Mathematical Methods in the Applied Sciences最新文献

In the paper, a nonlinear fourth-order difference and a linear compact difference operator are used to discretize the nonlinear fourth-order nonlocal partial integro-differential equations (PIDEs) with a weakly singular kernel. The first-order convolution quadrature is used to approximate the Riemann–Liouville (R-L) fractional integral terms. The Caputo derivative term in temporal direction is deal with L1 formula. We construct a new nonlinear fourth-order difference operator to discretize the nonlinear convection term. The fourth-order term is treated by a linear fourth-order compact difference method. Then, we further prove the existence, stability, convergence, and uniqueness of the solution of the proposed fourth-order nonlinear difference scheme. Finally, two examples are given to verify the correctness of the theory.

This paper presents an analysis of Hilfer fractional stochastic differential equations defined on Banach spaces, where the associated linear operators have nondense domains. The existence and uniqueness of mild solutions are established by applying the Banach fixed-point theorem, in combination with techniques from fractional calculus and semigroup theory. To demonstrate the applicability of the theoretical results, a representative example is presented.

This study investigates the propagation characteristics of Rayleigh waves in a layered medium consisting of a nonlocal fiber-reinforced elastic layer over a nonlocal elastic substrate, incorporating surface corrugation and imperfect interface conditions. Utilizing a harmonic wave analysis combined with the variable separable technique, analytical expressions for displacement fields in both media are derived. Numerical simulations are performed using Mathematica software, where the dispersion relation is derived through the application of appropriate boundary conditions solved and graphically represented. These simulations elucidate the influence of nonlocal elasticity parameters, fiber orientation, interface stiffness, corrugation amplitude, and layer thickness on the wave propagation characteristics. The findings demonstrate that nonlocal elasticity and interfacial imperfections markedly influence wave behavior, causing significant reductions in phase velocity and inducing intricate dispersion phenomena. This novel framework, which integrates size-dependent elasticity with complex interface effects, provides critical insights for applications in nondestructive evaluation, structural health monitoring, and the development of advanced wave-based sensing and vibration mitigation technologies.

This paper introduces a robust and efficient numerical framework for solving nonlinear time-fractional partial integro-differential equations (NLTFPIDEs). Temporal discretization is performed using the weighted and shifted Grünwald–Letnikov formula, incorporating the fractional trapezoidal rule and the second-order backward differentiation formula (BDF2). Spatial discretization leverages Chebyshev nodes as discretization points, with the Lagrange-collocation method applied to approximate partial derivatives. For irregular computational domains, the framework utilizes the finite block method (FBM) in two dimensions. Nonlinearities in the equations are handled through the quasilinearization technique. A comprehensive stability and convergence analysis using the energy method confirms the reliability of the proposed schemes. Numerical experiments validate the theoretical findings, highlighting the accuracy and efficiency of the approach.

In this paper, a SEAIR epidemic model with saturated incidence in a spatially heterogeneous environment is investigated. Firstly, by defining a noncompact measure, the well-posedness including the global existence of the nonnegative solution and the existence of the global attractor are established. Then, using the next generation operator method, the basic reproduction number is given as the threshold of disease dynamics. Some properties about basic reproduction number are also given. The result of the threshold dynamics shows that the disease-free equilibrium is globally asymptotically stable for $��{�R�}_{�0�}�<�1�$, while the disease is uniformly persistent for $��{�R�}_{�0�}�>�1�$. Additionally, with the diffusion rate of the susceptible population tending to 0 and $��\infty �$, the asymptotic profiles of positive steady states are investigated. Finally, through numerical examples, the research results are verified and biological significance is revealed.

This paper is concerned with a class of singular matrix difference systems of mixed order. By transforming them into the corresponding singular Hamiltonian systems, the limit point and limit circle classification of them is obtained, relationships among the associated maximal, pre-minimal, and minimal subspaces are derived, the singular part of the essential spectrum of such a system is characterized in terms of properties of that of the corresponding Hamiltonian system. In addition, several sufficient conditions are obtained for points to be in the regular part of the essential spectrum.

In this paper, we consider the chemotaxis-shallow water system in $��{�ℝ�}^{�2�}�$. We establish the local existence of classical solution without assuming the initial height is small or has a small perturbation near a constant. The far field behavior of the height is a constant which could be either vacuum or nonvacuum. The initial data is allowed vacuum and the spatial measure of the set of vacuum can be arbitrarily large.

In this paper, we study stability and convergence of variable step-size IMEX BDF2 method for solving numerically nonlinear partial integro-differential equations (PIDEs) with a penalty term which describe the jump–diffusion American option pricing model in finance. To avoid the full matrix due to the discretization of nonlocal integral operators, we explicitly discretize the integral operator and implicitly discretize the rest of the operators. The monotonicity of the nonlinear operator plays key roles in showing the stability of the variable step-size IMEX BDF2 method for abstract nonlinear PIDEs. Based on this stability result, the global error bounds for the variable step-size IMEX BDF2 method are provided. By combining fixed-point iteration and finite difference method for spatial discretization, the nonlinear PIDEs are effectively solved. Numerical results illustrate the effectiveness of the proposed method for American options under jump–diffusion models.

This paper presents novel results on the asymptotic stability of mild solutions in the $��p�$th moment for Riemann–Liouville fractional stochastic neutral integro-differential systems (abbreviated as Riemann–Liouville FSNIDSs) of order $��\alpha �\in �\left(�\frac{�1�}{�2�}�,�1�\right)�$, using Banach's contraction mapping principle. The central contribution lies in deriving the mild solution of FSNIDSs with a Riemann–Liouville fractional time derivative by applying a stochastic version of the. variation of constants formula. The analysis draws upon the theory of fractional differential equations, properties of Mittag-Leffler functions, and techniques from asymptotic analysis, under the assumption that the corresponding fractional stochastic neutral dynamical system is asymptotically stable.