Publication detail

# Boundary value problem for elliptic equations, a differential transformation approach

REBENDA, J. ŠMARDA, Z.

Original Title

Boundary value problem for elliptic equations, a differential transformation approach

English Title

Boundary value problem for elliptic equations, a differential transformation approach

Type

conference paper

Language

en

Original Abstract

In this paper, we propose an idea how the differential transformation, a semi-analytical approach based on Taylor’s theorem, can be used to find an approximation of the unique solution to the boundary value problem for partial differential equations of elliptic type. We focus on two-dimensional equations with initial conditions given at the sides of a square. The considered differential equation is transformed into a recurrence relation in two variables. Solving the recurrence relation using the boundary conditions leads to an infinite system of linear equations with infinitely many variables. Approximation of the solution is given in the form of two-dimensional Taylor polynomial whose coefficients are determined by solution to a truncated system of linear equations. Boundary value problem consisting of the Laplace equation and Dirichlet boundary conditions has been chosen to demonstrate relevance of the idea to the given problem.

English abstract

In this paper, we propose an idea how the differential transformation, a semi-analytical approach based on Taylor’s theorem, can be used to find an approximation of the unique solution to the boundary value problem for partial differential equations of elliptic type. We focus on two-dimensional equations with initial conditions given at the sides of a square. The considered differential equation is transformed into a recurrence relation in two variables. Solving the recurrence relation using the boundary conditions leads to an infinite system of linear equations with infinitely many variables. Approximation of the solution is given in the form of two-dimensional Taylor polynomial whose coefficients are determined by solution to a truncated system of linear equations. Boundary value problem consisting of the Laplace equation and Dirichlet boundary conditions has been chosen to demonstrate relevance of the idea to the given problem.

Keywords

Differential transformation; boundary value problem; partial differential equation

Released

25.07.2019

ISBN

9780735418547

Book

International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2018.

Pages from

1

Pages to

4

Pages count

4

URL

Documents

BibTex

```
@inproceedings{BUT157985,
author="Josef {Rebenda} and Zdeněk {Šmarda}",
title="Boundary value problem for elliptic equations, a differential transformation approach",
annote="In this paper, we propose an idea how the differential transformation, a semi-analytical approach based on Taylor’s theorem, can be used to find an approximation of the unique solution to the boundary value problem for partial differential equations of elliptic type. We focus on two-dimensional equations with initial conditions given at the sides of a square. The considered differential equation is transformed into a recurrence relation in two variables. Solving the recurrence relation using the boundary conditions leads to an infinite system of linear equations with infinitely many variables. Approximation of the solution is given in the form of two-dimensional Taylor polynomial whose coefficients are determined by solution to a truncated system of linear equations. Boundary value problem consisting of the Laplace equation and Dirichlet boundary conditions has been chosen to demonstrate relevance of the idea to the given problem.",
booktitle="International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2018.",
chapter="157985",
doi="10.1063/1.5114320",
howpublished="online",
number="310013",
year="2019",
month="july",
pages="1--4",
type="conference paper"
}
```