first Melnikov function is identically null and then, under this assumption, they provide the expression of the second Melnikov function. Denote by \(Z(m)\) the maximum number of limit cycles that bifurcate from the period annulus of the previous system with \(\epsilon=0\) by using up to the second-order Melnikov function. The authors prove the following results:
\begin{itemize}
\item[(i)] If \(m\) is odd, then \(Z(m)=15\);
\item[(ii)] \(Z(2) \, = \, 9\);
\item[(iii)] \(13 \leq Z(4) \leq 19\);
\item[(iv)] \(Z(2k) \geq 14\) for \(k \geq 3\).
\end{itemize}
These results improve some previously published theorems.
Reviewer: Maite Grau (Lleida)Exact and optimal quadratization of nonlinear finite-dimensional nonautonomous dynamical systemshttps://zbmath.org/1541.340592024-09-27T17:47:02.548271Z"Bychkov, Andrey"https://zbmath.org/authors/?q=ai:bychkov.andrey"Issan, Opal"https://zbmath.org/authors/?q=ai:issan.opal"Pogudin, Gleb"https://zbmath.org/authors/?q=ai:pogudin.gleb-a"Kramer, Boris"https://zbmath.org/authors/?q=ai:kramer.borisSummary: Quadratization of polynomial and nonpolynomial systems of ordinary differential equations (ODEs) is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling, and mathematical analysis. A quadratization reveals new variables and structures of a model, which may be easier to analyze, simulate, and control, and provides a convenient parametrization for learning. This paper presents novel theory, algorithms, and software capabilities for quadratization of nonautonomous ODEs. We provide existence results, depending on the regularity of the input function, for cases when a quadratic-bilinear system can be obtained through quadratization. We further develop existence results and an algorithm that generalizes the process of quadratization for systems with arbitrary dimension that retain the nonlinear structure when the dimension grows. For such systems, we provide dimension-agnostic quadratization. An example is semidiscretized PDEs, where the nonlinear terms remain symbolically identical when the discretization size increases. As an important aspect for practical adoption of this research, we extended the capabilities of the \textsf{QBee} software towards both nonautonomous systems of ODEs and ODEs with arbitrary dimension. We present several examples of ODEs that were previously reported in the literature, and where our new algorithms find quadratized ODE systems with lower dimension than the previously reported lifting transformations. We further highlight an important area of quadratization: reduced-order model learning. This area can benefit significantly from working in the optimal lifting variables, where quadratic models provide a direct parametrization of the model that also avoids additional hyperreduction for the nonlinear terms. A solar wind example highlights these advantages.Reflecting function and a generalization of the notion of first integralhttps://zbmath.org/1541.340632024-09-27T17:47:02.548271Z"Mironenko, V. I."https://zbmath.org/authors/?q=ai:mironenko.vladimir-ivanovich"Mironenko, V. V."https://zbmath.org/authors/?q=ai:mironenko.vladimir-vladimirovichThe reflection function of the differential system
\[
\frac{dx}{dt}= X(t,x), t\in \mathbb{R},x\in D\subset \mathbb{R}^{n}, \tag{*}
\]
whose right-hand side is continuously differentiable and \( D \) is a domain in \(\mathbb{R}^{n}\) , is a function \( F(t,x) \) that has the following properties: for each solution \( x(t) \) of (*), is defined on an interval \( (-\alpha,\alpha) \) and is symmetric with respect to zero, \( F(t,x(t))\equiv x(-t) \) for all \( t\in (-\alpha,\alpha), \) and \( F(0,x)\equiv x. \)
The notion of reflecting function of a differential system first appeared in the paper by \textit{V. I. Mironenko} [Differ. Uravn. 20, No. 9, 1635--1638 (1984; Zbl 0568.34029)]. Subsequently, the theory of reflecting function was studied in the monograph by \textit{V. I. Mironenko} [Отражающая функция и периодические решения дифференциал'ных уравнений (Russian). Minsk: Izdatel'stvo ``Universitetskoe'' (1986; Zbl 0607.34038)] and \textit{Z. Zhou} [The theory of reflecting function of differential equations and applications. Beijing (2014)].
A first integral of (*) is generally defined as a differentiable function \( U(t,x) \) that is constant on each solution \( x(t) \) of (*). Any differentiable function \( U(t,x) \) that is not identically constant is called a generaized first integral of (*) if the function \( U(t,x(t)) \) is even for any solution \( x(t) \) of (*) and is defined on a zero-symmetric interval \( (-\alpha, \alpha) \).
In this paper, the authors trace the relationship between the notion of generalized first integral and the notion of reflecting function and Poincaré map (periodic map) for periodic differential systems. The notion of generalized first integral is used to study questions of the existence and stability of periodic solutions of periodic differential systems. Here the centre focus problem is also analyzed.
Reviewer: Narahari Parhi (Bhubaneswar)Around the asymptotic properties of a two-dimensional parametrized Euler flowhttps://zbmath.org/1541.340642024-09-27T17:47:02.548271Z"Briane, Marc"https://zbmath.org/authors/?q=ai:briane.marcSummary: We study the two-dimensional Euler flow \(X(\cdot,x)\) for \(x\) in the torus \(\mathbb{T}^2: = \mathbb{R}^2/2\pi\mathbb{Z}^2\), solution to the ODE: \(\partial_t X(\cdot,x) = b(X(\cdot,x))\) for \(t\geq 0\), with \(X(0,x) = x\), where \(b\) is the vector field defined on \(\mathbb{T}^2\) by:
\[
b(x) = b(x_1,x_2): = (- A\cos x_1-B\sin x_2, \ A\sin x_1+B\cos x_2),\ \ A,B\in\mathbb{R}\setminus\{0\}.
\]
We derive for any \(x\in\mathbb{T}^2\), the asymptotics of \(X(t,x)\) as \(t\) tends to \(\infty\), depending on whether \(|A| = |B|\) or \(|A|\neq|B|\). In the first case, the orbits of the flow are all bounded. In the second case, it turns out that one of the coordinates of \(X(t,x)\) is bounded with an explicit bound, while the other one is equivalent to \(a(x)t\). The function \(a\) does not vanish in \(\mathbb{T}^2\) and satisfies uniform bounds which depend on parameters \(A\), \(B\). When \(|A|\neq|B|\), we also prove that for any global first integral \(u\) of the flow \(X\) with a periodic gradient, \(\nabla u\) has at least a cluster point of roots in \(\mathbb{T}^2\), which implies the non-existence of any real-analytic first-integral of the flow.Darboux theory of integrability on the Clifford \(n\)-dimensional torushttps://zbmath.org/1541.340652024-09-27T17:47:02.548271Z"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Valls, Claudia"https://zbmath.org/authors/?q=ai:valls.claudiaSummary: For the polynomial vector fields on a Clifford \(n\)-dimensional torus, we develop a Darboux theory of integrability. Moreover, we study the optimal maximum number of invariant meridians in terms of the degree of the polynomial vector field.Dynamics of a piecewise-linear Morris-Lecar model: bifurcations and spike addinghttps://zbmath.org/1541.340682024-09-27T17:47:02.548271Z"Penalva, J."https://zbmath.org/authors/?q=ai:penalva.jose-s|penalva.julia"Desroches, M."https://zbmath.org/authors/?q=ai:desroches.mathieu"Teruel, A. E."https://zbmath.org/authors/?q=ai:teruel.antonio-e"Vich, C."https://zbmath.org/authors/?q=ai:vich.catalinaSummary: Multiple-timescale systems often display intricate dynamics, yet of great mathematical interest and well suited to model real-world phenomena such as bursting oscillations. In the present work, we construct a piecewise-linear version of the Morris-Lecar neuron model, denoted PWL-ML, and we thoroughly analyse its bifurcation structure with respect to three main parameters. Then, focusing on the homoclinic connection present in our PWL-ML, we study the slow passage through this connection when augmenting the original system with a slow dynamics for one of the parameters, thereby establishing a simplified framework for this slow-passage phenomenon. Our results show that our model exhibits equivalent behaviours to its smooth counterpart. In particular, we identify canard solutions that are part of spike-adding transitions. Focusing on the one-spike and on the two-spike scenarios, we prove their existence in a more straightforward manner than in the smooth context. In doing so, we present several techniques that are specific to the piecewise-linear framework and with the potential to offer new tools for proving the existence of dynamical objects in a wider context.Model analysis and data validation of structured prevention and control interruptions of emerging infectious diseaseshttps://zbmath.org/1541.340702024-09-27T17:47:02.548271Z"Zhou, Hao"https://zbmath.org/authors/?q=ai:zhou.hao"Sha, He"https://zbmath.org/authors/?q=ai:sha.he"Cheke, Robert A."https://zbmath.org/authors/?q=ai:cheke.robert-a"Tang, Sanyi"https://zbmath.org/authors/?q=ai:tang.sanyiSummary: The design of optimized non-pharmaceutical interventions (NPIs) is critical to the effective control of emergent outbreaks of infectious diseases such as SARS, A/H1N1 and COVID-19 and to ensure that numbers of hospitalized cases do not exceed the carrying capacity of medical resources. To address this issue, we formulated a classic SIR model to include a close contact tracing strategy and structured prevention and control interruptions (SPCIs). The impact of the timing of SPCIs on the maximum number of non-isolated infected individuals and on the duration of an infectious disease outside quarantined areas (i.e. implementing a dynamic zero-case policy) were analyzed numerically and theoretically. These analyses revealed that to minimize the maximum number of non-isolated infected individuals, the optimal time to initiate SPCIs is when they can control the peak value of a second rebound of the epidemic to be equal to the first peak value. More individuals may be infected at the peak of the second wave with a stronger intervention during SPCIs. The longer the duration of the intervention and the stronger the contact tracing intensity during SPCIs, the more effective they are in shortening the duration of an infectious disease outside quarantined areas. The dynamic evolution of the number of isolated and non-isolated individuals, including two peaks and long tail patterns, have been confirmed by various real data sets of multiple-wave COVID-19 epidemics in China. Our results provide important theoretical support for the adjustment of NPI strategies in relation to a given carrying capacity of medical resources.Fractional evolution equations with nonlocal initial conditions and superlinear growth nonlinear termshttps://zbmath.org/1541.340762024-09-27T17:47:02.548271Z"Chen, Pengyu"https://zbmath.org/authors/?q=ai:chen.pengyu.1|chen.pengyu"Feng, Wei"https://zbmath.org/authors/?q=ai:feng.wei|feng.wei.1In this article, there is considered the following fractional evolution equation with nonlocal initial condition and nonlinear function with superlinear growth in a Banach space
\[
\begin{cases}
& ^C D^\alpha_t u(t)= A u(t) + f(t,u(t)) \text{ for a.e.} \ t \in [0,T], \\
& u(0)=H(u) \in X,
\end{cases}\tag{1}
\]
where \(^C D^\alpha_t\) denotes the fractional Caputo derivative of order \(\alpha \in (0,1)\) and lower terminal \(0\), \(X \subseteq Y\) are two Banach spaces, \(A : \mathcal{ D}(A) \subset X \to X\) is a linear operator generating a compact \(C_0\)-semigroup of contractions, \(f : [0, T] \times X \to Y\) is a Caratheodory map and \(H(u) : C([0, T]; X) \to X\) is a given continuous function.
As main result the authors have found sufficient conditions for the existence of at least one mild solution of the problem (1).
The proof is based on an obtained compactness result and an introduced approximation technique, combined with the Leray-Schauder continuation principle. The first advantage of this approach is that using the compactness of the semigroup generated by the linear operator, the authors neither assume any Lipschitz property of the nonlinear term nor the compactness of the nonlocal initial conditions. The second advantage is that the approximation technique coupled with the Hartmann-type inequality argument allows the treatment of nonlinear terms with superlinear growth.
At the end, three examples are provided to illustrate how the results obtained are applied to fractional parabolic equations with continuous nonlinearities with superlinearly growth and nonlocal initial conditions in the particular cases of periodic or antiperiodic conditions, multipoint conditions and integral-type conditions respectively.
Reviewer: Hristo S. Kiskinov (Plovdiv)Exact controllability for a class of fractional semilinear system of order \(1 < q < 2\) with instantaneous and noninstantaneous impulseshttps://zbmath.org/1541.340772024-09-27T17:47:02.548271Z"Chu, Yunhao"https://zbmath.org/authors/?q=ai:chu.yunhao"Liu, Yansheng"https://zbmath.org/authors/?q=ai:liu.yanshengSummary: This paper is mainly concerned with the existence of mild solutions and exact controllability for a class of fractional semilinear system of order \(q\in(1, 2)\) with instantaneous and noninstantaneous impulses. First, combining the Kuratowski measure of noncompactness and the Mönch fixed point theorem, we investigated the existence result for the considered system. It is remarkable that our assumptions for impulses and the nonlinear term are weaker than the Lipschitz conditions. Next, on this basis, the exact controllability for the considered system is determined. In the end, an example is provided to support the main findings.Comparison theorems for evolution inclusions with maximal monotone operators. \(L^2\)-theoryhttps://zbmath.org/1541.340782024-09-27T17:47:02.548271Z"Tolstonogov, A. A."https://zbmath.org/authors/?q=ai:tolstonogov.alexander-aIn this paper, the following evolution inclusion with time-dependent family of maximal monotone operators in a separable Hilbert space is considered.
\[
-x'(t)\in A(t) x(t) +f(t),
\]
\[
x(0)=x^0,
\]
where \(f(\cdot)\in L^1(T,H)\), \(A(t):D(A(t))\subset H\multimap H, t \in T\) is a family of maximal monotone operators with domain \(D(A(t))\subset H\), \(x^0\in D(A(0))\), \(T=[0,a] (a>0)\) and \(H\) is a separable Hilbert space
Consider also the sweeping process
\[
-x'(t)\in N(D(t))x(t) + \varphi(t),
\]
\[
x(0) = x^0\in D(A(0)),
\]
where \(\varphi(\cdot)\in L^1(T,H)\) and \(N(D(t))x\) is the normal cone to \(D(A(t))\) (in the sense of convex analysis) at the point \(x\in D(A(t))\).
It is shown that if the sweeping process is well-defined and has a solution for each single-valued perturbation from the space of integrable functions, then the evolution inclusion with the maximal monotone operators and single-valued perturbations from the space of integrable functions is also solvable. Quite general conditions in terms of the properties of the family of maximal monotone operators that ensure the existence of solutions for the sweeping process are presented.
Reviewer: Adrian Petruşel (Cluj-Napoca)Some qualitative results for nonlocal dynamic boundary value problem of thermistor typehttps://zbmath.org/1541.341102024-09-27T17:47:02.548271Z"Georgiev, Svetlin G."https://zbmath.org/authors/?q=ai:georgiev.svetlin-georgiev"Khuddush, Mahammad"https://zbmath.org/authors/?q=ai:khuddush.mahammad"Tikare, Sanket"https://zbmath.org/authors/?q=ai:tikare.sanket-aSummary: This paper is concerned with second-order nonlocal dynamic thermistor problem with two-point boundary conditions on time scales. By utilizing the fixed point theorems due to Schaefer and Rus, we establish some sufficient conditions for the existence and uniqueness of solutions. Further, we discuss the continuous dependence of solutions and four types of Ulam stability. We provide examples to support the applicability of our results.The dynamical analysis of nonlinear Ambartsumian equation via tempered \(\Xi\)-Hilfer fractional derivative on time scaleshttps://zbmath.org/1541.341112024-09-27T17:47:02.548271Z"Manikandan, S."https://zbmath.org/authors/?q=ai:manikandan.sreekanth-k|manikandan.sreenath-k"Sivasundaram, S."https://zbmath.org/authors/?q=ai:sivasundaram.seenith"Kanagarajan, K."https://zbmath.org/authors/?q=ai:kanagarajan.kuppusamy"Vivek, D."https://zbmath.org/authors/?q=ai:vivek.devarajSummary: In this chapter, we examine a new class of Ambartsumian equations of the fractional type with tempered \(\Xi\)-Hilfer fractional derivative with boundary conditions. The provided problem is transformed into an equivalent fixed point problem, which is then solved by using the Banach and Krasnosel'skii fixed point theorems. Ulam stability is investigated. An example is included to verify the theoretical results.
For the entire collection see [Zbl 1522.34006].A coupled system of differential-algebraic equation and hyperbolic partial differential equation. Analysis and optimal controlhttps://zbmath.org/1541.350032024-09-27T17:47:02.548271Z"Groh, Dennis"https://zbmath.org/authors/?q=ai:groh.dennisPublisher's description: Coupled systems of differential-algebraic equations (DAEs) and partial differential equations (PDEs) appear in various fields of applications such as electrical engineering, bio-mathematics, or multi-physics. They are of particular interest for the modeling and simulation of flow networks, for instance energy transport networks. In this thesis, we discuss a system in which an abstract DAE and a second order hyperbolic PDE are coupled through nonlinear coupling functions.
The analysis presented is split into two parts: In the first part, we introduce the concept of matrix-induced linear operators which arise naturally in the context of abstract DAEs but have surprisingly not been discussed in literature on abstract DAEs so far. We also present a novel index-1-like criterion that allows to separate dynamical and non-dynamical parts of the abstract DAE while allowing for a considerable reduction of required assumptions, compared to existing theoretical results for abstract DAEs.
In the second part, we build upon the developed techniques. We show how to combine the theoretical frameworks for abstract DAEs and second order hyperbolic PDEs in a way such that both parts of the solution are of similar regularity. We then use a fixed-point approach to prove existence and uniqueness of local as well as global solutions to the coupled system.
In the last part of this thesis, we throw a glance at a related optimal control problem and prove existence of a global minimizer.Conservation laws with nonlocal velocity: the singular limit problemhttps://zbmath.org/1541.353092024-09-27T17:47:02.548271Z"Friedrich, Jan"https://zbmath.org/authors/?q=ai:friedrich.jan"Göttlich, Simone"https://zbmath.org/authors/?q=ai:gottlich.simone"Keimer, Alexander"https://zbmath.org/authors/?q=ai:keimer.alexander"Pflug, Lukas"https://zbmath.org/authors/?q=ai:pflug.lukasThis paper deals with the nonlocal conservation law
\[
\partial_t q(t,x)= - \partial_x\bigl( q(t,x)\mathcal{W} \bigl[ \gamma ,V(q)\bigr] (t,x)\bigr),\tag{1}
\]
where \(V'\le0\), and the nonlocal operator is defined as follows
\[
\mathcal{W} \bigl[ \gamma ,V(q)\bigr] (t,x)=\frac{1}{\eta}\int_x^\infty \gamma\left(\frac{y-x}{\eta}\right)V(q(t,y)dy,
\]
\(\eta\) is a positive constant and \(\gamma \in L^\infty \cap L^1(\mathbb{R})\) is positive, decreasing, and \(\| \gamma \|_{L^1}=1\).
The authors augment (1) with the nonnegative inital condition
\[
q(0,\cdot)=q_0\in BV(\mathbb{R}),
\]
and study the well-posedness of (1) and the convergence of the solutions of (1) to the entropy ones of
\[
\partial_t q(t,x)= - \partial_x\bigl( q(t,x)V(q (t,x))\bigr),
\]
as \(\eta\to0\).
Reviewer: Giuseppe Maria Coclite (Bari)Closing lemma for piecewise smooth vector fields with a recurrent pointhttps://zbmath.org/1541.370222024-09-27T17:47:02.548271Z"Antunes, A. A."https://zbmath.org/authors/?q=ai:antunes.andre-amaral"Carvalho, T."https://zbmath.org/authors/?q=ai:de-carvalho.tiago"Gomide, O. M. L."https://zbmath.org/authors/?q=ai:gomide.otavio-m-lSummary: In this paper we provide a positive answer for the \(C^0\)-Closing Lemma in the context of \(n\)-dimensional piecewise smooth vector fields governed by the Filippov's rules. So, given a model presenting a nontrivially recurrent point it is possible to consider a \(C^0\)-close perturbation of it possessing a closed trajectory. Also, we conclude the paper proving the existence of a closed orbit around a T-singularity.New results for fractional Hamiltonian systemshttps://zbmath.org/1541.370682024-09-27T17:47:02.548271Z"Barhoumi, Najoua"https://zbmath.org/authors/?q=ai:barhoumi.najouaSummary: In this paper, we study the multiplicity of weak nonzero solutions for the following fractional Hamiltonian systems:
\[
\begin{cases}
_t D_\infty^\alpha(_{-\infty}D_t^\alpha u(t)) - L(t)u + \lambda u+ \nabla W(t, u) = 0,\\
u\in H^\alpha(\mathbb{R}, \mathbb{R}^N), \quad t\in\mathbb{R},
\end{cases}
\]
where \(\alpha\in(\frac{1}{2}, 1]\), \(\lambda \in\mathbb{R}\), \(_{-\infty}D^\alpha_t\) and \(_tD^\alpha_\infty\) are left and right Liouville-Weyl fractional derivatives of order \(\alpha\) on real line \(\mathbb{R}\), the matrix \(L(t)\) is not necessarily coercive nor uniformly positive definite and \(W: \mathbb{R}\times\mathbb{R}^N\rightarrow\mathbb{R}\) satisfies some new general and weak conditions. Our results are proved using new symmetric mountain pass theorem established by \textit{R. Kajikiya} [J. Funct. Anal. 225, No. 2, 352--370 (2005; Zbl 1081.49002)]. Some recent results in the literature are generalized and significantly improved and some examples are also given to illustrate our main theoretical results.Hyers-Ulam stability and existence of solutions for high-order fractional \(q\)-difference equations on infinite intervalshttps://zbmath.org/1541.390052024-09-27T17:47:02.548271Z"Wang, Jufang"https://zbmath.org/authors/?q=ai:wang.jufang"Zhang, Jinye"https://zbmath.org/authors/?q=ai:zhang.jinye"Yu, Changlong"https://zbmath.org/authors/?q=ai:yu.changlong(no abstract)The Peano-Sard theorem for Caputo fractional derivatives and applicationshttps://zbmath.org/1541.410212024-09-27T17:47:02.548271Z"Fernandez, Arran"https://zbmath.org/authors/?q=ai:fernandez.arran"Buranay, Suzan Cival"https://zbmath.org/authors/?q=ai:buranay.suzan-civalSummary: The classical Peano-Sard theorem is a very useful result in approximation theory, bounding the errors of approximations that are exact on sets of polynomials. A fractional version was developed by \textit{K. Diethelm} [Numer. Funct. Anal. Optim. 18, No. 7--8, 745--757 (1997; Zbl 0892.41018)] for fractional derivatives of Riemann-Liouville type, which we here extend to fractional derivatives of Caputo type. We indicate some applications to quadrature and interpolation formulae. These results will be useful in the approximate solution of fractional differential equations involving Caputo-type operators, which are often said to be more natural for applications.On Laplace transforms with respect to functions and their applications to fractional differential equationshttps://zbmath.org/1541.440042024-09-27T17:47:02.548271Z"Fahad, Hafiz Muhammad"https://zbmath.org/authors/?q=ai:fahad.hafiz-muhammad"ur Rehman, Mujeeb"https://zbmath.org/authors/?q=ai:ur-rehman.mujeeb"Fernandez, Arran"https://zbmath.org/authors/?q=ai:fernandez.arranSummary: An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called \(\Psi\)-fractional calculus. The operational calculus approach has proved useful for understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in practice. We prove several important properties of these generalised Laplace transforms, including an inversion formula, and apply it to solve some fractional differential equations, using the operational calculus approach for efficient solving.
{{\copyright} 2021 John Wiley \& Sons, Ltd.}Optimal control for Hilfer fractional neutral integrodifferential evolution equations with infinite delayhttps://zbmath.org/1541.490072024-09-27T17:47:02.548271Z"Kavitha, Krishnan"https://zbmath.org/authors/?q=ai:kavitha.krishnan"Vijayakumar, Velusamy"https://zbmath.org/authors/?q=ai:vijayakumar.velusamyThe authors deal with the neutral integro-differential equation
\begin{align*}
&D^{\alpha,\beta}_{0^+}[x(t)-g(t,x_t)]=Ax(t)+B(t)u(t)+f(t,x_t,\int_0^th(t,s,x_s)ds),\ t\in (0,b],\\
&I_{0^+}^{(1-\alpha)(1-\beta)}x(t)|_{t=0}=x_0,\ t\in (-\infty,0],
\end{align*}
where \(A\) is the infinitesimal generator of an analytical semigroup.
The Lagrange problem
\[
\mathcal{L}(u_0)=\int_0^b\mathcal{K}(t,x_t^0,x^0(t),u_0(t))dt\le \mathcal{L}(u),\ u\in \mathcal{U}
\]
is connected with the system above. The existence theorems for both the state and optimal control problems are the main results of the paper.
Reviewer: Igor Bock (Bratislava)A stochastic Galerkin method for the direct and inverse random source problems of the Helmholtz equationhttps://zbmath.org/1541.650062024-09-27T17:47:02.548271Z"Guan, Ning"https://zbmath.org/authors/?q=ai:guan.ning"Chen, Dingyu"https://zbmath.org/authors/?q=ai:chen.dingyu"Li, Peijun"https://zbmath.org/authors/?q=ai:li.peijun.1|li.peijun.2"Zhong, Xinghui"https://zbmath.org/authors/?q=ai:zhong.xinghuiSummary: This paper investigates a novel approach for solving both the direct and inverse random source problems of the one-dimensional Helmholtz equation with additive white noise, based on the generalized polynomial chaos (gPC) approximation. The direct problem is to determine the wave field that is emitted from a random source, while the inverse problem is to use the boundary measurements of the wave field at various frequencies to reconstruct the mean and variance of the source. The stochastic Helmholtz equation is reformulated in such a way that the random source is represented by a collection of mutually independent random variables. The stochastic Galerkin method is employed to transform the model equation into a two-point boundary value problem for the gPC expansion coefficients. The explicit connection between the sine or cosine transform of the mean and variance of the random source and the analytical solutions for the gPC coefficients is established. The advantage of these analytical solutions is that the gPC coefficients are zero for basis polynomials of degree higher than one, which implies that the total number of the gPC basis functions increases proportionally to the dimension, and indicates that the stochastic Galerkin method has the potential to be used in practical applications involving random variables of higher dimensions. By taking the inverse sine or cosine transform of the data, the inverse problem can be solved, and the statistical information of the random source such as the mean and variance can be obtained straightforwardly as the gPC basis functions are orthogonal. Numerical experiments are conducted to demonstrate the efficiency of the proposed method.Chebyshev-quasilinearization method for solving fractional singular nonlinear Lane-Emden equationshttps://zbmath.org/1541.650542024-09-27T17:47:02.548271Z"Mohammadi, Amir"https://zbmath.org/authors/?q=ai:mohammadi.amir-hossein-mousavi|mohammadi.amir|mohammadi.amir-amjad.1"Ahmadnezhad, Ghader"https://zbmath.org/authors/?q=ai:ahmadnezhad.ghader"Aghazadeh, Nasser"https://zbmath.org/authors/?q=ai:aghazadeh.nasserSummary: In this paper, we propose a method for solving some classes of the singular fractional nonlinear Lane-Emden type equations. The method is proposed by utilizing the second-kind Chebyshev wavelets in conjunction with the quasilinearization technique. The operational matrices for the second-kind Chebyshev wavelets are used. The method is tested on the fractional standard Lane-Emden equation, the fractional isothermal gas spheres equation, and some other examples. We compare the results produced by the present method with some well-known results to show the accuracy and efficiency of the method.Designing stable neural networks using convex analysis and ODEshttps://zbmath.org/1541.650552024-09-27T17:47:02.548271Z"Sherry, Ferdia"https://zbmath.org/authors/?q=ai:sherry.ferdia"Celledoni, Elena"https://zbmath.org/authors/?q=ai:celledoni.elena"Ehrhardt, Matthias J."https://zbmath.org/authors/?q=ai:ehrhardt.matthias-joachim"Murari, Davide"https://zbmath.org/authors/?q=ai:murari.davide"Owren, Brynjulf"https://zbmath.org/authors/?q=ai:owren.brynjulf"Schönlieb, Carola-Bibiane"https://zbmath.org/authors/?q=ai:schonlieb.carola-bibianeSummary: Motivated by classical work on the numerical integration of ordinary differential equations we present a ResNet-styled neural network architecture that encodes non-expansive (1-Lipschitz) operators, as long as the spectral norms of the weights are appropriately constrained. This is to be contrasted with the ordinary ResNet architecture which, even if the spectral norms of the weights are constrained, has a Lipschitz constant that, in the worst case, grows exponentially with the depth of the network. Further analysis of the proposed architecture shows that the spectral norms of the weights can be further constrained to ensure that the network is an averaged operator, making it a natural candidate for a learned denoiser in Plug-and-Play algorithms. Using a novel adaptive way of enforcing the spectral norm constraints, we show that, even with these constraints, it is possible to train performant networks. The proposed architecture is applied to the problem of adversarially robust image classification, to image denoising, and finally to the inverse problem of deblurring.The application of fuzzy transform method to the initial value problems of linear differential-algebraic equationshttps://zbmath.org/1541.650602024-09-27T17:47:02.548271Z"Mirzajani, S."https://zbmath.org/authors/?q=ai:mirzajani.saeed"Bahrami, F."https://zbmath.org/authors/?q=ai:bahrami.fariba"Shahmorad, S."https://zbmath.org/authors/?q=ai:shahmorad.sedaghatSummary: In this study, we propose a method based on the direct and inverse fuzzy transforms \((\mathcal{F} \)-transforms) to approximate solutions of the system of linear differential-algebraic equations. We obtain an analytical solution to the initial value problem in terms of basic functions using this method. Since the basic functions have compact support, by employing fuzzy transform to the problem, we obtain the system of linear algebraic equations with lower Hessenberg coefficient matrix. We discuss the solvability of this system and give the error analysis using the consistency of the initial value. We present several examples to illustrate the efficiency and the performance of this new method.Fractional integration via Picard method for solving fractional differential-algebraic systemshttps://zbmath.org/1541.650612024-09-27T17:47:02.548271Z"Mohammad, Susan H."https://zbmath.org/authors/?q=ai:mohammad.susan-h"Mohammed Al-Rozbayani, Abdulghafor"https://zbmath.org/authors/?q=ai:mohammed-al-rozbayani.abdulghaforSummary: In this paper, we applied an efficient integrative method called the fractional Picard method to find the approximate solution to a system of fractional algebraic differential equations (SFADEs). By comparing the results with the exact solution, it was found that this method is highly efficient for finding solutions. Also, a genetic algorithm (GA) was used to speed up the approximate solutions when choosing the best values for the fractional derivative, which increased the efficiency of Picard's fractional integral method in finding the best solutions. The MATLAB program was used to find approximate solutions.Sliding at first-order: higher-order momentum distributions for discontinuous image registrationhttps://zbmath.org/1541.684152024-09-27T17:47:02.548271Z"Bao, Lili"https://zbmath.org/authors/?q=ai:bao.lili"Lu, Jiahao"https://zbmath.org/authors/?q=ai:lu.jiahao"Ying, Shihui"https://zbmath.org/authors/?q=ai:ying.shihui"Sommer, Stefan"https://zbmath.org/authors/?q=ai:sommer.stefanSummary: In this paper, we propose a new approach to deformable image registration that captures sliding motions. The large deformation diffeomorphic metric mapping (LDDMM) registration method faces challenges in representing sliding motion since it per construction generates smooth warps. To address this issue, we extend LDDMM by incorporating both zeroth- and first-order momenta with a nondifferentiable kernel. This allows us to represent both discontinuous deformation at switching boundaries and diffeomorphic deformation in homogeneous regions. We provide a mathematical analysis of the proposed deformation model from the viewpoint of discontinuous systems. To evaluate our approach, we conduct experiments on both artificial images and the publicly available DIR-Lab 4DCT dataset. Results show the effectiveness of our approach in capturing plausible sliding motion.A second-order nonstandard finite difference method for a general Rosenzweig-MacArthur predator-prey modelhttps://zbmath.org/1541.920012024-09-27T17:47:02.548271Z"Manh Tuan Hoang"https://zbmath.org/authors/?q=ai:manh-tuan-hoang."Ehrhardt, Matthias"https://zbmath.org/authors/?q=ai:ehrhardt.matthias-joachim|ehrhardt.matthiasSummary: In this paper, we consider a general Rosenzweig-MacArthur predator-prey model with logistic intrinsic growth of the prey population. We develop the Mickens' method to construct a dynamically consistent second-order nonstandard finite difference (NSFD) scheme for the general Rosenzweig-MacArthur predator-prey model. The second-order NSFD method is based on a novel nonlocal approximation using right-hand side function weights and nonstandard denominator functions.
Through rigorous mathematical analysis, we show that the NSFD method not only preserves two important and prominent dynamical properties of the continuous-time model, namely positivity and asymptotic stability independent of the values of the step size, but also is convergent of order 2. Therefore, it provides a solution to the contradiction between the dynamic consistency and high-order accuracy of NSFD methods.
The proposed NSFD method improves positive and elementary stable nonstandard numerical schemes constructed in a previous work of \textit{D. T. Dimitrov} and \textit{H. V. Kojouharov} [J. Comput. Appl. Math. 189, No. 1--2, 98--108 (2006; Zbl 1087.65068)]. Moreover, the present approach can be extended to construct second-order NSFD methods for some classes of nonlinear dynamical systems.
Finally, the theoretical insights and advantages of the constructed NSFD scheme are supported by some illustrative numerical simulations.A simplified longitudinal model for the development of type 2 diabetes mellitushttps://zbmath.org/1541.920302024-09-27T17:47:02.548271Z"De Gaetano, Andrea"https://zbmath.org/authors/?q=ai:de-gaetano.andrea"Nagy, Ilona"https://zbmath.org/authors/?q=ai:nagy.ilona"Kiss, Daniel"https://zbmath.org/authors/?q=ai:kiss.daniel"Romanovski, Valery G."https://zbmath.org/authors/?q=ai:romanovskii.valerii-georgievich"Hardy, Thomas A."https://zbmath.org/authors/?q=ai:hardy.thomas-aSummary: Obesity and diabetes are a progressively more and more deleterious hallmark of modern, well fed societies. In order to study the potential impact of strategies designed to obviate the pathological consequences of detrimental lifestyles, a model for the development of type 2 diabetes geared towards large population simulations would be useful. The present work introduces such a model, representing in simplified fashion the interplay between average glycemia, average insulinemia and functional beta-cell mass, and incorporating the effects of excess food intake or, conversely, of physical activity levels. Qualitative properties of the model are formally established and simulations are shown as examples of its use.Cytokine storm mitigation for exogenous immune agonistshttps://zbmath.org/1541.920332024-09-27T17:47:02.548271Z"Kareva, Irina"https://zbmath.org/authors/?q=ai:kareva.irina"Gevertz, Jana L."https://zbmath.org/authors/?q=ai:gevertz.jana-lSummary: Cytokine storm is a life-threatening inflammatory response characterized by hyperactivation of the immune system. It can be caused by various therapies, auto-immune conditions, or pathogens, such as respiratory syndrome coronavirus 2 which causes coronavirus disease COVID-19. Here we propose a conceptual mathematical model describing the phenomenology of cytokine-immune interactions when a tumor is treated by an exogenous immune cell agonist which has the potential to cause a cytokine storm, such as CAR T cell therapy. Numerical simulations reveal that as a function of just two model parameters, the same drug dose and regimen could result in one of four outcomes: treatment success without a storm, treatment success with a storm, treatment failure without a storm, and treatment failure with a storm. We then explore a scenario in which tumor control is accompanied by a storm and ask if it is possible to modulate the duration and frequency of drug administration (without changing the cumulative dose) in order to preserve efficacy while preventing the storm. Simulations reveal existence of a ``sweet spot'' in protocol space (number versus spacing of doses) for which tumor control is achieved without inducing a cytokine storm. This theoretical model, which contains a number of parameters that can be estimated experimentally, contributes to our understanding of what triggers a cytokine storm, and how the likelihood of its occurrence can be mitigated.Spatiotemporal dynamics of a multi-delayed prey-predator system with variable carrying capacityhttps://zbmath.org/1541.920612024-09-27T17:47:02.548271Z"Anshu"https://zbmath.org/authors/?q=ai:anshu."Dubey, Balram"https://zbmath.org/authors/?q=ai:dubey.balram(no abstract)Impact of Allee and fear effects in a fractional order prey-predator system with group defense and prey refugehttps://zbmath.org/1541.920732024-09-27T17:47:02.548271Z"Tan, Wenhui"https://zbmath.org/authors/?q=ai:tan.wenhui"Tian, Hao"https://zbmath.org/authors/?q=ai:tian.hao"Song, Yanjie"https://zbmath.org/authors/?q=ai:song.yanjie"Duan, Xiaojun"https://zbmath.org/authors/?q=ai:duan.xiaojun(no abstract)Multistability and chaos in SEIRS epidemic model with a periodic time-dependent transmission ratehttps://zbmath.org/1541.920802024-09-27T17:47:02.548271Z"Brugnago, Eduardo L."https://zbmath.org/authors/?q=ai:brugnago.eduardo-l"Gabrick, Enrique C."https://zbmath.org/authors/?q=ai:gabrick.enrique-c"Iarosz, Kelly C."https://zbmath.org/authors/?q=ai:iarosz.kelly-c"Szezech, José D. jun."https://zbmath.org/authors/?q=ai:szezech.jose-danilo-jun"Viana, Ricardo L."https://zbmath.org/authors/?q=ai:viana.ricardo-luiz"Batista, Antonio M."https://zbmath.org/authors/?q=ai:batista.antonio-marcos"Caldas, Iberê L."https://zbmath.org/authors/?q=ai:caldas.ibere-l(no abstract)Impact of fractional conformable derivatives on A(H1N1) infection modelhttps://zbmath.org/1541.920852024-09-27T17:47:02.548271Z"Eid, Hind Ali Ahmad"https://zbmath.org/authors/?q=ai:eid.hind-ali-ahmad"Alkord, Mohammed N."https://zbmath.org/authors/?q=ai:alkord.mohammed-n-aSummary: In this study, the conformable fractional derivative (CFD) of order \(\varrho\) in conjunction with the LC operator of order \(\rho\) is used to develop the model of the transmission of the A(H1N1) influenza infection. A brand-new A(H1N1) influenza infection model is presented, with a population split into four different compartments. Fixed point theorems were used to prove the existence of the solutions and uniqueness of this model. The basic reproduction number \(R_0\) was determined. The least and most sensitive variables that could alter \(R_0\) were then determined using normalized forward sensitivity indices. Through numerical simulations carried out with the aid of the Adams-Moulton approach, the study also investigated the effects of numerous biological characteristics on the system. The findings demonstrated that if \(\varrho <1\) and \(\rho <1\) under the CFD, also the findings demonstrated that if \(\varrho =1\) and \(\rho =1\) under the CFD, the A(H1N1) influenza infection will not vanish.Fractal and fractional SIS model for syphilis datahttps://zbmath.org/1541.920892024-09-27T17:47:02.548271Z"Gabrick, Enrique C."https://zbmath.org/authors/?q=ai:gabrick.enrique-c"Sayari, Elaheh"https://zbmath.org/authors/?q=ai:sayari.elaheh"Souza, Diogo L. M."https://zbmath.org/authors/?q=ai:souza.diogo-l-m"Borges, Fernando S."https://zbmath.org/authors/?q=ai:borges.fernando-s"Trobia, José"https://zbmath.org/authors/?q=ai:trobia.jose"Lenzi, Ervin K."https://zbmath.org/authors/?q=ai:kaminski-lenzi.ervin"Batista, Antonio M."https://zbmath.org/authors/?q=ai:batista.antonio-marcos(no abstract)The effect of demographic stochasticity on Zika virus transmission dynamics: probability of disease extinction, sensitivity analysis, and mean first passage timehttps://zbmath.org/1541.920982024-09-27T17:47:02.548271Z"Maity, Sunil"https://zbmath.org/authors/?q=ai:maity.sunil-kumar"Mandal, Partha Sarathi"https://zbmath.org/authors/?q=ai:mandal.partha-sarathi|mandal.partha-sarathi.2(no abstract)Mathematical model and optimal control analysis of coffee berry borer with temperature and rainfall variabilityhttps://zbmath.org/1541.921012024-09-27T17:47:02.548271Z"Melese, Abdisa Shiferaw"https://zbmath.org/authors/?q=ai:melese.abdisa-shiferawSummary: This study focuses on a nonlinear deterministic mathematical model for coffee berry borer \textit{(Hypothenemus hampei)} with temperature and rainfall variability. In the model analysis, CBB free and coexistence equilibria are computed. The basic reproduction numbers at a minimum and maximum temperature and rainfall are derived. The qualitative analysis of the model revealed the scenario for equilibria together with basic reproduction numbers. The local stability of equilibria is established through the Jacobian matrix and the Routh-Hurwitz criteria, while the global stability of equilibria is demonstrated using an appropriate Lyapunov function. The normalized sensitivity analysis has also been performed to observe the impact of different parameters on basic reproduction numbers. The proposed model is extended into an optimal control problem by incorporating two control variables, namely, the preventive measure variable based on the separation of susceptible coffee berries from contacting the pests based on biological control and an increase in the death rate of colonizing females of CBB based on chemical control. Optimal disease control analysis is examined using Pontryagin's minimum principle. Finally, the numerical simulations are performed based on analytical results and are discussed quantitatively. Furthermore, the cost-effectiveness of control strategies to determine the best approach to minimize the CBB burden was studied. The study is significant in providing reliable information on how one can use mathematical modeling to improve the roles of control strategies and prevention in CBB transmission in a coffee farm. The outcome of the study may guide public agriculture policymakers on optimal control strategies to control the pests. In particular, using chemical pesticides is very effective to combat pests with minimum costs.Modeling the transmission routes of hepatitis E virus as a zoonotic disease using fractional-order derivativehttps://zbmath.org/1541.921072024-09-27T17:47:02.548271Z"Osman, Shaibu"https://zbmath.org/authors/?q=ai:osman.shaibu"Lassong, Binandam Stephen"https://zbmath.org/authors/?q=ai:lassong.binandam-stephen"Dasumani, Munkaila"https://zbmath.org/authors/?q=ai:dasumani.munkaila"Boateng, Ernest Yeboah"https://zbmath.org/authors/?q=ai:boateng.ernest-yeboah"Onsongo, Winnie Mokeira"https://zbmath.org/authors/?q=ai:onsongo.winnie-mokeira"Diallo, Boubacar"https://zbmath.org/authors/?q=ai:diallo.boubacar.1"Makinde, Oluwole Daniel"https://zbmath.org/authors/?q=ai:makinde.oluwole-danielSummary: Hepatitis E virus (HEV) is one of the emerging zoonotic diseases in Sub-Saharan Africa. Domestic pigs are considered to be the main reservoir for this infectious disease. A third of the world's population is thought to have been exposed to the virus. The zoonotic transmission of the HEV raises serious zoonotic and food safety concerns for the general public. This is a major public health issue in both developed and developing countries. The World Health Organization (WHO) estimated that 44,000 people died in 2015 as a result of HEV infection. East and South Asia have the highest prevalence of this disease overall. In this study, we proposed, developed, and analyzed the transmission routes of the infection using a fractional-order derivative approach. The existence, stability, and uniqueness of solutions were established using the approach and concept in Banach space. Local and global stability was determined using the Hyers-Ulam (HU) stability approach. Numerical simulation was conducted using existing parameter values, and it was established that, as the susceptible human population declines, the number of infected human populations rises with a change in fractional order \(\widehat{\theta}\). When the susceptible pig population increases, the number of infected pig populations rises with a change in \(\widehat{\theta}\). It was observed that a few variations in the fractional derivative order did not alter the function's overall behavior with the results of numerical simulations. Moreover, as the number of recovered human populations increases, there is a corresponding increase in the population of recovered pigs with a change in \(\widehat{\theta}\). The exponential increase in the infected pig population can be controlled by treatment of the infected pigs and prevention of the susceptible pigs. The authors recommend policymakers, and stakeholders prioritize the fight against the virus by enforcing the prevention of humans and treatment of infected pigs. The model can be extended to optimal control and cost-effectiveness analysis to determine the most effective control strategy that comes with less cost in the combat of the disease.Solving dynamic optimization problems to a specified accuracy: an alternating approach using integrated residualshttps://zbmath.org/1541.930932024-09-27T17:47:02.548271Z"Nie, Yuanbo"https://zbmath.org/authors/?q=ai:nie.yuanbo"Kerrigan, Eric C."https://zbmath.org/authors/?q=ai:kerrigan.eric-cEditorial remark: No review copy delivered.On the converse safety problem for differential inclusions: solutions, regularity, and time-varying barrier functionshttps://zbmath.org/1541.931542024-09-27T17:47:02.548271Z"Maghenem, Mohamed"https://zbmath.org/authors/?q=ai:maghenem.mohamed-adlene"Sanfelice, Ricardo G."https://zbmath.org/authors/?q=ai:sanfelice.ricardo-gEditorial remark: No review copy delivered.Impulse-controllability of system classes of switched differential algebraic equationshttps://zbmath.org/1541.931672024-09-27T17:47:02.548271Z"Wijnbergen, Paul"https://zbmath.org/authors/?q=ai:wijnbergen.paul"Trenn, Stephan"https://zbmath.org/authors/?q=ai:trenn.stephanSummary: In this paper, impulse-controllability of system classes containing switched DAEs is studied. We introduce several notions of impulse-controllability of system classes and provide a characterization of strong impulse-controllability of system classes generated by arbitrary switching signals. For a system class generated by switching signals with a fixed-mode sequence, it is shown that either almost all systems are impulse-controllable, or almost all systems are impulse-uncontrollable. Sufficient conditions for all systems in this system class to be impulse-controllable or impulse-uncontrollable are presented. Furthermore, it is observed that even if all systems in a system class are impulse-controllable, knowledge of all the switching times is generally necessary to construct an input that ensures impulse-free solutions. Consequently, it is impossible to design a controller in real time if the switching times in the future are unknown. This phenomenon can be regarded as a causality issue. Therefore, the concept of (quasi-) causal impulse-controllability is introduced, and system classes which are (quasi-) causal are characterized. Finally, necessary and sufficient conditions for a system class to be causal given some dwell-time are stated.The difference between port-Hamiltonian, passive and positive real descriptor systemshttps://zbmath.org/1541.931692024-09-27T17:47:02.548271Z"Cherifi, Karim"https://zbmath.org/authors/?q=ai:cherifi.karim"Gernandt, Hannes"https://zbmath.org/authors/?q=ai:gernandt.hannes"Hinsen, Dorothea"https://zbmath.org/authors/?q=ai:hinsen.dorotheaSummary: The relation between passive and positive real systems has been extensively studied in the literature. In this paper, we study their connection to the more recently used notion of port-Hamiltonian descriptor systems. It is well-known that port-Hamiltonian systems are passive and that passive systems are positive real. Hence it is studied under which assumptions the converse implications hold. Furthermore, the relationship between passivity and KYP inequalities is investigated.Dynamic optimization of complementarity systemshttps://zbmath.org/1541.932232024-09-27T17:47:02.548271Z"Stechlinski, Peter"https://zbmath.org/authors/?q=ai:stechlinski.peter-gEditorial remark: No review copy delivered.