A finite difference method for quasi-linear and nonlinear differential functional parabolic equations with Dirichlet's condition

Lucjan Sapa¹

¹ Faculty of Applied Mathematics AGH University of Science and Technology Al. Mickiewicza 30 30-059 Krak/ow, Poland

Annales Polonici Mathematici, Tome 93 (2008) no. 2, pp. 113-133.

Résumé

We deal with a finite difference method for a wide class of nonlinear, in particular strongly nonlinear or quasi-linear, second-order partial differential functional equations of parabolic type with Dirichlet's condition. The functional dependence is of the Volterra type and the right-hand sides of the equations satisfy nonlinear estimates of the generalized Perron type with respect to the functional variable. Under the assumptions adopted, quasi-linear equations are a special case of nonlinear equations. Quasi-linear equations are also treated separately. It is proved that our numerical methods are consistent, convergent and stable. Error estimates are given. The proofs are based on the comparison technique. Examples of physical applications and numerical experiments are presented.

Détail

DOI : 10.4064/ap93-2-2

Keywords: finite difference method wide class nonlinear particular strongly nonlinear quasi linear second order partial differential functional equations parabolic type dirichlets condition functional dependence volterra type right hand sides equations satisfy nonlinear estimates generalized perron type respect functional variable under assumptions adopted quasi linear equations special nonlinear equations quasi linear equations treated separately proved numerical methods consistent convergent stable error estimates given proofs based comparison technique examples physical applications numerical experiments presented