ANNALES POLONICI MATHEMATICI 92.2 (2007) Difference methods for parabolic functional differential problems of the Neumann type by K. Kropielnicka (Gda´ nsk) Abstract. Nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type are considered. A general class of difference methods for the problem is constructed. Theorems on the convergence of difference schemes and error estimates of approximate solutions are presented. The proof of the stability of the difference functional problem is based on a comparison technique. Nonlinear estimates of the Perron type with respect to the functional variable for given functions are used. Numerical examples are given. 1. Introduction. For any two metric spaces X and Y we denote by C (X, Y ) the class of all continuous functions defined on X and taking values in Y . Let M [n] denote the set of all n × n real matrices. We will use vectorial inequalities, understanding that the same inequalities hold be- tween the corresponding components. Let E = [0,a] × [−b, b], where a> 0, b =(b 1 ,...,b n ), b i > 0 for 1 ≤ i ≤ n, and ∂ 0 E = [0,a] × ([−b, b] \ (−b, b)). Write Σ = E × C (E, R) × R n × M [n] and ∂ 0 E j = {(t, x) ∈ ∂ 0 E : x j = b j }∪{(t, x) ∈ ∂ 0 E : x j = −b j }, 1 ≤ j ≤ n, and suppose that f : Σ → R, ϕ :[−b, b] → R, ϕ j : ∂ 0 E j → R, 1 ≤ j ≤ n, are given functions. We consider the functional differential equation (1) ∂ t z (t, x)= f (t, x, z, ∂ x z (t, x),∂ xx z (t, x)) together with the initial boundary condition of Neumann type 2000 Mathematics Subject Classification : 65M12, 35R10. Key words and phrases : functional differential equations, difference methods, stability and convergence. [163] c  Instytut Matematyczny PAN, 2007