Taylor polynomial solutions of second order linear partial differential equations
type: Item
instance of: scholarly article
title: Taylor polynomial solutions of second order linear partial differential equations
zbMATH Open document ID: 1055.35032
DOI: 10.1016/S0096-3003(03)00532-0
author: Cenk Keşan
published in: Applied Mathematics and Computation
publication date: 2004-09-24T00:00:00Z
review text: Polynomial approximation for the solutions of second order linear PDE with two variables and variable coefficients of the form \[ A(x,y)\frac{\partial^2u}{\partial x^2}+ B(x,y)\frac{\partial^2u}{\partial x\partial y}+ C(x,y)\frac{\partial^2u}{\partial y^2}+ D(x,y)\frac{\partial u}{\partial x}+ E(x,y)\frac{\partial u}{\partial y}+F(x,y)u=G(x,y), \] where \(A(x,y)\), \(B(x,y)\), \(C(x,y)\), \(D(x,y)\), \(E(x,y)\), \(F(x,y)\), and \(G(x,y)\) are functions having Taylor expressions on the domain \(a\leq x,y\leq b\), under the conditions: \[ \sum_{i=0}^1\left(u^{(0,i)}(x,a)+ u^{(0,i)}(x,b)+ u^{(0,i)}(x,c_1)\right)= f(x), \] \[ \sum_{i=0}^1\left(u^{(i,0)}(a,y)+ u^{(i,0)}(b,y)+ u^{(i,0)}(c_0,y)\right)= g(y), \] \(u(a,a)+ u(a,b)+u(a,c_1)+ u(b,a)+u(b,b)+u(b,c_1)+u(c_0,a)+ u(c_0,b)+ u(c_0,c_1)=\lambda\) is considered. The truncated Taylor series of the functions from the above equation are taken and then substituted into the equation and given conditions. Thereby the considered problem for the PDE is reduced to a matrix equation. Two illustrative examples are considered.
Mathematics Subject Classification ID: 35C05
Mathematics Subject Classification ID: 35C10
Mathematics Subject Classification ID: 35G15
zbMATH DE Number: 2102853
zbMATH Keywords: truncated Taylor series
zbMATH Keywords: approximate solutions
reviewed by: Valery V. Karachik
MaRDI profile type: Publication
cites work: Q3746168
full work available at URL: s0096-3003(03)00532-0
OpenAlex ID: W4250745079
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description: scientific article; zbMATH DE number 2102853