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/lp/de-gruyter/numerical-solutions-to-the-darboux-problem-with-the-functional-FcIdXqjzki
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- Publisher
- de Gruyter
- Copyright
- © 1998 Plenum Publishing Corporation
- ISSN
- 1072-947X
- eISSN
- 1072-9176
- DOI
- 10.1515/GMJ.1998.71
- Publisher site
- See Article on Publisher Site
Abstract
The paper deals with the Darboux problem for the equation D xy z ( x, y ) = f ( x, y, z ( x , y )) where z ( x , y ) is a function defined by z ( x , y )( t, s ) = z ( x + t , y + s ), ( t, s ) ∈ – a 0, 0 × – b 0, 0. We construct a general class of difference methods for this problem. We prove the existence and uniqueness of solutions to implicit functional difference equations by means of a comparison method; moreover we give an error estimate. The convergence of explicit difference schemes is proved under a general assumption that given functions satisfy nonlinear estimates of the Perron type. Our results are illustrated by a numerical example.
Journal
Georgian Mathematical Journal – de Gruyter
Published: Feb 1, 1998
Keywords: Volterra condition; differential-integral equation; implicit functional-difference equation; comparison method; nonlinear estimate