Quadratic Singular Perturbation Problems With Nonmonotone Transition Layer Properties

In this article, we consider the quadratic singular perturbation problems with Nonmonotone Transition Layer Properties. Under certain conditions, solutions are shown to exhibit nonmonotone transition layer behavior at turning point t=0. The formal approximation of problems is constructed using composite expansions, and then approximation solutions of left and right sides at t=0 are joined by joint method which exhibits spike layer behavior and boundary layer behavior respectively. As a result, an approximate solution is formed which exhibits nonmonotone transition layer behavior. In addition, the existence and asymptotic behavior of solutions are proved by the theory of differential inequalities.

Solving a System of Singular Perturbation Problems via Neural Networks

In this work, we will introduce a new procedure to solve a system of singular perturbation problems (SSPPs) via artificial neural networks. The neural networks use the code of back propagation with altered training algorithms such as quasi-Newton, Levenberg-Marquardt, and Bayesian regularization. In our research, we provide examples of two different types of systems, showing the accuracy, speed, resolution, and convergence of the new technology, the effectiveness of using the network techniques for solving this type of equations. The convergence properties of the technique and accuracy of the interpolation technique are considered.