Implicit difference schemes for mixed problems related to parabolic functional differential equations

2011 ◽  
Vol 100 (3) ◽  
pp. 237-259 ◽  
Author(s):  
Milena Netka
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Difference Schemes ◽  
Mixed Problems ◽  
Implicit Difference Schemes ◽  
Functional Differential

scholarly journals Implicit difference schemes for quasilinear parabolic functional equations

2012 ◽  
Vol 45 (4) ◽  
Author(s):  
Milena Matusik
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Initial Boundary ◽  
Difference Schemes ◽  
New Class ◽  
Implicit Difference Schemes ◽  
Functional Differential ◽  
The Stability ◽  
Quasilinear Parabolic

AbstractWe 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.

Explicit and Implicit Difference Methods for Quasilinear First Order Partial Functional Differential Equations

2014 ◽  
Vol 14 (2) ◽  
pp. 151-175
Author(s):  
Zdzisław Kamont ◽  
Anna Szafrańska
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Difference Schemes ◽  
Implicit Methods ◽  
Implicit Difference Methods ◽  
Difference Methods ◽  
Functional Differential ◽  
Explicit Difference Schemes

Abstract. Initial boundary value problems of the Dirichlet type for quasilinear functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both classes of the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. Error estimates for both methods are presented. Interpolating operators corresponding to functional variables are constructed.

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