DIFFERENCE METHODS FOR INFINITE SYSTEMS OF QUASILINEAR PARABOLIC FUNCTIONAL DIFFERENTIAL EQUATIONS

Classical solutions of initial boundary value problems for infinite systems of quasilinear parabolic differential functional equations are considered. Two type of difference schemes are constructed. We prove that solutions of infinite difference schemes approximate solutions of our differential functional problem. In the second part of the paper we show that solutions of infinite differential functional systems can be approximated by solutions of difference systems with initial boundary conditions and the systems are finite. A complete convergence analysis for the methods is presented. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given functions. Mathematics Subjects Classification: 65M10, 35K50, 35R10.


Introduction
For any metric spaces X and Y we denote by C(X, y) the class of all continuous functions from X into Y. Let N and Z be the sets of natural numbers and integers respectively. We will use vectorial inequalities with the understanding that the same inequalities hold between their corresponding components. Denote by l°° the class of all real sequences p -such that Iblloo = sup {Ifvl : H € N} < oo. Let the symbol M nxn denotes the set of all real n x n matrices. For A = [aij]i,j=i,...,n we write ||A|| nxri = \ a v\-Denote by M°° the set of all and n

z(t,x) = ip(t,x) on E 0 Ud 0 E.
We consider classical solutions of the above problem. A function v : i2 -> v = {w^j^eN, is a classical solution of problem (1), (2) if (i) v 6 C(i2,l°°), derivatives dtv^, d Xi v^, 1 < i, j < n, exist and they are continuous on E for all fx £ N, (ii) v satisfies (1) on E and the condition (2) holds.
We are interested in establishing a method of approximating of classical solutions to problem (1), (2) by solutions of associated difference equations. The classical difference methods for partial differential equations or systems consist of replacing partial derivatives by suitable difference operators. Solutions of difference functional equations are defined on the mesh. Thus we need also some interpolating operators. Then solutions of difference equations or systems approximate, under suitable assumptions on given functions and on the mesh, solutions of the original problem.
The main task in these investigations is to find a finite difference approximation which is stable. The method of difference inequalities and theorems on linear recurrent inequalities are used in the investigation of the stability of difference schemes. Convergence results are also based on a general theorem on an error estimate of approximate solutions to functional difference equations of the Volterra type with initial or inital boundary conditions and with an unknown function of several variables.
The problems mention above have an extensive bibliography. It is not our aim to show a full review of papers concerning difference methods for parabolic differential functional problems. We will mention only those which contain such reviews. They are [2], [4], [6], [7].
Theorems on the existence and uniqueness of solutions to infinite systems of parabolic differential functional problems can be found in [1], [8], [9]. Two type of difference schemes for problem (1), (2) are constructed. We show that solutions of infinite difference schemes approximate solutions of our differential functional problem. In the second part of the paper we show that solutions of infinite differential functional systems can be approximated by solutions of difference systems with initial boundary conditions and the systems are finite. Results presented in the paper are new also in the case of infinite systems without a functional dependence.
The present paper is organized in the following way. In Section 2 we construct the infinite system of functional difference equations generated by (1), (2). The difference operators approximating mixed derivatives depend on local properties of coefficients of the differential equations. We prove a theorem on the convergence of the difference method. In Section 3 we consider finite systems of functional differential equations. Components of the unknown function 2 = {z^J^eN with the numbers greater than fixed k e N, are replaced by the respective components of an extention of the initial boundary function. The difference method used to such differential systems has the following property. The solution of the difference scheme approximates the solution of differential problem (1), (2) if the step of the mesh is tending to zero and if the number of equations used in this scheme is increasing to infinity.

Infinite systems of difference equations
We formulate a difference problem corresponding to (1), (2). We denote by T (A,B) the class of all functions defined on A and taking values in B, where A and B are arbitrary sets. We define a mesh on the set i? in the following way. Suppose that (ho,h') where h! = (hi,... ,h n ), hi > 0, 0 < i < n, stand for steps of the mesh. We write \\h'\\ = h\ + ... + h n . For h = (ho,h') and (r,m) € Z 1+n where m = (mi,...,mn) we define nodal points as follows:   The motivation for the definition of the set E' h is the following. Approximate solutions of problem (1), (2) are functions Uh defined on Eh-We will write a difference system generated by (1) at each point of the set E' h .
In the sequel we will need the following interpolating operator Th : OhH =n(^H • and we take 0° = 1 in the above formulas. If to = 0 and x^ < x < x( m + 1? ) then we put Then we have defined T^w on D. The above interpolating operator has been proposed in [5] for the construction of difference schemes corresponding to first order partial differential functional equations. We will use the following property of LEMMA 1. Suppose that w : D -»l°° and

j<n. Then \\r h w h -u>\\ D <cho + C\\h'\\ 2 where Wh is the restriction ofw to the set D^.
The proof of the above lemma is similar to the proof of Theorem 3.18 in [5]. Details are omitted. Put Write ei = (0,..., 0,1,0,..., 0) € R n with 1 standing on i-th place. We define the difference operators ¿o, $ = (¿1, • • • ) and , ¿f, 1 < i < n, where the difference operators dij, 1 < i, j < n, are define as follows. Write The difference expressions for (i,j) G J are defined in the following way: We will approximate solutions of (1), (2) by means of solutions of the problem We first prove that solutions of (11), (12) are uniformly bounded on

THEOREM 2. Suppose that Assumptions H[g, f, A] and H[g, f, a] are satisfied and 1) the function v : ft -> , v = (v^^eN)
is a classical solution of (I), (2) and there is co G R+ such that \dXixjVfi{t,x)\ < co, (t,x) G E, 1 < i, j < n, p G N,

Finite systems of difference equations
We consider the problem (1), (2). Let dp : Q -> dp = {(/^¿»eN, be such that  1(t,x), (t, x) e Eq U d0E, 1 < /z < k. we write r < f if rij)\j\i < 0 for any A = (Ai,..., A n ) € R n .   ASSUMPTION H[q, f,a, If we assume that for each /X G N there are AC^ G such that for (i, x) € E
We are ready to prove the main theorem in this part of the paper. Assumptions H[g, f, a, tp] and H[g, f, A] are satisfied and 1) the function v : Q -» v = {v^l^eNj is a classical solution of (1), (2) and there is CQ € R+ such that l^x-v^i, x)| < CQ on E, 1 < i,j < n, neN, 2) for each k G N the function is a solution of (26), (27) and the constant ck € R+ is such that \dXixjU^(t,x)\ <ck on E, 1 < i, j < n, 1 < /i < k, Thus we obtain the assertion (30) with 7t fc '(/i) = cjj^ (a) and = w' fc '(a). •