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

Amila J Maldeniya,Sanath K Boralugoda,Kaushika De Silva,Naleen C Ganegoda,Raúl E Curto

doi:10.1155/2021/5511283

Abstract

In this paper, we present some properties of integrable distributions which are continuous linear functional on the space of test functionDℝ2. Here, it uses two-dimensional Henstock–Kurzweil integral. We discuss integrable distributional solution for Poisson’s equation in the upper half spaceℝ+3with Dirichlet boundary condition.

Highlights

We find an integrable distributional solution for Poisson’s equation in the upper half space
Partial differential equations (PDEs) are more difficult to solve than ordinary differential equations (ODEs). erefore, numerical approximations are widely used in application. ere are standard numerical methods such as finite difference and finite element etc. to solve (1)
Numerical integration should be done with respect to s and t while fixing x, y, and z, details in [26]. is reduces solving the partial differential equation into applying numerical integration for the double integral (27)

Summary

Introduction

We find an integrable distributional solution for Poisson’s equation in the upper half space. 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. The same author obtains useful results of Poisson kernel in the unit disk via HK-integral with application into Dirichlet problem [11]. Both articles [10, 11] prove its results in R2 and do not extend into R3. Ere it uses Poisson Kernel, and proof of integrable distributional solution for the Laplace equation is not given.

Integrable Distributions

Multiplier for Integrable Distributions

Integrable Distributional Solution for Poisson Equation

Discussion