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

Tom 93 / 2008

Lucjan Sapa Annales Polonici Mathematici 93 (2008), 113-133 MSC: 65M12, 65M15, 65M06, 35R10. DOI: 10.4064/ap93-2-2


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.


  • Lucjan SapaFaculty of Applied Mathematics
    AGH University of Science and Technology
    Al. Mickiewicza 30
    30-059 Krak/ow, Poland

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