Numerical Solution of Some Differential Equations with Henstock–Kurzweil Functions

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.

Stable Identification of Sources Located on Interface of Nonhomogeneous Media

“…It is applied to obtain a solution with highly oscillatory functions. Special quadratures have been applied to calculate the integrals that appear in the weak formulation of the finite element method [32]. In this study, we applied the trapezoidal rule to calculate integrals that appear in the applications of the finite element method.…”

Sturm–Liouville Differential Equations Involving Kurzweil–Henstock Integrable Functions

“…The Finite Element method (FEM) has been used to give approximations of the solution of a differential equation when the functions involved are continuous or square integrable. León-Velasco et al in [9] used the FEM to find numerical approximations of the solution of certain Sturm-Liouville-type differential equations involving Henstock-Kurzweil integrable functions. The existence and uniqueness of the problems given in [9], as well as the convergence of the FEM are not studied in that paper.…”

“…León-Velasco et al in [9] used the FEM to find numerical approximations of the solution of certain Sturm-Liouville-type differential equations involving Henstock-Kurzweil integrable functions. The existence and uniqueness of the problems given in [9], as well as the convergence of the FEM are not studied in that paper. In Section 5, we give conditions for the existence and uniqueness of elliptic problems on a bounded interval.…”

Solving Poisson Equation by Distributional HK-Integral: Prospects and Limitations

“…Finite element method can be used to solve elliptic problems with Henstock-Kurzweil integrable functions. Such an approximation is used in [3] for ODEs, where existence and uniqueness of the solution are not discussed. e proposed analytical approach in our paper will direct to a solution, where if its explicit form is not available, one has to only do a numerical integration.…”