Mathematical Methods in the Applied Sciences最新文献

Computer viruses have caused great harm and nasty effects on people's lives. Therefore, analyzing and establishing the corresponding mathematical model can help explore and study the propagation law of computer viruses. On this basis, we apply the theory of computer virus propagation dynamics by combining computer viruses and the variability of their removal rates of different states in the network. The study presents a new Susceptible-Exposed-Infected-Quarantine-Recovered-Susceptible (SEIQRS) computer virus propagation model with two delays. In addition, we introduce impulse control, an optimal control method, to explore the effect of the impulse period on the model. We first derive the basic reproduction number of the model. Then, the characteristic equations and eigenvalues are discussed analytically by controlling the delay parameters variation in order to analyze local stability and Hopf bifurcation for the computer virus model. Next, we add impulse control and find that it could better control the propagation of computer viruses. Firstly, we analyze the dynamical behavior of this model using the relevant theories of time delay differential equations to provide a theoretical basis for the effective elimination and control of virus propagation in computer networks. Secondly, we use Matlab to conduct numerical simulations, which verify the correctness of theoretical results. Finally, we synthesize the theoretical and experimental analysis of the present model to provide some scientific suggestions for better control of the spreading of computer viruses.

This paper introduces a new algorithmic framework for solving pseudomonotone equilibrium problems and demicontractive fixed-point problems. Unlike conventional methods that incorporate a single inertial step, our approach employs dual inertia to accelerate convergence while preserving stability. The proposed method combines the viscosity approximation technique with the extragradient method to guarantee strong convergence. Initially, the extragradient method is applied under a Lipschitz continuity assumption on the equilibrium bifunction. Subsequently, this condition is relaxed by adopting a self-adaptive step size strategy, allowing variable step sizes to be updated iteratively based on the current iterates information. Notably, the algorithm operates without requiring prior knowledge of the Lipschitz constants or any line search procedures. Strong convergence is established under mild condition, and its applicability to variational inequality problems is demonstrated. Numerical experiments validate the effectiveness of the proposed approach, showcasing its capability to handle large-scale and complex problems efficiently while outperforming traditional single-inertia methods.

This paper introduces a novel numerical algorithm for solving pantograph differential equations and Volterra functional integro-differential equations including the piecewise fractional derivative. The proposed algorithm combines the piecewise Gegenbauer functions, the Ritz method, and operational matrices. These functions have great flexibility to match the problems containing piecewise fractional derivatives. Due to the problems, we calculate the operational matrix of derivatives, the operational matrix of the piecewise fractional derivative, and the pantograph operational matrix. By employing the proposed numerical algorithm, we transform the problems into a system of algebraic equations. Moreover, we discuss the error estimation of the approximate solution and residual error. Finally, we provide details on the implementation of this technique through several numerical examples to demonstrate its applicability.

Installment options, as path-dependent contingent claims, involve paying the premium discretely or continuously in installments, rather than as a lump sum at the time of purchase. In this paper, we applied the PDE approach to price European continuous-installment option and consider Heston stochastic volatility model for the dynamics of the underlying asset. We proved the existence and uniqueness of the weak solution for our pricing problem based on the two-dimensional finite element method. Due to the flexibility to continue or stop paying installments, installment options pricing can be modeled as an optimal stopping time problem. This problem is formulated as an equivalent free boundary problem and then as a linear complementarity problem (LCP). We wrote the resulted LCP in the form of a variational inequality and used the finite element method for the discretization. Then the resulting time-dependent LCPs are solved by using a projected successive over relaxation iteration method. Finally, we implemented our numerical method. The numerical results verified the efficiency and accuracy of the proposed numerical method.

In this paper, we first derive the algorithm of calculating the normal form of spatially nonhomogeneous Hopf bifurcation and Turing bifurcation for the general reaction–diffusion system with spatial average. Then we investigate the spatiotemporal dynamics of nonlocal Lotka–Volterra competitive model and nonlocal Holling–Tanner predator–prey model. It has been shown that the spatial average is a new mechanism to induce the patterns. By calculating the normal form, the type of bifurcation and stability of spatiotemporal patterns bifurcating from the constant equilibrium are investigated. For the nonlocal Lotka–Volterra competitive model, the coexistence of two spatially nonhomogeneous spatial patterns is found. For the nonlocal Holling–Tanner predator–prey model, we found not only the coexistence of two spatially nonhomogeneous spatial patterns but also the stable spatially nonhomogeneous periodic patterns.

The current study addresses the issue of synchronization in competitive neural networks that are based on memristors and involve an inertial term, parameter uncertainty, and two delay components using sampled-data control. To achieve synchronization, appropriate Lyapunov-Krasovskii functionals (LKFs) are constructed, which include double and triple integral terms that capture the information of time delay cross terms. Some sufficient synchronization conditions are derived in terms of linear matrix inequalities (LMIs) using an improved reciprocally convex combination inequality and generalized free weighting matrices inequality. The effectiveness of the proposed findings is highlighted through an illustrative example, validating the robustness and reliability of the developed synchronization approach.