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"
}