Mathematical Methods in the Applied Sciences最新文献

This work addresses the numerical solution of fractional time-dependent partial differential equations (FTPDEs) that involve the Riesz fractional derivative in space. Motivated by the effectiveness of space-fractional operators in modeling anomalous diffusion and dispersion phenomena in mathematical physics, we extend this framework to describe classical Brownian motion using a fractional-order formulation based on the Riesz derivative. To this end, we develop a high-order, robust, and efficient numerical scheme for approximating the Riesz derivative, which combines both the left- and right-sided Riemann–Liouville derivatives in a symmetric formulation. We perform a comprehensive analysis of the proposed method, particularly examining its stability and convergence. Furthermore, we apply this method to explore the complex dynamics of pattern formation in two important fractional reaction-diffusion equations, which remain of significant interest in the field. Our experimental results, presented for various fractional parameter values, highlight the method's effectiveness and reveal the intricate behaviors of the system. By utilizing the Riesz fractional derivative, our approach captures the nonlocal and memory effects characteristic of fractional dynamics. This allows for more accurate modeling of phenomena where standard integer-order methods fall short, particularly in capturing the subtleties of anomalous diffusion and pattern formation. The high-order approximation scheme not only ensures numerical accuracy but also enhances computational efficiency, making it a valuable tool for researchers dealing with fractional partial differential equations.

In this paper, we investigate a linearized three-layer finite difference scheme for solving the variable-coefficient nonlinear Kuramoto-Tsuzuki (K-T) complex equation. The proposed scheme employs the standard central finite difference method for spatial discretization, while a combination of the central averaging method and the central difference method is applied in the temporal direction. The nonlinear term is treated using a semi-implicit linearization approach, which effectively enhances stability and reduces computational cost. We provide a rigorous analysis to establish the second-order accuracy of the scheme, as well as the boundedness and uniqueness of the solution are proved by utilizing an energy-based analytical method in conjunction with mathematical induction. Finally, numerical experiments are performed to validate the theoretical analysis.

In this study, we introduce a self-adjoint fractional Sturm–Liouville problem involving Prabhakar fractional derivative, resulting in a unified framework that encompasses Riemann–Liouville and classical settings. By establishing integration-by-part identities for Prabhakar derivative, we demonstrate the essential spectral properties of the fractional Prabhakar Sturm–Liouville problem, including orthogonality of eigenfunctions. Further, the problem's well-posedness is studied by proving coercivity and boundedness properties.

In this research study, a fully discretized scheme based on the Jacobi–Galerkin approach to solve the vanishing–nonvanishing delayed Volterra integral equations has been proposed. Difficulty of this research study is because of the terms of the vanishing and nonvanishing delays. Therefore, one can apply some associated transformations on the variables of the original equation and obtain its numerical solution efficiently by using the Jacobi–Galerkin approach in fully discretized form (i.e., using high accurate Gauss–Legendre quadrature rule). The convergence analysis is discussed in detail by using the important lemmas, which proves that the error of the approximated solution will be decayed very fast in both of the infinity and weighted space norms, respectively, with the condition of sufficiently smooth solutions. Otherwise, as seen in the last example, the rate of convergence will be algebraic instead of exponential. Some test problems are presented to prove the theoretical investigations experimentally and the efficiency of the approach. Superior results can be observed in the case of errors with respect to a recently proposed approach in the literature.

Based on the dynamic management of insulin, we consider the effects of insulin, glucose, and glucocorticoid concentration on the dynamics of the diabetes model with nonlinear impulsive control to simulate the physiological process of exogenous insulin therapy. Firstly, if $��{�\sigma �}_{�1�}�=�0�$, in type 1 diabetes mellitus (T1DM), the existence of positive periodic solution of the system is proved by using fixed point theory for difference equations, whereas the global asymptotic stability of positive periodic solution is proved by using the Floquet multiplier theory and comparison theorem. Next, if $��{�\sigma �}_{�1�}�>�0�$, in type 2 diabetes mellitus (T2DM), the permanence of the solution of the system is proved by using the comparison theorem. Furthermore, simulation results are performed to verify the correctness of the theoretical findings and to study the system's dynamic behavior under various conditions. By simulating the feedback regulation mechanism between insulin and glucose in a closed-loop system, the model not only elucidates the dynamic equilibrium relationship between insulin and glucose but also offers a novel and feasible strategy for the treatment and control of diabetes, thereby demonstrating substantial theoretical and practical significance.

The main objective of this study is to model the HIV-1 infection using ordinary differential equations (ODEs) by considering factors that include infectivity rate, antiretroviral therapy (ART), logistic growth, intracellular time delay for the incidence of the uninfected cells and the infected cells, latency period, and immune responses. The theoretical frameworks such as stability, bifurcation, and the suitable control parameter are investigated to treat the disease progression for the HIV-1 infection model. Based on the stability theory, the steady-state analysis for the HIV-1 model is performed for three types of equilibrium: (1) disease-free equilibrium, (2) immune-free equilibrium, and (3) equilibrium with infection. Utilizing a next-generation matrix approach, the reproduction number is theoretically derived to ensure the virulence of the spread. The Routh–Hurwitz criterion is employed to perform the stability analysis for the model with and without delay for the infection equilibrium point. The qualitative changes in the behaviors of the system called bifurcations, investigating intracellular delay as a bifurcation parameter, Hopf bifurcation conditions, are derived. The threshold value for the delay is analytically obtained; below the threshold value, the model remains stable; if the threshold value exceeds, there are oscillations in the cell population. Furthermore, the model is structured as an optimal control problem choosing therapy as a control parameter based on the Hamilton–Lagrangian approach and the Pontryagin maximum principle. Numerical simulations are conducted to validate the effectiveness of the proposed stability conditions, bifurcation conditions, and optimal control design for the therapy.

In this note, a brief introduction to the physical and mathematical background of the two-component Ginzburg–Landau theory is given. From this theory, we derive a boundary value problem whose solution can be obtained in part by solving a minimization problem using the technique of variational method, except that two of the eight boundary conditions cannot be satisfied. To overcome the difficulty of recovering the full set of boundary conditions, we employ a variety of methods, including the uniform estimation method and the bounded monotonic theorem, which may be applied to other complicated vortex problems in gauge field theories. The twisted vortex solutions are obtained as energy-minimizing cylindrically symmetric field configurations. We also give the sharp asymptotic estimates for the twisted vortex solutions at the origin and infinity.

In this work, we introduce a novel numerical technique for solving geometric differential equations, termed the geometric sinc method (G-Sinc method). This method constructs highly accurate approximations by leveraging both the geometric structure of the underlying problem and the analytical strength of the sinc function. To accommodate the geometric features of the solution domain, we define a modified G-Sinc basis and develop a corresponding computational framework. Several illustrative examples are presented to demonstrate the method's accuracy and efficiency. Comparative results with known exact solutions and conventional numerical approaches show that the G-Sinc method offers several key advantages: improved accuracy with fewer grid points, better handling of boundary singularities, and enhanced stability for complex geometries. These results indicate that the G-Sinc method is a powerful and promising tool for solving a broad class of geometric differential equations with superior precision and computational efficiency.