Mathematical Methods in the Applied Sciences最新文献

Recently, there have been many communicable diseases reported worldwide. Studying a variety of these diseases is therefore necessary, particularly double- and multi-epidemics with fluctuating environmental factors. This paper addresses the existence of the positive solution of an epidemic model with two separate hypotheses, namely, the susceptible-infected-recovered (SIR) and susceptible-infected-recovered-susceptible (SIRS) transmission mechanisms, to focus on this particular case. In this instance, white noise describes the surrounding fluctuations, and a saturated incidence rate carries the disease(s) that are being transmitted. Sufficient conditions are investigated for the disease(s) to persist and eventually go extinct. An ergodic stationary distribution is established under specific sufficient conditions for the considered scenario. Lastly, a few numerical examples are provided to support the theory developed.

Traditional vegetation models are typically limited to simulating specific localized regions or discrete points, failing to capture the spatial heterogeneity of vegetation–water interactions in complex terrains. This study explores a novel network-organized vegetation model for semiarid areas, which extends the Klausmeier framework to network topology through Laplacian matrix quantification of interpatch diffusion, overcoming the uniform terrain limitations of conventional continuum models. The model integrates finite soil carrying capacity with delayed water absorption mechanisms, unveiling a new pathway through which time delay $��\tau �$ drives spatial pattern transitions via the Hopf bifurcation. Furthermore, it establishes stability criteria for delay systems and develops a center manifold-based approach for analyzing periodic solutions. Theoretical analysis demonstrates that the equilibrium remains stable without delay but loses stability through the Hopf bifurcation when $��\tau �$ exceeds critical thresholds. Numerical simulation not only supports the theoretical results but also further demonstrates the evolution process of the system under various parameter conditions, especially exploring the impact of key parameter changes such as soil bearing capacity $��p�$ and time delay $��\tau �$ on system stability. This work provides a novel framework for modeling semiarid ecosystems and explains the mechanisms by which landscape fragmentation influences vegetation patterns under delay-driven dynamics.

This paper introduces a novel and efficient numerical method for pricing European options. Our approach leverages the characteristic function framework and approximates the risk-neutral density of the underlying asset price process using a series expansion in cardinal Hermite interpolant multiwavelets. We first detail the construction and key properties of these multiwavelets, including their interpolation capabilities and suitability for function approximation on bounded intervals. A primary challenge in collocation methods is their inherent sensitivity to the selection of collocation points. To address this, we formulate the pricing problem as a rectangular system of linear equations, which is then solved robustly using a least squares technique, thereby enhancing the method's stability and accuracy. We provide a rigorous theoretical analysis of the proposed scheme, establishing its convergence and deriving the associated order of convergence. The analytical findings are strongly supported by extensive numerical experiments. These results not only validate the theoretical convergence rates but also demonstrate the high computational efficiency and accuracy of our method compared to established benchmarks.

The paper considers a fractional model described by the Bolza problem of optimal control with delay in the phase variable depending on the fractional Caputo derivative of order $��0�<�\alpha �<�1�$ and the fractional Riemann–Liouville integral of order $��\beta �>�0�$. For the problem posed, a necessary optimality condition in the form of the Pontryagin maximum principle is formulated. Moreover, for the control, which is singular, a necessary optimality condition of a higher order is obtained for the first time. Unlike previous works, the results are formulated for both conditions $��\beta �\ge �\alpha �$ and $��\beta �<�\alpha �$. The effectiveness of the available results is illustrated by specific examples.

This article presents an efficient numerical solution for specific integro-differential equations of all fractional orders in the Caputo sense within the interval $��(�0�,�1�]�$. We propose a new numerical scheme to evaluate the approximate solution of a nonlinear system of integro-fractional differential equations of the Volterra–Hammerstein type by utilizing a quadratic B-spline curve, while a linear B-spline curve is employed as a predictor at the initial step to initiate the scheme. First, we establish the Caputo-type fractional-order derivative of the quadratic B-spline curve and apply it to transform the considered problem into a system of nonlinear algebraic equations, where the Gauss–Legendre quadrature formula is applied for integral evaluations. The resulting system is then solved using Newton's method. Additionally, the linear B-spline curve is utilized as a predictor to determine the values of the unknown functions at a single node and to compute one of the unknown control points. The presented method enhances the accuracy and efficiency of numerical computations, making it a useful technique for addressing complex fractional-order models in applied mathematics and engineering. Some illustrative examples are presented, with results displayed in graphs and tables, to validate the method's effectiveness and robustness. MATLAB 9.2 was used to execute and perform the computations for the proposed algorithms.