Implicit difference methods for parabolic functional differential equations
by
Karolina Kropielnicka
Institute of Mathematics, University of Gdańsk, Wit Stwosz Str. 57, 80-952 Gdańsk, Poland

Implicit difference methods for parabolic functional differential equations

Karolina Kropielnicka

We present a new class of numerical methods for parabolic functional differential equations with initial boundary conditions of the Dirichlet type. Classical solutions of equations

∂tz(t, x)= n å i, j=1  fij(t, x, za(t, x))∂xixjz(t, x)+ n å i=1  gi(t, x, za(t, x))∂xiz(t, x)+G(t, x, za(t, x))

are approximated by solutions of implicit (with respect to the time variable) difference schemes

d0z(r, m)= n å i, j=1  fij(P(r, m)[z])dij z(r+1, m)+ n å i=1  gi(P(r, m)[z])diz(r+1, m)+G(P(r, m)[z]).

We prove that under natural assumptions on f_h there exists exactly one solution of the problem consisting of the above difference functional equation and a suitable initial boundary condition.

We give a complete convergence analysis for the methods and we show by examples that the new methods are considerable better than the explicit schemes.

Differential equations with deviated variables and differential integral problems can be obtained as particular cases from a general model by specializing given operators.

Date received: March 31, 2007