Solving singular second-orderinitial/boundary value problems in reproducing kernel Hilbert space

Abstract

In this paper, we presents a reproducing kernel method for computing singular second-order initial/boundary value problems (IBVPs). This method could deal with much more general IBVPs than the ones could do, which are given by the previous researchers. According to our work, in the first step, the analytical solution of IBVPs is represented in the RKHS which we constructs. Then, the analytic approximation is exhibited in this RKHS. Finally, the n-term approximation is proved to converge to the analytical solution. Some numerical examples are displayed to demonstrate the validity and applicability of the present method. The results obtained by using the method indicate the method is simple and effective. Mathematics Subject Classification (2000) 35A24, 46E20, 47B32.

Iterative Multistep Reproducing Kernel Hilbert Space Method for Solving Strongly Nonlinear Oscillators

“…Definition 4 (see [24]). The inner space 3 2 [ , ] = { : ( ) ( ) are absolutely continuous real-valued functions on…”

“…Reproducing kernel functions possess some important properties such as being symmetric, unique, and nonnegative. The reader is asked to refer to [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] in order to know more details about reproducing kernel functions, including their mathematical and geometric properties, their types and kinds, and their applications and method of calculations.…”

“…The RKHS method was successfully used by many authors to investigate several scientific applications side by side with their theories. The reader is kindly requested to go through [21][22][23][24][25][26][27][28][29][30][31][32] in order to know more details about RKHS method, including its history, its modification for use, its scientific applications, its kernel functions, and its characteristics.…”

A Unified Reproducing Kernel Method and Error Estimation for Solving Linear Differential Equation with Functional Constraints

“…Wang et al [3] not only got the exact solution described as series for a class of boundary value problems of ordinary differential equations, but also proposed iterative method for the approximate solutions. Gao et al [4] and Li and Wu [5] investigated the reproducing kernel methods to solve singular second-order initial/boundary value problems and multipoint boundary value problems. Zhang [6] and Wu and Lin [7] specially introduced the reproducing kernel methods of solving linear differential equation based on reproducing kernel theory.…”