On the solution of partial differential equations with specified boundary conditions
type: Item
instance of: scholarly article
title: On the solution of partial differential equations with specified boundary conditions
zbMATH Open document ID: 0678.65084
DOI: 10.1016/0022-247X(89)90084-X
author: G. Adomian
author: M. Elrod
author: Randolph Rach
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: By a certain decomposition, explicit solutions to linear partial differential equations are computed. As an example, the equation \(\partial_{xx}u+\partial_{yy}u=g(x,y)\) is written as \[ u=c_ 1k_ 1(y)+c_ 2k_ 2(y)x+\partial^{-1}_{xx}g-\partial^{- 1}_{xx}\partial_{yy}u, \] \[ u=c_ 3k_ 3(x)+c_ 4k_ 4(x)y+\partial^{-1}_{yy}g-\partial^{-1}_{yy}\partial_{xx}u. \] This is used iteratively with appropriate boundary conditions to compute the solution \(u(x,y)=x^ 2y^ 2/2.\) Several other (more complicated) examples are considered.
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 35C05
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 4111961
zbMATH Keywords: specified boundary conditions
zbMATH Keywords: Poisson equation
zbMATH Keywords: decomposition
zbMATH Keywords: explicit solutions
reviewed by: Martin Brokate
MaRDI profile type: Publication
cites work: Nonlinear equations with mixed derivatives
cites work: A new approach to boundary value equations and application to a generalization of Airy's equation
cites work: The noisy convergence phenomena in decomposition method solutions
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description: scientific article; zbMATH DE number 4111961