# Method of lines for nonlinear first order partial functional differential equations.

1 Introduction

The method of lines for evolution functional differential equations is obtained by replacing partial derivatives with respect to spatial variables with difference expressions. Differential equations contain functional variables which are elements of the set of continuous functions defined on subsets of a finite dimensional space. Then we need some interpolating operators. This leads to initial problems for systems of ordinary functional differential equations. Such obtained differential difference problems satisfy consistency condition on sufficiently regular solutions of original problems. The main question in these considerations is to find sufficient conditions for the stability of the numerical method of lines. Methods of differential inequalities and comparison techniques are used in the investigations of the stability.

There is an ample literature on the numerical method of lines for evolution differential or functional differential equations. The monographs [9], [10], [11], [20], [21], [22], [26] contain a large bibliography on theoretical investigations and applications.

The papers [5], [15] initiated a theory of the numerical method of lines for functional differential equations. Nonlinear parabolic functional differential equations with initial boundary value conditions were investigated in [13], [14], [16], [18], [27]. Results concerning the stability of the method of lines were obtained in these papers by using a comparison technique.

The papers [1], [2], [6], [7], [12], [28] concern equations with first order partial derivatives. Initial problems with solutions defined on the Haar pyramid and initial boundary value problems were considered. Error estimates implying the convergence of the method are obtained by using a method of differential inequalities. It is assumed that given operators satisfy nonlinear estimates of the Perron type with respect to functional variables.

The monograph [11] contains an exposition of the method of lines for hyperbolic functional differential problems.

The method is also treated as a tool for proving existence theorems for differential problems corresponding to parabolic equations [22] - [24] or hyperbolic problems [3], [4], [8], [17], [19].

The aim of the paper is to construct a method of lines for nonlinear first order partial functional differential equations with initial conditions and solutions defined on the Haar pyramid. Our results are based on the following idea. The original problem is transformed into a system of quasilinear functional differential equations for an unknown function and for their partial derivatives with respect to spatial variables. The numerical method of lines is constructed for systems such obtained.

All the results on the numerical method of lines given in [1], [2], [5]-[7], [12] [14], [27], [28] have the following property. The authors have assumed that given operators satisfy the Lipschitz condition or satisfy nonlinear estimates of the Perron type with respect to functional variables and these conditions are global with respect to all variables. Our assumptions on regularity of given functions are more general. We construct estimates of solutions of initial problems for first order partial functional differential equations and solutions of differential difference systems. We assume that nonlinear estimates of the Perron type and suitable inequalities are local with respect to functional variables. It is clear that there are differential equations with deviated variables and differential integral equations such that local estimates of the Perron type hold and global inequalities are not satisfied.

We use in the paper general ideas for functional differential equations and inequalities which were introduced in [11], [25].

We formulate our functional differential problems. For any metric spaces X and Y, by C(X, Y) we denote the class of all continuous functions from X into Y. We use vectorial inequalities with the understanding that the same inequalities hold between their corresponding components.

Let E be the Haar pyramid

E = {(t,x) [member of] [R.sup.1+n] : t [member of] [0,a], -b + Mt [less than or equal to] x [less than or equal to] b - Mt}

where a > 0, b, M [member of] [R.sup.n.sub.+], b = ([b.sub.1], ... , [b.sub.n]), M = ([M.sub.1], ... , [M.sub.n]) and b > Ma. Write

[E.sub.0] = [-[b.sub.0],0] x [-b,b].

For (t, x) [member of] E we define

D[t,x] = {([tau],y) [member of] [R.sup.1+n] : [tau] [less than or equal to] 0, (t + [tau],x + y) [member of] [E.sub.0] [union] E}.

Then the set D[t, x] is a sum of the following sets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let B = [-[b.sub.0] - a,0] x [-2b,2b] then D[t,x] [subset] B for (t,x) [member of] E. For a function z : [E.sub.0] [union] E [right arrow] R and for a point (t,x) [member of] E we define [Z.sub.(tx)] : D[t, x][right arrow] R by

[z.sub.(t,x)]([tau],y) = z(t + [tau],x + y), ([tau],y) [member of] D[t,x].

The function [Z.sub.(tx)] is the restriction of z to the set ([E.sub.0] [union] E) [intersection] ([-[b.sub.0],t] x [R.sup.n]) and this restriction is shifted to the set D[t, x].

Let [[phi].sub.0] : [0, a] [right arrow] R and [phi] : E [right arrow] [R.sup.n],[phi] = ([[phi].sub.1], ... , [[phi.sub.n]), be given functions. The requirements on [[phi].sub.0] and [phi] are that 0 [less than or equal to] [[phi].sub.0] [less than or equal to] t for t [member of] [0,a] and ([[phi].sub.0](t),[phi](t,x)) [member of] E for (t,x) [member of] E. Write [phi](t,x) = ([[phi].sub.0](t),[phi](t,x)) on E.

Put [OMEGA] = E x C(B, R) x C(B, R) x [R.sup.n] and suppose that f : [OMEGA][right arrow]R,[psi] : [E.sub.0] [right arrow] R are given functions. We will say that f satisfies condition (V) if for each (t, x, q) [member of] E x [R.sup.n] and for v, [??], w, [??] [member of] C(B, R) such that v([tau],y) = [??]([tau],y) for ([tau],y) [member of] D[t,x] and w([tau]t,y) = [??]([tau],y) for ([tau],y) [member of] D[[phi](t,x)] we have f (t,x,v,w,q) = f (t, x, [??], [??], q). Note that the condition (V) means that the value of f at the point (t, x, v, w, q) [member of] [OMEGA] depends on (t, x, q) and on the restrictions of v and w to the sets D[t,x] and D[[phi](t,x)] only.

Let z be an unknown function of the variables (t,x) = (t,[x.sub.1], ... ,[x.sub.n]). We consider the functional differential equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

with the initial condition

z(t, x) = [psi](t, x) on [E.sub.0] (2)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In the paper we assume that f satisfies the condition (V) and we consider classical solutions of (1), (2).

Our concern is the method of lines for problem (1), (2). In the first step we construct a quasilinear system of functional differential equations for z and u = [[partial derivative].sub.x]z. We use a discretization with respect to spatial variable x for such obtained system. Then we associate with (1), (2) a net of Cauchy problems for ordinary functional differential equations. Solutions of such systems are considered as approximate solution of (1), (2). Then we estimate the difference between the exact and approximate solutions of (1), (2) and, as a consequence, we prove that approximate solutions converge to the classical solution of (1), (2). We present a complete convergence analysis for the method and we give numerical examples.

The paper is organized as follows. In Section 2 we formulate a numerical method of lines for (1), (2). In the next section we prove that there exists exactly one solution of the Cauchy problem for differential difference equations generated by (1), (2). We give estimates of solutions of (1), (2) and solutions of ordinary functional differential equations. A convergence result and an error estimate of approximate solutions are presented in Section 4. Examples are given in the last part of the paper.

2 Differential difference problems

We denote by [M.sub.nxn] the class of all n x n matrices with real elements. If [union] [member of] [M.sub.nxn] then [[union].sup.T] is the transpose matrix. For x,y [member of] [R.sup.n], x = ([x.sub.1], ... , [x.sub.n]), y = ([y.sub.1], ... ,[y.sub.n]) and [union] [member of] [M.sub.kxn], [union] = [[u.sub.ij].sub.i,j]=1, ... ,n, we put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We denote by CL(B, R) the set of all linear and continuous real functions defined on C(B, R) and by [parallel] * [[parallel].sub.*] the norm in CL(B, R) generated by the maximum norm in C(B, R).

For each (t,x)[member of] E we define the sets [I.sub.0][t,x], [I.sub.-][t,x], [I.sub.+][t,x] [subset] {1, ... ,n} as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We need assumptions on [phi], f and [psi].

Assumption H [[phi]]. The functions [[phi].sub.0] : [0, a] [right arrow] R and [phi] : E [right arrow] [R.sub.n], [phi] = ([[phi].sub.1], ... , [[phi.sub.n]), are continuous and

1) 0 [less than or equal to] [[phi].sub.0] less than or equal to] t, for t [member of] [0,a] and [phi](t,x) = ([[phi].sub.0](t,x),[phi](t,x)) for (t,x) [member of] E,

2) partial derivatives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3) Q [member of] [R.sub.+] is defined by the relation [parallel][[partial derivatives].sub.x][phi](t, x)[parallel] [less than or equal to] Q on E.

Assumption H[f,[psi]]. The function f : [OMEGA] [right arrow] R of the variables (t,x,v,w,q),x = ([x.sub.1], ... ,[x.sub.n]), q = ([q.sub.1], ... ,[q.sub.n]), is continuous and satisfies the condition (V), moreover

1) the partial derivatives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the Frechet derivatives [[partial derivatives].sub.v]f (P), [[partial derivatives].sub.w]f (P) exist for P = (t, x, v, w, q) [member of] [OMEGA],

2) [[partial derivatives].sub.x]f, [[partial derivatives].sub.q]f[member of] C([OMEGA], [R.sup.n]), [[partial derivatives].sub.v]f, [[partial derivatives].sub.w]f [member of] CL(B, R),

3) the function [[partial derivatives].sub.q]f satisfies the conditions:

(i) if x [member of] [-b, b] \ [-b + Ma, b - Ma] and (t, x, v, w, q) [member of] [OMEGA] then

x [??][[partial derivatives].sub.q]f (t, x, v, w, q) [less than or equal to] [0.sub.[n]] (3)

where [0.sub.[n]] = (0, ... ,0) [member of] [R.sup.n],

(ii) if x [member of] [-b + Ma, b - Ma] then the function

sign [[partial derivatives].sub.q]f (*, x, *) : [0,a] x C(B, R) x C(B, R) x [R.sup.n] [right arrow] [R.sup.n], (4)

sign [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

is constant,

4) [psi] : [E.sub.0] [right arrow] R is of class [C.sup.2].

We define a mesh on the set [E.sub.0] [union] E with respect to the spatial variable. Let h = ([h.sub.1], ... , [h.sub.n]), [h.sub.i] > 0 for 1 [less than or equal to] i [less than or equal to] n, stand for steps of the mesh. Let us denote by H the set of all h such that there is K = ([K.sub.1], ... , [K.sub.n]) [member of] [N.sup.n] with the property K [??] h = b. For h [member of] H and for m [member of] [Z.sup.n], m = ([m.sub.1], ... , [m.sub.n]), we put

[x.sup.(m)] = m [??] h, [x.sup.(m)] = ([x.sup.(m1).sub.1], ... , [x.sup.(mn).sub.n]).

Write

[R.sup.1+n.sub.t,h] = {(t, [x.sup.(m)]) : t [member of] R, m [member of] [Z.sup.1+n]}

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Elements of the set [E.sub.0.h] [union] [E.sub.h] will be denoted by (t,[x.sup.(m)]) or (t,x). By [F.sub.c]([B.sub.h],R) we denote the class of all w : [B.sub.h] right [arrow] R such that w(-,[x.sup.(m)]) [member of] C([-[b.sub.0] - a,0],R) for -K [less than or equal to] m [less than or equal to] K. In a similarly way we define the space [F.sub.c]([B.sub.h],[R.sub.n]). For a functions z : [E.sub.0.h] [union] [E.sub.h][right arrow] R, u : [E.sub.0.h] [union] [E.sub.h] [right arrow] [R.sub.n], u = ([u.sub.1], ... , [u.sub.n]), we write [z.sup.(m)] (t) = z(t,[x.sup.(m)]), [u.sup.(m)] (t) = u(t,[x.sup.(m)]).

Suppose that Assumption H[f, [psi]] is satisfied. For [x.sup.(m)] [member of] (-b, b) we put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We construct the numerical method for (1), (2). Write [e.sub.i] = (0, ... , 0,1,0, ... ,0) [member of] [R.sup.n] with 1 standing on the i-th place. For functions z : [E.sub.0.h] [union] [E.sub.h] [right arrow] R, u : [E.sub.0.h] [union] [E.sub.h] [right arrow] [R.sup.n], u = ([u.sub.1], ... , [u.sub.n]), we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

and we put i = 1, ... , n in above definitions. Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since equation (1) contains the functional variables [Z.sub.(t,x)] and [z.sub.[phi](t,x)] which are elements of the spaces C(D[[phi]t,x],R) and C(D[p(t,x)],R) then we need an interpolating operator [T.sub.h] : [F.sub.c] ([B.sub.h],R) [right arrow] C(B,R). For a simplicity we write [T.sub.h][Z.sub.[tm]] instead of [T.sub.h][Z.sup.(t[x.sup.(m)])] and [T.sub.h][z.sub.[phi][[tm]] instead of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where z : [E.sub.0.h] [union] [E.sub.h] [right arrow] R. Let us denote

P[[z, u].sup.(m)](t) = (t, [x.sup.(m)], [T.sub.h] [z.sub.[tm]], [T.sub.h] [z.sub.[phi][t,m]], [u.sup.(m)](t)).

Write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [G.sub.h] = ([G.sub.h.1,.... ,[G.sub.h.n]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where P [member of] [OMEGA]. We consider the system of functional differential equations

d/dt [z.sup.(m)] (t) = [F.sub.h] [[z,u].sup.(m)](t), (9)

d/dt [u.sup.(m)] (t) = [G.sub.h] [[z,u].sup.(m)](t) (10)

with initial conditions

[z.sup.(m)] (t) = [psi].sup.(m).sub.h](t), [u.sup.(m)](t) = [PSI].sup.(m).sub.h] (t) on [E.sub.0.h] (11)

where [[psi].sub.h] : [E.sub.0.h] [right arrow] R and [[PSI].sub.h] : [E.sub.0.h] [right arrow] [R.sup.n], [[PSI].sub.h] = ([[PSI].sub.h.1], ... , [[PSI].sub.h.n]), are given functions. The differential difference problem (9) - (11) is called a method of lines for (1), (2). This method is obtained in the following way.

We use a method of quasilinearization for (1), (2). It means that we transform the nonlinear initial problem (1), (2) into a system of quasilinear differential equations with unknown functions (z, u) where u = [[partial derivative].sub.x]z. Suppose that Assumption H[f, [psi]] is satisfied. We consider the following linearization of equation (1) with respect to u

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where U[t,x] = (t,x,z(tx),[z.sub.[phi](tx)], u(t,x)). Differential equations for u are obtained by differentiating equation (1) with respect to the spatial variable

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

With equations (12), (13) we consider the following initial condition

z(t, x) = [psi](t, x), u(t, x) = [PSI](t, x) on (t, x)[member of] [E.sub.0]. (14)

Under natural assumptions on given functions the above problem has the properties:

(i) If ([??], [??]) is a solution of (12)-(14) then [[partial derivative].sub.x]z = [??] and z is a solution of (1), (2).

(ii) If [??] is a solution of (1), (2) and [[partial derivative].sub.x] [??] = [??] then (z, [??]) is a solution of (12)-(14).

The differential difference problem (9)-(11) is discretization with respect to the spatial variable of (12)-(14).

3 Solutions of functional differential problems

For functions z [member of] C([E.sub.0] [union] E,R), u [member of] C([E.sub.0] [union] E, [R.sup.n]) and [z.sub.h] [member of] [F.sub.c] ([E.sub.0.h] [union] [E.sub.h], R), [u.sub.h] [member of] [F.sub.c] ([E.sub.0.h] [union] [E.sub.h], [R.sup.n]) we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where t [member of] [0, a]. We need the following assumptions.

Assumption H[[T.sub.h]]. The operator [T.sub.h] : [F.sub.c]([B.sub.h],R) right arrow] C(B,R) satisfies the conditions

1) for w, [bar.w] [member of] [F.sub.c]([B.sub.h], R) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2) if w : B [right arrow] R is of class [C.sup.1] and [w.sub.h] is the restriction of w to [B.sub.h] then there is [gamma] : H [right arrow] [R.sub.+] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3) if [[theta].sub.h] [member of] [F.sub.c]([B.sub.h], R) is given by [[theta].sub.h]([tau],y) = 0 on [B.sub.h] then ([T.sub.h] [[theta].sub.h])([tau], y) = 0 for ([tau], y) [member of] B.

Example of the interpolating operator which satisfies the above assumptions can be found in [11], Chapter VI.

Assumption H[f, [??]]. The functions [phi] and f, [psi] satisfy Assumptions H[[phi]] and H[f, [[phi]], moreover

1) there is [??] [member of] C([0, a) x [R.sub.+] x [R.sub.+], [R.sub.+]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the function q is nondecreasing with respect to the last two variables,

2) the constant A [member of] [R.sub.+] is defined by the relation

[absolute value of f (t, x, [theta], [theta],[0.sub.[n]])] [less than or equal to] A, (t, x) [member of] E,

where [theta] [member of] C(B,R) and [theta]([tau],s) = 0 for ([tau],s) [member of] B,

3) there is [A.sub.0] [member of] R such that for a point P = (t, x, v, w, q) [member of] O[OMEGA] we have

[parallel][[partial derivative].sub.v] f (P)[[parallel].sub.*], [parallel][[partial derivative].sub.w] f (P)[[parallel].sub.*] [less than or equal to] [A.sub.0],

4) for P = (t, x,v,w,q) [member of] [OMEGA] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

5) for every ([mu],v) [member of] [R.sub.+] x [R.sub.+] there exists on [0, a] the maximal solution ([[omega].sub.0](; [mu],v),[omega](*,[mu],v)) of the problem

[lambda]'(t) = A + 2[A.sub.0] [lambda] (t) + 2 [parallel]M[[parallel].sub.[eta]] (t), (15)

[eta]'(t) = [??](t, [lambda](t), [eta](t)) = [A.sub.0] (1 + Q)[eta](t), (16)

([lambda](0), [eta](0)) = ([mu], v), (17)

6) [[psi].sub.h] : [E.sub.0.h] [union] [E.sub.h] right arrow R, [[PSI].sub.h] : [E.sub.0.h] [union] [E.sub.h] [right arrow] [R.sup.n] and there are [[alpha].sub.0], [alpha] : H [right arrow] [R.sub.+] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Suppose that Assumption H [f, [??]] is satisfied. Let ft, v [member of] [R.sub.+] be defined by the relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

we will assume nonlinear estimates of Perron type for [[partial derivative].sub.x]f, [[partial derivative].sub.v]f, [[partial derivative].sub.w]f, [[partial derivative].sub.q]f on subspace of [OMEGA]. Now we construct this subspace.

Suppose that Assumption H[f, [??]] is satisfied and [bar.[mu]], [bar.v] [member of] [R.sub.+] are defined by (19), (20). Let us denote by ([[omega].sub.0](*, [bar.[mu]],[bar.v]), [omega](*, [bar.[mu]],[bar.v])) the maximal solution of (15) (17) with [mu] = [bar.[mu]], v = [bar.v]. Set [bar.c] = [[omega].sub.0](a, [bar.[mu]], [bar.v]), [??] = [omega](a, [bar.[mu]],[bar.v]), C = ([bar.c], [??]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Assumption H[f, [sigma]]. The functions [phi] and f, [psi] satisfy Assumptions H[[phi]], H[f,[psi] ],

H [f, [??]] and

1) [sigma] : [0, a] x [R.sub.+] [right arrow] [R.sub.+] is continuous and it is nondecreasing with respect to the second variable,

2) for each c [greater than or equal to] 1 the maximal solution of the Cauchy problem

[omega]'(t) = c\[omega](t) + [sigma](t,[omega](t))], omega](0) = 0 (21) is [??](t) = 0 for t [member of] [0, a],

3) the expressions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 3.1. It is important that we have assumed nonlinear estimates of Perron type for [[partial derivative].sub.x]f, [[partial derivative].sub.v]f, [[partial derivative].sub.w]f, [[partial derivative].sub.q]f on [OMEGA] [C]. There are differential equations with deviated variables and differential integral equations such that condition 3) of Assumption H [f, [sigma]] is satisfied and global estimates for [[partial derivative].sub.x]f, [[partial derivative].sub.v]f, [[partial derivative].sub.w]f, [[partial derivative].sub.q]f are not satisfied. We give comments on such equations.

Set [??] = E x [R.sup.2] x [R.sup.n] and suppose that the function G :[??] [right arrow] R of the variables (t, x, p, r, q) satisfies the conditions

1) G [member of] C([??], R) and for each (t, x) [member of] E the function G(, t, x, *) : [R.sup.2] x [R.sup.n] [right arrow] R is of class [C.sup.2],

2) there is [??] [member of] [R.sub.+] such that [absolute value of [[partial derivative].sub.p]G(P)] [less than or equal to] [??],[absolute value of [[partial derivative].sub.v]G(P)] [less than or equal to] [??] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where P = (t, x, v, w, q) [member of] [??],

3) there is [??] [member of] C([0, a] x [R.sub.+] x [R.sub.+], [R.sub.+]) such that

(i) for each t [member of] [0, a] the function Q(t, *, *) is nondecreasing,

(ii) condition 5) of Assumption H [f, [??]] is satisfied and

[parallel][[partial derivative].sub.x]G(t,x,v,w,q)[[parallel].sub.[infinity]] [less than or equal to] [??] (t, max{[absolute value of v], [absolute value of w]}, [parallel]q[[parallel].sub.[infinity]]) on [??]

Consider the operator f defined by

f (t,x,v,w,q) = [member of](t, x, v(0,[0.sub.[n]]), w(0,[0.sub.[n]]), q) on [OMEGA]. (22)

Then (1) reduces to the differential equation with deviated variables

[[partial derivative].sub.t]z(t, x) = G(t, x, z(t, x), z([phi](t, x)), [[partial derivative].sub.x]z(t, x)).

Then there is L [member of] [R.sub.+] such that the operator f given by (22) satisfies Assumption H [f, [sigma]] for [sigma](t, p) = Lp, (t, p) [member of] [0,a] [member of] [R.sub.+].

Set

f(t,x,v,w,q) = G (t, x, [[integral].sub.D[t,x]] v([tau],y) dyd[tau],w (0, [0.sub.[0]]),q) on [OMEGA]. (23)

Then (1) reduces to the functional differential equation

[[partial derivative].sub.t]z(t,x) = G(t,x, [[integral].sub.D[t,x]] z(t + [tau], x + y) dyd[tau],z ([phi] (t, x)), [[partial derivative].sub.x] z (t,x)).

There is L [member of] [R.sub.+] such that the operator given by (23) satisfies Assumption H [f, [sigma]] for [sigma](t, p) = Lp, (t, p) [member of] [0,a] x [R.sub.+].

It is important in the above examples that we do not assume that the partial derivatives of the second order of [member of](t, x, *) are bounded on [??].

We give estimates of solutions of (12) - (14).

Lemma 3.1. Suppose that Assumption H [f, [??]] is satisfied and ([bar.z], [bar.u]) : [E.sub.0] [union] E [right arrow] [R.sup.1+n], [bar.u] = ([[bar.u].sub.1], ... , [[bar.u].sub.n]), are the solutions of (12) -(14) then

[parallel][bar.z][[parallel].sub.t] [less than or equal to] [[omega].sub.0] (t, [bar.[mu]], [bar.v]), [[absolute value of [bar.u]]] t [less than or equal to] [omega] (t, [bar.[mu]], [bar.v]), (24)

where ([[omega].sub.0](*, [bar.[mu],[bar.v]), [omega](*, [bar.[mu]], [bar.v],)) is the maximal solution of (15)-(17) with ([mu],v) = ([bar.[mu]], [bar.v]).

Proof. Write

[bar.[lambda]] (t) = [parallel][bar.z][[parallel].sub.t], [bar.[eta]] (t) = [[[absolute value of [bar.u]]] t, t [member of] [0,a]

Let us denote by ([[omega].sub.0](*, [bar.[mu]],[bar.v], [epsilon]) the maximal solution of the initial problem

[lambda]' (t) = A + 2[A.sub.0] [lambda] (t) + 2 [parallel]M[parallel] [eta] (t) + [epsilon], (25)

[eta]' (t) = [??](t,[lambda] (t),[mu](t)) + [A.sub.0] (1 + Q) [eta] (t) + [epsilon] (26)

([lambda](0), [eta](0)) =([bar.[mu]] + [epsilon], [bar.v] + [epsilon]) (27)

where [epsilon] > 0. There is [??] > 0 such that for 0 < [epsilon] < [??] the solution of (25)-(27) is defined on [0, a] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

uniformly on [0, a]. We prove that for 0 < [epsilon] < [??] we have

[bar.[lambda]](t) < [[omega].sub.0](t, [bar.[mu]], [bar.v], [epsilon]), [bar.[eta]] (t) < [omega](t, [bar.[mu]], [bar.v], [epsilon]) (29)

where t [member of] [0, a].

It is clear that there is [??] [member of] (0, a] such that inequalities (29) are satisfied on [0, [bar.t]). Suppose by contradiction that estimates (29) are not satisfied on [0, a]. Then there is t [member of] (0, a] such that

[bar.[lambda]]([tau]) < [[omega].sub.0]([tau], [bar.[mu]], [bar.v], epsilon]) and [bar.[eta]([tau]) < [omega]([tau], [bar.[mu]], [bar.v], [epsilon]) for [tau] [member of] [0, t)

and

[bar.[lambda]](t) = [[omega].sub.0](t, [bar.[mu]], [bar.v], [epsilon]) or [bar.[eta]](t) = [omega](t, [bar.[mu]],[bar.v], [epsilon]). Suppose that [bar.[eta]](t) = [omega](t, [bar.[mu]], [bar.v], [epsilon]). Then we have

[D.sub.-][bar.[eta]](t) [greater than or equal to] [omega]'(t, [bar.[mu]], [bar.v], [epsilon]). (30)

There are ([bar.t], x) [member of] E, [bar.t] <[less than or equal to]t, and j [member of] {1, ... ,n} such that [bar.[eta]] (t) = [absolute value of [[bar.u].sub.j] ([bar.t], x)]. Suppose that [bar.t] < t. Then [D.sub.- ][bar.[eta]](t) = 0 which contradicts (30). If [bar.t] = t then we have (i) [bar.[eta]](t) = u[bar.[eta]](t, x) or (ii) [bar.[eta]](t) = -[bar.u][bar.[eta]](t,x). Let us consider the first case. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

Let us consider the function [gamma] : [0, t] [right arrow] [R.sup.n], [gamma] = ([gamma]1, ... , [gamma]n), defined as follows:

[[gamma].sub.i] ([tau]) = [x.sub.i] for i [member of] [I.sub.0] [t, x],

[[gamma].sub.i] ([tau]) = -[b.sub.i] + [M.sub.i][tau] for i [member of] [I.sub.-] [t,x],

[[gamma].sub.i] ([tau]) = [b.sub.i] - [M.sub.i][tau] for i [member of] [I.sub.+][t,x].

Set [xi]([tau]) = [[bar.u].sub.j]([tau], [gamma]([tau])) for [tau] [member of] [0, t]. Then we have [xi]([tau]) [less than or equal to] [bar.[eta]]([tau]) for [tau] [member of] [0, t) and [xi](t) = [bar.[eta]] (t). This gives

[D.sub.-] [bar.[eta]](t) [less than or equal to] [xi]'(t)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Set

[bar.U] (t,x,[[bar.z].sub.t,x]), [[bar.z].sub.[phi](t,x)],[bar.u](t,x)).

It follows from (13) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from condition 1) - 4) Assumption H [f, [??]] and from (31)-(33) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts (30). The case (ii) can be treated in a similar way.

We can use the same reasoning for the relation [bar.[lambda]](t) = [[omega].sub.0](t, [bar.[mu],[bar.v], [epsilon]).

From (29) we obtain in the limit, letting c tend to zero, inequalities (24). This completes the proof.

Lemma 3.2. If Assumptions H[f, [sigma]] and H [[T.sub.h]] are satisfied then there exists exactly one solution ([z.sub.h], [u.sub.h]) : [E.sub.0.h] [union] [E.sub.h][right arrow] [R.sup.1+n], [u.sub.h] = ([u.sub.h.1], ... , [u.sub.h.n]), of the Cauchy problem (12)-(14) and

[parallel][z.sub.h][[parallel].sub.h.t] [less than or equal to] [[omega].sub.0] (t, [bar.[mu]], [bar.v]), [[[absolute value of [u.sub.h]]].sub.h.t] [less than or equal to] [omega] (t, [bar.[mu]], [bar.v]) (34)

where ([[omega].sub.0](t, [bar.[mu]], [bar.v]), [omega](t, [bar.[mu]], [bar.v])) is the maximal solution of (15)-(17) with ([mu], v) = ([bar.[mu], [bar.v]).

Proof. It is clear that there is [??] > 0 such that the solution ([z.sub.h], [u.sub.h]) of (12) (14) is defined on ([E.sub.0.h] [union] [E.sub.h]) [intersection]([-[b.sub.0],[??]] x [R.sup.n]). Suppose that ([z.sub.h], [u.sup.h]) is defined on ([E.sub.0.h] [union] [E.sub.h]) n ([-[b.sub.0], [??]) x [R.sup.n]), [??] > 0, and it is non continuable. For [epsilon] > 0 we denote by ([[omega].sub.0](*,[bar.[mu]],[bar.v],[epsilon]),[omega](*,[bar.[mu]],[bar.v],[epsilon])) the maximal solution of (15) - (17). There is [[epsilon].sub.0] > 0 such that for 0 < [epsilon] < [[epsilon].sub.0] the functions ([[omega].sub.0](*,[bar.[mu]],[bar.v],[epsilon]),[omega](*,[bar.[mu]],[bar.v],[epsilon)) are defined on [0, [??]) and condition (28) is satisfied. Set

[[xi].sub.h](t) = [parallel][z.sub.h][[parallel].sub.h.t], [[chi].sub.h](t) = [[[absolute value of [u.sub.h]].sub.h.t], t [member of [0, [??]).

We prove that

[[xi].sub.h](t) < [[omega].sub.0] (t, [bar.[mu]], [bar.v], [epsilon]) and [chi].sub.h] (t) < [omega] (t, [bar.[mu]], [bar.v], [epsilon])

where t [member of] [0, [??]). It is clear that there is [??] > 0 such that estimates (35) are satisfied on [0, [??]). Suppose by contradiction that (35) fails to be true on [0, [??]). Then there is t [member of] (0, [??]) such that

[[??].sub.h] ([tau]) < [[omega].sub.0] ([tau], [bar.[mu]], [[bar.v], [epsilon]) and [[chi].sub.h]([tau]) < [omega] ([tau], [bar.[mu]], [bar.v],[epsilon]) for [tau] [member of] (0,t)

and

[[??].sub.h] (t) < [[omega].sub.0] (t, [bar.[mu]], [[bar.v], [epsilon]) or [[chi].sub.h](t) < [omega] ([tau], [bar.[mu]], [bar.v],[epsilon]).

Suppose that [[chi].sub.h](t) = [omega](t, [bar.[mu]], [bar.v], [epsilon]). Then we have

[D.sub.-][[chi].sub.h](t) [greater than or equal to] [omega]'(t,bar.[mu]],[bar.v],[epsilon]). (36)

There are ([bar.t], [x.sup.(m)])[member of] [E.sub.h], [bar.t] [less than or equal to] t, and j [member of] {1, ..., n} such that [[chi].sub.h](t) = [absolute value of [u.sup.(m).sub.h.j])([bar.t])] . If [bar.t] < t then [D.sub.-][[chi].sub.h](t) = 0 which contradicts (36). Let us consider the case when [bar.t] = t. Then we have (i) [[chi].sub.h] (t) = [u.sup.(m).sub.h.j(t) or [[chi].sub.h](t) = -[u.sup.(m).sub.h.j] (t). We consider the first case. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from condition 3) of Assumption H[f, [psi]] and from (7), (8) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We thus get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts (36). The case (ii) can be treated in a similar way.

A similar proof remains valid for case [[xi].sub.h](t) = [[omega].sub.0](t,[bar.[mu]], [bar.v], [epsilon]). The inequalities (35) are satisfied on [0, [??]). From (35) we obtain in the limit, letting c tend to zero, that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

Suppose that (t,[x.sup.(m)]), (t,[x.sup.(m)]) [member of] [E.sub.h], t, [bar.t] [member of] (0,[bar.a]). It follows from Assumptions H[[T.sub.h] ], H[f, [??] and from (37) that there are [[??].sub.h.0][[??].sub.h] [member of] C([0,[bar.a]], R) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then there are the limits

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the solution ([z.sub.h], [u.sub.h]) is defined on ([E.sub.0.h] [union] [E.sub.h]) [intersection] ([-[b.sub.0], [bar.a]] x [R.sup.n]). This contradicts our assumption that ([z.sub.h], [u.sub.h]) is defined on ([E.sub.0.h] [union] [E.sub.h]) n ([-[b.sub.0], [??]), [R.sup.n]) and it is non continuable.

If [??] < a then there is [epsilon] > 0 such that the solution ([z.sub.h], [u.sub.h]) exists on ([E.sub.0. h] [union] [E.sub.h]) [intersection] ([-[b.sub.0], [??] + [epsilon]) x [R.sup.n]) and inequalities (37) are satisfied for t [member of] [0, [??] + epsilon]). It follows from the above considerations that ([z.sub.h], [u.sub.h]) is defined on [E.sub.0.h] [union] [E.sub.h] and estimates (34) are satisfied.

Suppose that ([z.sub.h], [u.sub.h]) and ([[??].sub.h], [[??].sub.h]) are solutions of (12) - (14). Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows from condition 3) of Assumption H [f, [sigma]] and from (38), (39) that there is [c.sub.h] [greater than or equal to] 1 such that the function [bar.[omega].sub.h] satisfies the differential inequality

[D.sub.-][bar.[omega].sub.h](t) [less than or equal to] [c.sub.h] [[bar.[omega].sub.h] (t) + [sigma](t, [bar.[omega].sub.h](t))], t [member of] (0,a],

and [bar.[omega].sub.h](0) = 0. It follows from condition 2) of Assumption H[f, [sigma]] that [bar.[omega].sub.h](t) = 0 for t [member of] (0,a].Then ([z.sub.h], [u.sub.h]) = ([z.sub.h], [u.sub.h]). This completes the proof of the lemma.

4 Convergence of the method of lines

Now we formulate the main result of the paper.

Theorem 4.1. Suppose that Assumptions H[f, [sigma]] and H[[T.sub.h]] are satisfied and

1) [bar.z] : [E.sub.0] [union] E [right arrow] R is a solution of(1), (2) and [bar.z] is of class [C.sup.2],

2) [bar.u] = [[partial derivative].sub.x][bar.z] and ([z.sub.h], [u.sub.h]) is the restriction of (z, u) to [E.sub.0.h] [union] [E.sub.h].

Then there is exactly one solution ([z.sub.h], [u.sub.h]) : [E.sub.0.h] [union] [E.sub.h] [right arrow] [R.sup.1+n] of (12) - (14) and there is [??] : H [right arrow] [R.sub.+] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

Proof. The existence and uniqueness of a solution of (12) - (14) follows from Lemma 3.2. Let [[GAMMA].sub.h.0] : [E.sub.0.h] [right arrow] R, [[GAMMA].sub.h] : [E.sub.h] [right arrow] [R.sup.n] be defined by the relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

There are [[gamma].sub.0], [gamma] : H [right arrow] [R.sub.+] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [c.sup.*] [member of] [R.sub.+] be defined by the relation

[c.sup.*] = max {[parallel][[partial derivative].sub.xx][bar.z] (t,x) [[parallel].sub.nxn] : (t, x) [member of] E}

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] It follows from Lemma 3.1 and 3.2 and from Assumption H[[T.sub.h]] that for (t,[x.sup.(m)]) [member of] [E.sub.h] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (42)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

For functions [zeta], [chi] : [0, a][right arrow] [R.sub.+] we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [bar.a] = 1 + [??](1 + Q) + [c.sup.*]. Let us denote by ([[omega].sub.h.0](*, [epsilon]),[[omega].sub.h](*, [epsilon])) the maximal solution of the Cauchy problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

[zeta](0)= [[alpha].sub.0](h)+ [epsilon], x(0)= [alpha](h) + [epsilon] (46)

where [[alpha].sub.0], [alpha] : H [right arrow] [R.sup.+] are given by (18). It follows from condition 2) of Assumption H[f, [sigma]] that there is [[epsilon].sub.0] > 0 such that for 0 < [epsilon] < [[epsilon].sub.0] the functions ([[omega].sub.h.0](*, [epsilon]),[epsilon])) are defined on [0,a] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where ([[omega].sub.h.0],[[omega].sub.h]) is the maximal solution (45), (46) with [epsilon] = 0. Write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We prove that for each 0 < [epsilon] < [[epsilon].sub.0] we have

[[lambda].sub.h](t) < [[omega].sub.h.0](t,[epsilon]) and [xi](t) < [[omega].sub.h](t,[epsilon]) (47)

where t [member of] [0,a]. It is clear that is [??] [member of] (0,a] such that inequalities (47) are satisfied on [0, [??]). Suppose by contradiction that (47) fails to be true on [0, a]. Then there is t [member of] (0, a] such that

[[gamma].sub.h]([tau]) < [[omega].sub.h.0] (t,[epsilon]) and [[??].sub.h([tau]) < [[omega].sub.h] ([tau], [epsilon]) for [tau] [member of] [0, t)

and

[[lambda].sub.h] (t) = [[omega].sub.h.0] (t,[epsilon]) or [[xi].sub.h](t) < [[omega].sub.h] (t, [epsilon]).

Suppose that [[lambda].sub.h](t) = [[omega].sub.h.0](t, [epsilon]). Then we have

[D.sub.-][[lambda].sub.h](t) [greater than or equal to] [omega]'.sub.h.0](t,[epsilon]). (48)

There is ([bar.t], [x.sup.(m)]) [member of] [E.sub.h], [bar.t] [less than or equal to] t, such that [[lambda].sub.h](t) = [absolute value of [z.sup.(m).sub.h] (t)-[[bar.z].sup.(m).sub.h]]. If [bar.t] < t then [D.sub.-][[lambda].sub.h](t) = 0 which contradicts (48). Suppose that [bar.t] = t. Then we have (i) [[lambda].sub.h](t) = [z.sup.(m).sub.h](t)-[[bar.z].sup.(m).sub.h](t) or (ii) [[lambda].sub.h](t) = -[[z.sup.(m).sub.h(t) - [[bar.z].sup.(m).sub.h](t)]. Let us consider the case (i). It follows from condition 3) of Assumption H[f, [sigma]] and from (42) - (44) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from conditions 3), 4) of Assumption H[f, [psi]] and from (5), (6) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts (48).The case (ii) can be treated in a similar way. Suppose that [[xi].sub.h](t) = [[omega].sub.h](t, [epsilon]). Then we have

[D.sub.-][[xi].sub.h](t) [greater than or equal to] [[omega].sup.'sub.h](t,[epsilon]). (49)

There are ([bar.t],[x.sup.(m)])[member of] [E.sub.h],[bar.t] [less than or equal to] t and j [member of] {1, ... ,n} such that [[xi].sub.h](t) = [absolute value of [u.sup.(m).sub.h.j]([bar.t]) -[[bar.u].sup.(m).sub.h]([bar.t])]. Suppose that [bar.t] < t. Then [D.sub.-] [[xi].sub.h](t) = 0 which contradicts (49). Suppose that [bar.t] = t. Then we have (i) [[xi].sub.h](t) = [u.sup.(m).sub.h.j](t)[bar.u].sup.(m).sub.j](t) or (ii) [[xi].sub.h](t) = -[[u.sup.(m).sub.h.j](t)-[[bar.u].sup.(m).sub.h.j(t)]. Let us consider the case (i). We deduce from condition 3) of Assumption H[f, [sigma]] and from (42) - (44) that the expressions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

may be estimated by [sigma](t, [[omega].sub.h.0](t, epsilon]) + [[omega].sub.h](t, [epsilon])). Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from condition 3) of Assumption H[f, p ] and from (7), (8) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts (49). The case (ii) can be treated in a similar way. Then inequalities (47) are satisfied on [0, a]. From (47) we obtain in the limit, letting [epsilon] tend to zero, the estimates

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

where ([[omega].sub.h.0], [[omega].sub.h]) is the maximal solution of (45), (46) with [epsilon] = 0. Let us denote by [[omega].sub.h] the maximal solution of the Cauchy problem

[omega]' (t) = d[omega] (t) + ([bar.a] + 2[??]) [sigma] (t,[omega] (t)) + [[gamma].sub.0] (h) + [gamma] (h), (51)

[omega](0) = [[alpha].sub.0](h) + [alpha](h) (52)

where d = max {[A.sub.0](3 + Q),2[parallel]M[parallel]}. We conclude from (50) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows that conditions (40), (41) are satisfied for [??](h) = [[bar.[omega]].sub.h](a). This completes the proof of the theorem.

Remark 4.1. Suppose that all the assumptions of Theorem 4.1 are satisfied with [sigma](t, p) = Lp, (t, p) [member of] [0, a] [member of] [R.sub.+], where L [member of] [R.sub.+] Then we have the following error estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [??] = d + ([bar.a] + 2[??]) L. The above inequality is obtained by solving problem (51), (52) with [sigma](t, p) = Lp.

Remark 4.2. It is assumed in [11] that the right hand sides of functional differential equations satisfy global estimates of Perron type. It follows from Theorem 4.1 that local estimates are sufficient for the convergence of the method of lines.

5 Numerical Examples

Example 5.1. Put n = 2 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consider the differential integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](53)

with the initial condition

z(0,x,y) = 1 for (x,y) [member of] [-2,2] x [-2,2] (54)

where

f (t, x, y) = [e.sup.-ty] - [e.sup.tx] + [(x - y).sup.et] (x-y).

The solution of the above problem is known, it is [bar.z](t,x,y) = [e.sup.t(x-y)]. Let us denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] approximate solutions of ordinary functional differential equations corresponding to (53), (54). They are obtained by using the explicit Euler difference method. Nodal points on [0,1] are obtained by [t.sup.(r)] = [rh.sub.0],r = 0,1, ... , [N.sub.0].

Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (55)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (56)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57)

where 0 [less than or equal to] r [less than or equal to] [N.sub.0]. Let us denote by [[??].sub.h] an approximate solution of (53), (54) which is obtained by using the Lax difference scheme. Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (58)

where 0 [less than or equal to] r [less than or equal to] [N.sub.0]. In the Table 1 we give experimental values of the errors ([[epsilon].sub.h], [[epsilon].sub.h.x], [[epsilon].sub.h.y]) and [[??].sub.h] for [h.sub.0] = 0.001, [h.sub.1] = [h.sub.2] = 0.05.

Note that errors of the classical difference method [[??].sup.(r).sub.h] are larger then the errors obtained by discretization of the numerical method of lines [[epsilon].sup.(r).sub.h] . This is due to the fact that Lax difference scheme has the following property: we approximate partial derivatives of z with respect to spatial variables by difference expressions which are calculated by using previous values of the approximate solutions. In our approach we approximate the partial derivatives [[partial derivative].sub.x]z and [[partial derivative].sub.y]z by using difference equations which are generated by the original problem.

Example 5.2. For n = 2 we put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consider the differential equation with deviated variables

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (59)

with the initial condition

z(0, x,y) = 1 for (x, y) [member of] [-2.5,2.5] x [-2.5,2.5] (60)

where

f (t, x,y) = xy (1 + 2t) exp {txy} -1 -exp {t/4 ([x.sup.2] - [y.sup.2]} sin exp {t/4 xy}.

The solution of the above problem is [bar.z](t, x, y) = [e.sup.txy] . Let us denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] approximate solutions of ordinary functional differential equations corresponding to (59), (60). They are obtained by using the implicit Euler method. Let ([[epsilon].sub.h], [[epsilon].sub.h.x], [[epsilon].sub.h.y]) be defined by (55)-(57).

Let us denote by [[??].sub.h] an approximate solution of (53), (54) which is obtained by using the Lax difference scheme. Denote by [[??].sub.h] errors of the method given by (58). In the Table 2 we give experimental values of the above defined errors for

[h.sub.0] = 0.01, [h.sub.1] = [h.sub.2] = 0.01.

In theorems on the convergence of explicit difference schemes for (1), (2) we need assumptions on the mesh. They are called the (CFL) condition.

The (CFL) condition for (59) and for the Lax difference method has the form

[h.sub.0] [less than or equal to] 0.1[h.sub.i], i = 1, 2.

Note that the steps [h.sub.0] = 0.01, [h.sub.1] = [h.sub.2] = 0.01 do not satisfy the above condition and classical Lax difference scheme is not applicable.

Remark 5.1. The result presented in the paper can be extended on weakly coupled functional differential systems

[[partial derivative].sub.t] [z.sub.i] (t, x) = [f.sub.i] (t, x, [z.sub.(t,x)] , [z.sub.[phi](t,x)], [[partial derivative].sub.x] [z.sub.i] (t,x), i = 1, ... ,k,

with the initial condition

z(t,x) = [psi](t,x), (t, x)[member of] [E.sub.0],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are given functions.

Received by the editors in July 2012-In revised form in December 2012.

Communicated by P. Godin.

References

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Department of Mathematical and Numerical Analysis

Gabriela Narutowicza Street 11-12

80-952 Gdansk, Poland

email: annak@mif.pg.gda.pl
```Table 1

[[epsilon].sup    [[epsilon].sup
[t.sup.(r)]     .(r).sub.h]     .(r).sub.h x x]

0.5              0.000935          0.000656
0.6              0.001491          0.000998
0.7              0.002155          0.001512
0.8              0.002915          0.002222
0.9              0.003755          0.003151
1.0              0.004657          0.004316

[[epsilon].sup       [[??].sup
[t.sup.(r)]   .(r).sub.h x y]     .(r).sub.h]

0.5              0.000609          0.143086
0.6              0.000865          0.184319
0.7              0.001218          0.228142
0.8              0.001668          0.272560
0.9              0.002213          0.315301
1.0              0.002849          0.354176

Table 2

[[epsilon].sup    [[epsilon].sup
[t.sup.(r)]     .(r).sub.h]     .(r).sub.h x x]

0.25             0.004843          0.004770
0.30             0.005078          0.005130
0.35             0.005109          0.005279
0.40             0.004971          0.005238
0.45             0.004702          0.005038
0.50             0.004341          0.004716

[[epsilon].sup       [[??].sup
[t.sup.(r)]   .(r).sub.h x y]     .(r).sub.h]

0.25             0.004637          0.296195
0.30             0.004961          0.417914
0.35             0.005082          6.239350
0.40             0.005019          257.9430
0.45             0.004807          5300.330
0.50             0.004483          46673.60
```