Reproducing kernel Hilbert space method for nonlinear second order singularly perturbed boundary value problems with time-delay

Abstract

Abstract The present paper aims to carry out a new scheme for solving a type of singularly perturbed boundary value problem with a second order delay differential equation. Getting through the solution, we used Reproducing Kernel Hilbert Space (RKHS) method as an efficient approach to obtain the analytical solution for ordinary or partial differential equations that appear in vast areas of science and engineering. A key of this method is to keeping the continuous form of problems. Indeed, without discretizing the continuous problem, we change it to an equivalent iterative form and proving its convergence. Also, we will present a construction of the reproducing kernel in Hilbert space that satisfying the homogeneous nonlinear boundary conditions of the considered problem. Accuracy amount of absolute error with respect to different parameters of singularity has been studied for the performance of this method by solving several hardly nonlinear problems. Error estimation and convergence analysis show that the approximate results have uniform convergence to the continuous solutions.