Implicit difference methods for parabolic functional differential equations

Abstract

We present a new class of numerical methods for the solution of quasilinear parabolic functional differential equations. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show by an example that the new methods are considerable better that the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators. Results obtained in the paper can be applied to differential integral problems and equations with retarded variables.

Implicit difference methods for evolution functional differential equations

“…The above relations and (33) imply that v h is the approximate solution of (3), (4). Thus, all assumptions of Theorem 2 are satisfied, and (45) is a consequence of Theorem 2.…”

“…Numerical examples given in [3][4][5][9][10][11] show that implicit difference methods are natural tools for numerical solution of evolution functional differential equations.…”

“…Initial problems in the Haar pyramid and initial boundary value problems were considered. Implicit difference schemes for nonlinear parabolic equations with initial boundary conditions of the Dirichlet type were studied in [4,11,12]. The convergence of a class of implicit difference methods for parabolic equations with initial boundary conditions of the Neumann type were investigated in [13].…”

Numerical methods for Hamilton Jacobi functional differential equations

“…The papers [7,11] initiated the theory of implicit difference schemes for parabolic functional differen tial equations. The papers [12,18] concern equations with first order partial derivatives and implicit dif ference schemes.…”

Implicit difference methods for parabolic functional differential problems of the Neumann type

“…The papers [13][14][15][16] initiated the theory of implicit difference methods for nonlinear parabolic differential equations. Classical solutions of initial boundary-value problems of the Dirichlet type for nonlinear equations without mixed derivatives were approximated in [14,15] by solutions of difference schemes that are implicit with respect to the time variable.…”