Numerical Solution of Some Differential Equations with Henstock–Kurzweil Functions

D A León-Velasco,T Pérez-Becerra,J A Escamilla-Reyna,J J Oliveros-Oliveros,M M Morín-Castillo

doi:10.1155/2019/8948570

Abstract

In this work, the Finite Element Method is used for finding the numerical solution of an elliptic problem with Henstock–Kurzweil integrable functions. In particular, Henstock–Kurzweil high oscillatory functions were considered. The weak formulation of the problem leads to integrals that are calculated using some special quadratures. Definitions and theorems were used to guarantee the existence of the integrals that appear in the weak formulation. This allowed us to apply the above formulation for the type of slope bounded variation functions. Numerical examples were developed to illustrate the ideas presented in this article.

Highlights

Is is a consequence of the weak formulation of the di erential equation
The authors of [2] proposed a numerical improvement to the Gauss–Lobatto quadrature. e quadratures used in this work are the Lobatto quadrature [1, 2, 4] and the open quadrature de ned for functions with a singularity [5]. e results obtained are compared with those given by the trapezoid quadrature. ese three quadrature methods are described below
Is work is organized as follows: In Section 2, the basic elements of the Finite Element Method (FEM) are given; in Section 3, the de nition of the function and some basic results, which allow the application of the FEM, are given; in Section 4, some quadratures for functions are described; in Section 5, numerical examples are presented in order to validate the proposed methodology

Summary

Introduction

Is is a consequence of the weak formulation of the di erential equation. As the second step, numerical methods of integration for functions must be used, in particular, for the case of the highly oscillatory functions. e trapezoid and Simpson methods are commonly used for the numerical calculation of integrals. 1. Introduction e main concern of this work consists of nding, using the Finite Element Method (FEM), the numerical solution of differential equations in which integrable Henstock–Kurzweil functions de ned on the interval [ , ] appear. To calculate the integral numerically, we need a partition of the interval [ , ] of − 1 subintervals and make the sum of the values of the function =

Results

Conclusion