Applied Mathematics and Computation最新文献

This paper introduces three local structure-preserving algorithms for the one-dimensional nonlinear Schrödinger equation with power law nonlinearity, comprising two local energy-conserving algorithms and one local momentum-conserving algorithm. Additionally, we extend these local conservation algorithms to achieve global conservation under periodic boundary conditions. Theoretical analyses confirm the conservation properties of these algorithms. In numerical experiments, we validate the advantages of these algorithms in maintaining long-term energy or momentum conservation by comparing them with a multi-symplectic Preissman algorithm.

In this paper we introduce an efficient numerical method in order to solve Volterra integral equations (VIE) of the second type. We are motivated by the fact that the coupled PDE-ODE model, used to describe the metastatic tumor growth, can be reformulated in terms of VIE, whose unknowns are biological observables, such as the cumulative number of metastases and the total metastatic mass. Here in particular we focused our attention on the 2D non autonomous case, where also the treatment is considered. After reformulating the model as a VIE and introducing and studying the numerical method, we first compare it with a method previously introduced by the authors for the 1D case, and extended to the 2D case only for the sake of comparison, in term of efficiency in the run time execution. Secondly, we present numerical results on the effectiveness of different treatment protocols on the total cumulative number of metastases and the total metastatic mass.

In the present paper, uniform hyperbolic polynomial (UHP) B-spline based collocation method is proposed for solving advection-diffusion equation (ADE) numerically. The Von-Neumann's criterion is used to perform stability analysis. It reveals that the proposed scheme is unconditionally stable. The proposed method is implemented on various examples and numerical outcomes which are reported in table. The numerical outcomes are compared with the other methods available in standard literature. The rate of convergence is also calculated numerically which is found to be closed to 2. The numerical investigation reveals that the developed scheme is efficient, accurate and easy to implement. The proposed method is also applied to solve two-dimensional and three-dimensional ADE to demonstrate the efficiency of proposed scheme.

Our study investigates the dynamics of disease interaction and persistence within populations, exploring various epidemic scenarios, including backward bifurcation and cross-immunity effects. We establish conditions under which the disease-free equilibrium of the model demonstrates local or global asymptotic stability, contingent on the efficacy of quarantine measures. Notably, we find that a strain with a quarantine reproduction number greater than 1 will out-compete a strain with a quarantine reproduction number less than 1, leading to its extinction under complete immunity conditions. Additionally, we identify scenarios where diseases persist in a sub-critical coexistence endemic equilibrium, despite one control reproduction number being below one. Our exploration of backward bifurcation reveals the model's capacity to accommodate the coexistence of the disease-free equilibrium with up to four endemic equilibria. Moreover, we demonstrate that the existence of cross-immunity enhances the coexistence of two strains. However, co-infections and imperfect quarantine measures pose significant challenges in containing outbreaks, sustaining the outbreak potential even with successful control of individual virus strains. Conversely, controlling outbreaks becomes more manageable in the absence of co-infections, especially with perfect quarantine measures. We conclude by advocating for public health strategies that address the complexities posed by co-infections, emphasizing the importance of simultaneously tackling multiple pathogens.

In this paper, we investigate and implement a numerical method that is based on the mimetic finite difference operator in order to solve the nonlinear Allen–Cahn equation with periodic and non-periodic boundary conditions. In addition, we also analyze the performance of this mimetic-based method by using the classical heat equation with a variety of boundary conditions. We assess the performance of the mimetic-based numerical method by comparing the errors of its solutions with those obtained by a classical finite difference method and the pdepde built-in Matlab function. We compute the errors by using the exact solutions when they are available or with reference solutions. We adapt and implement the mimetic-based numerical method by using the MOLE (Mimetic Operators Library Enhanced) library that includes some built-in functions that return representations of the curl, divergence and gradient operators, in order to deal with the Allen-Cahn and heat equations. We present several results with regard to errors and numerical convergence tests in order to provide insight into the accuracy of the mimetic-based numerical method. The results show that the numerical method based on the mimetic difference operator is a reliable method for solving the Allen–Cahn and heat equations with periodic and non-periodic boundary conditions. The numerical solutions generated by the mimetic-based method are relatively accurate. We also proposed a new method based on the mimetic finite difference operator and the convexity splitting approach to solve Allen-Cahn equation in 2D. We found that, for small time step sizes the solutions generated by the mimetic-based method are more accurate than the ones generated by the pdepe Matlab function and similar to the solutions given by a finite difference method.

This study is concerned with reachable set bounding of delayed second-order memristive neural networks (SMNNs) with bounded input disturbances. By applying an analytic method, some inequality techniques and an adaptive control strategy, a sufficient condition of reachable set estimation criterion is derived to guarantee that the states of delayed SMNNs are bounded by a compact ellipsoid. A non-reduced order method is employed to investigate the reachable set bounding problem instead of the reduced order method by variable substitution. In addition, the proposed result is presented in algebraic form, which is easy to test. Finally, a simulation is performed to demonstrate the validity of the proposed algorithm.

We discuss a new high accuracy compact exponential scheme of order four in space and two in time to solve the three-dimensional quasi-linear parabolic partial differential equations. The derived half-step discretization based scheme is implicit in nature and demands only two levels for computation. The generalization of the proposed exponential scheme for the system of the quasi-linear parabolic PDEs is also represented. We generate unconditionally stable alternating direction implicit scheme for the linear parabolic equation in general form. The accuracy and the theoretical results of the proposed scheme are verified for high Reynolds number by several numerical problems like linear and non-linear convection-diffusion equation, coupled Burgers' equations, Navier-Stokes equations, quasi-linear parabolic equation, etc.

This research introduces an extended application of neural networks for solving nonlinear partial differential equations (PDEs). A neural network, combined with a pseudo-arclength continuation, is proposed to construct bifurcation diagrams from parameterized nonlinear PDEs. Additionally, a neural network approach is also presented for solving eigenvalue problems to analyze solution linear stability, focusing on identifying the largest eigenvalue. The effectiveness of the proposed neural network is examined through experiments on the Bratu equation and the Burgers equation. Results from a finite difference method are also presented as comparison. Varying numbers of grid points are employed in each case to assess the behavior and accuracy of both the neural network and the finite difference method. The experimental results demonstrate that the proposed neural network produces better solutions, generates more accurate bifurcation diagrams, has reasonable computational times, and proves effective for linear stability analysis.

Corruption of third-party judges seriously undermines the level of cooperation. Without intervention, more corruptors and defectors would emerge, disrupting social harmony. Therefore, introducing an anti-corruption mechanism is crucial for the evolution of cooperation. In this paper, we propose a social monitoring mechanism to monitor third-party judges so that their payoffs are affected by the proportions of cooperators. Monte Carlo simulations on periodic boundary lattices. The results show that the social monitoring mechanism is effective in promoting cooperation and inhibiting corruption, and enhances the effectiveness of zealots in promoting cooperation. This facilitation effect is not only manifested in the Prisoner's Dilemma Game but also in the Snowdrift Game, which confirms the robustness of the results. Our research provides new insights for solving social dilemmas and curbing corruption.

In this paper, we construct a nonlinear evolutionary game model to analyze the cooperation mechanisms of the population based on a nonlinear relationship among environment and strategies. In the model, replicator dynamics and aspiration dynamics are used to explore the evolutionary outcomes of collective decision, respectively. The results suggest that the environment tends to become progressively more affluent as the number of cooperators increases, if there is a smaller intensity of environmental destruction of defectors. Interestingly, the enriched environments may attract more defectors. Hence, the population requires a higher level of vigilance against plentiful environments in response to the emergence of defectors. As opposed to replicator dynamics, aspiration dynamics can avoid the persistent oscillatory loops due to the level of aspiration. Further, we investigate the effect of complexity between the population strategy and the environment on the evolutionary outcomes. It is found that higher level of complexity can drive the environment closer to a state of affluence, but the population's strategy structure will not be modified. These insights into the relationship between environment and strategies further our understanding of the evolutionary mechanism of population and society.