Existence of the inverse of a linear stochastic operator
type: Item
instance of: scholarly article
title: Existence of the inverse of a linear stochastic operator
zbMATH Open document ID: 0596.60068
DOI: 10.1016/0022-247X(86)90065-X
author: G. Adomian
author: K. Malakian
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: This paper shows that the inverse \(L^{-1}\) can be computed where L is an nth-order linear stochastic differential operator in \(Ly=x\). Somehow the assumption made in the proof that initial conditions were zero disappeared in the publication process. With this necessary statement, the result is correct. It is best to write equation (1) of the paper as \(y=\sum^{\infty}_{n=0}(-1)^ n(L^{-1}R)^ ny_ 0\), where \(y_ 0=\phi +L^{-1}x\) with \(\phi\) representing the homogeneous solution. This yields a correct solution for \(Ly=x\) but then we cannot say \(L^{- 1}\) is the inverse. We can generalize the computation to the two-, three- , or four-dimensional case, again getting an algorithm for solution (even for a nonlinear case if the first author's \(A_ n\) polynomials are used for nonlinear terms), however not an inverse unless all initial/boundary conditions vanish.
Mathematics Subject Classification ID: 60H25
zbMATH DE Number: 3960700
zbMATH Keywords: existence of the inverse
zbMATH Keywords: linear stochastic differential operator
MaRDI profile type: Publication
cites work: Inversion of nonlinear stochastic operators
full work available at URL: 0022-247x(86)90065-x
OpenAlex ID: W2080077528
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description: scientific article; zbMATH DE number 3960700