Implicit difference schemes for quasilinear parabolic functional equations. (English) Zbl 1302.65196
Summary: We present a new class of numerical methods for quasilinear parabolic
functional differential equations with initial boundary conditions of the Robin type. 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 that the new
methods are considerable better than the explicit schemes. The proof of the stability is
based on a comparison technique with nonlinear estimates of the Perron type for given
functions with respect to functional variables. Results obtained in the paper can be applied
to differential equations with deviated variables and to differential integral problems.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35K55 | Nonlinear parabolic equations |
| 35R10 | Partial functional-differential equations |
Keywords:
quasilinear parabolic functional differential equations; implicit difference methods; nonlinear estimates of the Perron typeReferences:
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