Applications of Mathematics最新文献

One of the fundamental applications of artificial neural networks is solving Partial Differential Equations (PDEs) which has been considered in this paper. We have created an effective method by combining the spectral methods and multi-layer perceptron to solve Generalized Fitzhugh-Nagumo (GFHN) equation. In this method, we have used Chebyshev polynomials as activation functions of the multi-layer perceptron. In order to solve PDEs, independent variables, which are collocation points, have been used as input dataset. Furthermore, the loss function has been constructed from the residual of the equation and its boundary condition. Minimizing the loss function has adjusted the appropriate values for the parameters of the network. Hence, the network has shown an outstanding performance not only on the training dataset but also on the unseen data. Some numerical examples and a comparison between the results of our proposed method and other existing approaches have been provided to show the efficiency and accuracy of the proposed method. For this purpose different cases such as linear, nonlinear and multi dimensional equations are considered.

HIV is known for causing the destruction of the immune system by affecting different types of cells, while SARS-CoV-2 is an extremely contagious virus that leads to the development of COVID-19. Understanding how these two viruses interact in coinfected individuals is essential, especially in populations under antiretroviral treatment. In this study, we develop and analyze a novel mathematical model capturing the coinfection dynamics of HIV and SARS-CoV-2 under the influence of highly active antiretroviral therapy (HAART). In contrast to previous models, our formulation includes the effect of HAART on both infections and derives the basic reproduction numbers for each virus. We prove that transcritical bifurcations occur when the basic reproduction numbers cross the threshold value of 1, and we establish the conditions for stability of the disease-free equilibria. Numerical simulations show that HAART, although designed to control HIV, also reduces SARS-CoV-2 proliferation in coinfected hosts, which, as far as we know, has not been fully addressed in previous models in the literature. These findings reveal a potentially beneficial indirect effect of antiretroviral therapy on SARS-CoV-2 dynamics, offering new theoretical insights into the control of viral coinfections.

We investigate a class of Kirchhoff-type equations characterized by critical growth within Orlicz-Sobolev spaces. The main result establishes the existence of infinitely many solutions with negative energy. Using an adapted concentration-compactness principle and advanced variational methods, we overcome key challenges such as non-compactness and non-differentiability to the associated functionals. This work extends existing results to more general functional spaces, offering new insights into nonlocal nonlinear equations.

We consider a projection-based model reduction approach to computing the maximal impact, one agent or a group of agents has on the cooperative system. As a criterion for measuring the agent-team impact on multi-agent systems, we use the H_∞ norm, and output synchronization is taken as the underlying cooperative control scheme. We investigate a projection-based model reduction approach that allows efficient H_∞ norm calculation. The convergence of this approach depends on initial interpolation points, so we present approaches to their determination. Since the analysis of multi-agent systems is important from different perspectives, several comparisons are presented in the section on numerical experiments. A graph Laplacian matrix of an inter-agent interaction graph is a foundational element in modeling and analyzing multi-agent systems. We consider various graph topology matrices, system parameters, and excitations of different agents. Different strategies for selecting initial interpolation points are also compared with baseline approaches for calculating the H_∞ norm.

We study the stability of a discontinuous Galerkin (DG) method applied to the numerical solution of traffic flow problems on networks. We discretize the Lighthill-Whitham-Richards equations on each road by DG. At traffic junctions, we consider two types of numerical fluxes that are based on Godunov’s numerical flux derived in a previous work of ours. These fluxes are easily constructible for any number of incoming and outgoing roads, respecting the drivers’ preferences. The analysis is split into two parts: in Part I, contained in this paper, we analyze the stability of the resulting numerical scheme in the L²-norm. The resulting estimates allow for a linear-in-time growth of the square of the L²-norm of the DG solution. This is observed in numerical experiments in certain situations with traffic congestions. Next, we prove that under certain assumptions on the junction parameters (number of incoming and outgoing roads and drivers’ preferences) the DG solution satisfies an entropy inequality where the square entropy is nonincreasing in time. Numerical experiments are presented. The work is complemented by the followup paper, Part II, where a maximum principle is proved for the DG scheme with limiters.

We prove the maximum principle for a discontinuous Galerkin (DG) method applied to the numerical solution of traffic flow problems on networks described by the Lighthill-Whitham-Richards equations. The paper is a followup of the preceding paper, Part I, where L² stability of the scheme is analyzed. At traffic junctions, we consider numerical fluxes based on Godunov’s flux derived in our previous work. We also construct a new Godunov-like numerical flux taking into account right of way at the junction to cover a wider variety of scenarios in the analysis. These fluxes are easily constructible for any number of incoming and outgoing roads, respecting the drivers’ preferences. We prove that the explicit Euler or SSP DG scheme with limiters satisfies a maximum principle on general networks. Numerical experiments demonstrate the obtained results.

We investigate a system of partial differential equations that models the motion of an incompressible double-diffusion convection fluid. The additional stress tensor is generated by a potential with p-structure. In a three-dimensional periodic setting and (p in [{{5} over {3}},2)), we employ a regularized approximation scheme in conjunction with the Galerkin method to establish the existence of regular solutions, provided that the forcing term is properly small. Furthermore, we demonstrate the existence of periodic regular solutions with period T when the external force exhibits periodicity in time with the same period T.

A new fifth-order weighted essentially nonoscillatory (WENO) scheme is designed to approximate Hamilton-Jacobi equations. As employing a fifth-order linear approximation and three third-order ones on the same six-point stencil as before, a newly considered WENO-Z methodology is adapted to define nonlinear weights and the final WENO reconstruction results in a simple and clear convex combination. The scheme has formal fifth-order accuracy in smooth regions of the solution and nonoscillating behavior nearby singularities. A full account is given of the key role of parameters in WENO reconstruction and their selection. The latter half describes the adaptive stage on WENO approximation in convergence framework, which enables us to get the numerical solution to converge still achieving high-order accuracy for the nonconvex problems where the pure WENO scheme fails to converge. Detailed numerical experiments are performed to demonstrate the ability of the proposed numerical methods.

We performe an exponential decay analysis for a Timoshenko-type system under the thermal effect by constructing the Lyapunov functional. More precisely, this thermal effect is acting as a mechanism for dissipating energy generated by the bending of the beam, acting only on the vertical displacement equation, different from other works already existing in the literature. Furthermore, we show the good placement of the problem using semigroup theory.

The existence of nontrivial solutions is considered for the fractional Schrödinger-Poisson system with double quasi-linear terms:

where (−Δ)^α is the fractional Laplacian for α = s, t ∈ (0, 1] with s < t and 2t + 4s > 3. Under assumptions on V and f, we prove the existence of positive solutions and negative solutions for the above system by using perturbation method and the mountain pass theorem.