# Implicit Difference Inequalities  Corresponding to First-Order Partial  Differential  Functional Equations

Authors: Z. Kamont; K. Kropielnicka
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2009)

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## Abstract

We give a theorem on implicit difference functional inequalities generated
by mixed problems for nonlinear systems of first-order partial
differential functional equations. We apply this result in the investigations
of the stability of difference methods. Classical solutions of mixed
problems are approximated in the paper by solutions of suitable implicit
difference schemes. The proof of the convergence of difference method
is based on comparison technique, and the result on difference functional
inequalities is used. Numerical examples are presented.


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## Body

## 1. Introduction

The papers [1, 2] initiated the theory of difference inequalities
generated by first-order partial differential equations. The results and the
methods presented in [1, 2] were extended in [3, 4] on functional differential   problems, and they were
generalized in [5–8] on parabolic differential
and differential functional equations. Explicit difference schemes were  considered in the above papers.Our purpose is to give a result on implicit difference
inequalities corresponding to initial boundary value problems for first-order
functional differential equations.We prove also that that there are implicit difference
methods which are convergent. The proof of the convergence is based on a
theorem on difference functional inequalities.We formulate our functional differential problems. For
any metric spacesX and Y we denote by C(X,Y) the class of all continuous functions from X into Y. We will use vectorial inequalities with the
understanding that the same inequalities hold between their corresponding
components. Write(1.1)E=[0,a]×(−b,b),D=[−d0,0]×[−d,d],where a>0, b=(b1,…,bn)∈ℝn, bi>0 for 1≤i≤n  and  d=(d1,…,dn)∈ℝ+n,  d0∈ℝ+, ℝ+=[0,+∞).  Let c=b+d and(1.2)E0=[−d0,0]×[−c,c],∂0E=[0,a]×([−c,c]∖(−b,b)),Ω=E∪E0∪∂0E.For a function z:Ω→ℝk, z=(z1,…,zk),
and for a point (t,x)∈E¯  where E¯ is the closure of E,
we define a function z(t,x):D→ℝk by z(t,x)(τ,y)=z(t+τ,x+y), (τ,y)∈D. Then z(t,x) is the restriction of z to the set [t−d0,t]×[x−d,x+d] and this restriction is   shifted to the set D. Write Σ=E×C(D,ℝk)×ℝn and suppose that f=(f1,…,fk):Σ→ℝk and φ:E0∪∂0E→ℝk, φ=(φ1,…,φk),
are given functions. Let us denote by z=(z1,…,zk) an unknown function of the variables (t,x), x=(x1,…,xn). Write(1.3)𝔽[z](t,x)=(f1(t,x,z(t,x),∂xz1(t,x)),…,fk(t,x,z(t,x),∂xzk(t,x)))and ∂xzi=(∂x1zi,…,∂xnzi), 1≤i≤k.  We consider the system of functional
differential equations(1.4)∂tz(t,x)=𝔽[z](t,x)with the initial boundary
condition(1.5)z(t,x)=φ(t,x)onE0∪∂0E.In the paper we consider
classical solutions of (1.4), (1.5).We give examples of equations which can be obtained
from (1.4) by specializing the operator f.Example 1.1.
Suppose that the functionα:E→ℝ1+n satisfies the condition: α(t,x)−(t,x)∈D for (t,x)∈E. For a given f˜=(f˜1,…,f˜k):E×ℝk×ℝk×ℝn→ℝk we put (1.6)f(t,x,w,q)=f˜(t,x,w(0,θ),w(α(t,x)−(t,x)),q)onΣ,where θ=(0,…,0)∈ℝn.
Then (1.4)   is reduced to the system of differential equations with deviated
variables(1.7)∂tzi(t,x)=f˜i(t,x,z(t,x),z(α(t,x)),∂xzi(t,x)),i=1,…,k.Example 1.2.
For the abovef˜ we define (1.8)f(t,x,w,q)=f˜(t,x,w(0,θ),∫Dw(τ,y)dydτ,q)onΣ.Then (1.4)  is equivalent to the
system of differential integral equations(1.9)∂tzi(t,x)=f˜i(t,x,z(t,x),∫Dz(t+τ,x+y)dydτ,∂xzi(t,x)),i=1,…,k.It is clear
that more complicated differential systems with deviated variables and
differential integral problems can be obtained from (1.4) by a suitable definition
of f. Sufficient conditions for the existence and
uniqueness of classical or generalized solutions of (1.4), (1.5) can be found in
[9, 10].Our motivations for investigations of implicit
difference functional inequalities and for the construction of implicit
difference schemes are the following. Two types of assumptions are needed in
theorems on the stability of difference functional equations generated by (1.4),
(1.5). The first type conditions concern regularity of f.
It is assumed that(i)
the functionf of the variables (t,x,w,q), q=(q1,…,qn),
is of class C1 with respect to q and the functions ∂qfi=(∂q1fi,…,∂qnfi), 1≤i≤k, are bounded,(ii)
f satisfies the Perron type estimates with
respect to the functional variable w.      The second type
conditions concern the mesh. It is required that difference schemes generated
by (1.4),
(1.5)  satisfy the condition(1.10)1−h0∑j=1n1hj|∂qjfi(t,x,w,q)|≥0onΣfori=1,…,k,where h0 and h′=(h1,…,hn) are steps of the mesh with respect to t and (x1,…,xn) respectively. The above assumption is known as
a generalized Courant-Friedrichs-Levy (CFL) condition for (1.4),
(1.5) (see
[11, Chapter 3]  and
[10, Chapter 5]). It is
clear that strong assumptions on relations between h0 and h′ are required in (1.10). It is important in our
considerations that assumption (1.10) is omitted in a theorem on difference
inequalities and in a theorem on the convergence of difference schemes.We show that there are implicit difference methods for
(1.4),
(1.5) which are convergent while the corresponding explicit difference
schemes are not convergent. We give suitable numerical examples.The paper is organized as follows. A theorem on
implicit difference functional inequalities with unknown function of several
variables is proved in Section2. We propose in Section 3 implicit difference
schemes for the numerical solving of functional differential equations.
Convergence results and error estimates are presented. A theorem on difference
inequalities is used in the investigation of the stability of implicit
difference methods. Numerical examples are given in the last part of the paper.We use in the paper general ideas for finite
difference equations which were introduced in [12–14]. For further bibliographic informations concerning
differential and functional differential inequalities and applications see the
survey paper [15] and
the monographs [16, 17].


## 2. Functional Difference Inequalities

For any two
setsU and W we denote by F(U,W) the class of all functions defined on U and taking values in W.
Let ℕ and ℤ be the sets of natural numbers and integers,
respectively. For x=(x1,…,xn)∈ℝn, p=(p1,…,pk)∈ℝk we put(2.1)∥x∥=|x1|+⋯+|xn|,∥p∥∞=max{|pi|:1≤i≤k}.We define a mesh on Ω in the following way.  Suppose that (h0,h′), h′=(h1,…,hn), stand for steps of the mesh. For (r,m)∈ℤ1+n where m=(m1,…,mn),
we define nodal points as follows:(2.2)t(r)=rh0,x(m)=(x1(m1),…,xn(mn))=(m1h1,…,mnhn).Let us denote by H the set of all h=(h0,h′) such that there are K0∈ℤ and K=(K1,…,Kn)∈ℤn satisfying the conditions: K0h0=d0 and (K1h1,…,Knhn)=d. Set(2.3)ℝh1+n={(t(r),x(m)):(r,m)∈ℤ1+n},Dh=D∩ℝh1+n,Eh=E∩ℝh1+n,E0.h=E0∩ℝh1+n,∂0Eh=∂0E∩ℝh1+n,Ωh=Eh∪E0.h∪∂0Eh.Let N0∈ℕ be defined by the relations: N0h0≤a<(N0+1)h0 and(2.4)Eh′={(t(r),x(m))∈Eh:0≤r≤N0−1}.For functions w:Dh→ℝk and z:Ωh→ℝk we write w(r,m)=w(t(r),x(m)) on Dh and z(r,m)=z(t(r),x(m)) on Ωh.
We need a discrete version of the operator (t,x)→z(t,x). For a function z:Ωh→ℝk and for a point (t(r),x(m))∈Eh we define a function z[r,m]:Dh→ℝk by(2.5)z[r,m](τ,y)=z(t(r)+τ,x(m)+y),(τ,y)∈Dh.Solutions of difference
equations corresponding to (1.4),
(1.5) are functions defined on the mesh. On the
other hand (1.4)  contains the functional variable z(t,x) which is an element of the space C(D,ℝk). Then we need an interpolating operator Th:F(Dh,ℝk)→C(D,ℝk). We define Th in the following way. Let us denote by (ϑ1,…,ϑn) the family of sets defined by(2.6)ϑi={0,1}ifdi>0,ϑi={0}ifdi=0,1≤i≤n.Set υ=(υ1,…,υn)∈ℤn and υi=0 if di=0, υi=1 if di>0 where 1≤i≤n. Write(2.7)Δ+={λ=(λ1,…,λn):λi∈ϑifor1≤i≤n}.Set ei=(0,…,0,1,0,…,0)∈ℝn with 1 standing on the ith place.Letw∈F(Dh,ℝk) and (t,x)∈D. Suppose that d0>0. There exists (t(r),x(m))∈Dh such    that (t(r+1),x(m+υ))∈Dh and t(r)≤t≤t(r+1), x(m)≤x≤x(m+υ). Write(2.8)Th[w](t,x)=(1−t−t(r)h0)∑λ∈Δ+w(r,m+λ)(x−x(m)h′)λ(1−x−x(m)h′)1−λ+t−t(r)h0∑λ∈Δ+w(r+1,m+λ)(x−x(m)h′)λ(1−x−x(m)h′)1−λ, where(2.9)(x−x(m)h′)λ=∏i=1n(xi−xi(mi)hi)λi,(1−x−x(m)h′)1−λ=∏i=1n(1−xi−xi(mi)hi)1−λi and we take 00=1 in the above formulas. If d0=0 then we put(2.10)Th[w](t,x)=∑λ∈Δ+w(r,m+λ)(x−x(m)h′)λ(1−x−x(m)h′)1−λ.Then we have defined Th[w] on D. It is easy to see that Th[w]∈C(D,ℝk). The above interpolating operator has been
first proposed in [10, Chapter 5].Forw,w¯∈F(Dh,ℝk) we write w≤w¯ if w(r,m)≤w¯(r,m) where (t(r),x(m))∈Dh. In a similar way we define the relation w≤w¯ for w,w¯∈C(D,ℝk) and the relation z≤z¯ for z,z¯∈F(Ωh,ℝk) and for z,z¯∈C(Ω,ℝk).We formulate an implicit difference scheme for (1.4),
(1.5). For x,y∈ℝn we write x⋄y=(x1y1,…,xnyn)∈ℝn.Assumption (H[f]).
The functionf=(f1,…,fk):Σ→ℝk of the variables (t,x,w,q), q=(q1,…,qn), is continuous and
(1)
the partial derivatives(∂q1fi,…,∂qnfi)=∂qfi, i=1,…,k,  exist on Σ  and the functions ∂qfi, i=1,…,k,  are continuous and bounded on  Σ,(2)
there isx˜∈(−b,b), x˜=(x˜1,…,x˜n), such that(2.11)(x−x˜)⋄∂qfi(t,x,w,q)≥θonΣfori=1,…,k,(3)
there isε0>0 such that for 0<h0<ε0 and w,w¯∈C(D,ℝk), w≤w¯,  we have(2.12)w(0,θ)+h0f(t,x,w,q)≤w¯(0,θ)+h0f(t,x,w¯,q),(t,x,q)∈E×ℝn.Remark 2.1.
The
existence theory of classical or generalized solutions to (1.4),
(1.5) is based on
a method of bicharacteristics. Suppose that z∈C(Ω,ℝk), u∈C(Ω,ℝn). Let us denote by (2.13)gi[z,u](⋅,t,x)=(gi.1[z,u](⋅,t,x),…,gi.n[z,u](⋅,t,x))the ith bicharacteristic of (1.4)
corresponding to (z,u). Then gi[z,u](⋅,t,x) is a solution of the Cauchy problem(2.14)y′(τ)=−∂qfi(τ,y(τ),z(τ,y(τ)),u(τ,y(τ))),y(t)=x.Assumption  (2.11)  states that the
bicharacteristics satisfy the following monotonicity conditions: If xj−x˜j≥0 the function gij[z,u](⋅,t,x) is non increasing. If xj−x˜j<0 then gij[z,u](⋅,t,x) is nondecreasing.
The same property of bicharacteristics is needed in a
theorem on the existence and uniqueness of solutions to (1.4),
(1.5) see [9]. It is important that our
theory of difference methods is consistent with known theorems on the existence
of solutions to (1.4),
(1.5).Remark 2.2.
Given the functionf˜=(f˜1,…,f˜k):E×ℝ×C(D,ℝk)×ℝn→ℝk of the variables (t,x,p,w,q). Write fi(t,x,w,q)=f˜i(t,x,wi(0,θ),w,q), i=1,…,k, on Σ. Then system (1.4) is equivalent
to (2.15)∂tzi(t,x)=f˜i(t,x,zi(t,x),z(t,x),∂xzi(t,x)),i=1,…,k.Note that the dependence of f˜ on the classical variable z(t,x) is distinguished in (2.15). Suppose that
(1)
f˜ is nondecreasing with respect to the
functional variable,(2)
there exists the derivative∂pf˜=(∂pf˜1,…,∂pf˜k) and ∂pf˜i(t,x,p,w,q)≥L   for i=1,…,k and 1+Lh0≥0.  Then the
monotonicity condition (3) of Assumption (H[f])  is satisfied.Let us denote byH⋆ the set of all h=(h0,h′)∈H such that(2.16)hi<min{bi−x˜i,x˜i+bi},i=1,…,n.Suppose that ω:Ωh→ℝ. We apply difference operators δ=(δ1,…,δn) given by(2.17)ifx˜j≤xj(mj)<bjthenδjω(r,m)=1hj[ω(r,m+ej)−ω(r,m)],if−bj<xj(mj)<x˜jthenδjω(r,m)=1hj[ω(r,m)−ω(r,m−ej)],and we put j=1,…,n in (2.17). Let δ0 be defined by(2.18)δ0ω(r,m)=1h0[ω(r+1,m)−ω(r,m)]and δ0z=(δ0z1,…,δ0zk). Write(2.19)𝔽h[z](r,m)=(f1(t(r),x(m),Thz[r,m],δz1(r+1,m)),…,fk(t(r),x(m),Thz[r,m],δzk(r+1,m))). Given φh:E0.h∪∂0Eh→ℝk, we consider the functional difference
equation(2.20)δ0z(r,m)=𝔽h[z](r,m)with the initial boundary
condition(2.21)z(r,m)=φh(r,m)onE0.h∪∂0Eh.The above problem is considered as an implicit
difference method for (1.4), (1.5). It is important that the difference expressions (δ1zi,…,δnzi), 1≤i≤k, are calculated at the point (t(r+1),x(m)) and the functional variable Thz[r,m] appears in a classical sense.We prove a theorem on implicit difference inequalities
corresponding to (2.20), (2.21). Note that results on implicit difference methods
presented in [18] are
not applicable to (2.20), (2.21).Theorem 2.3.
Suppose that Assumption(H[f])  is satisfied and
(1)
h∈H⋆, h0<ε0 and the functions u,v:Ωh→ℝk satisfy the difference functional
inequality(2.22)δ0u(r,m)−𝔽h[u](r,m)≤δ0v(r,m)−𝔽h[v](r,m)onEh′,(2)
the initial boundary estimateu(r,m)≤v(r,m) holds on E0.h∪∂0Eh.     Then(2.23)u(r,m)≤v(r,m)onEh.Proof.
We prove
(2.23) by induction on r.
It follows from  assumption (2) that estimate (2.23) is satisfied for r=0 and (t(0),x(m))∈Eh. Assume that u(j,m)≤v(j,m) for (t(j),x(m))∈Eh∩([0,t(r)]×ℝn). We prove that u(r+1,m)≤v(r+1,m) for (t(r+1,m),x(m))∈Eh. Write(2.24)Ui(r,m)=ui(r,m)+h0fi(t(r),x(m),Thu[r,m],δui(r+1,m))−vi(r,m)−h0fi(t(r),x(m),Thv[r,m],δui(r+1,m)),i=1,…,k. It follows from (2.22) that(2.25)(ui−vi)(r+1,m)≤Ui(r,m)+h0[fi(t(r),x(m),Thv[r,m],δui(r+1,m))−fi(t(r),x(m),Thv[r,m],δvi(r+1,m))], where i=1,…,k. The monotonicity condition (3) of Assumption (H[f])  implies the inequalities Ui(r,m)≤0 for (t(r),x(m))∈Eh, i=1,…,k. Then we have(2.26)(ui−vi)(r+1,m)≤h0∑j=1n∫01∂qjfi(Qi(r,m)[v,τ])dτδj(ui−vi)(r+1,m),where i=1,…,k and(2.27)Qi(r,m)[v,τ]=(t(r),x(m),Thv[r,m],δvi(r+1,m)+τδ(ui−vi)(r+1,m)).Write(2.28)Γ+(m)={j∈{1,…,n}:xj(mj)∈[x˜j,bj)},Γ−(m)={1,…,n}∖Γ+(m).It follows from (2.11), (2.17)
that(2.29)(ui−vi)(r+1,m)[1+h0∑j=1n1hj∫01|∂qjfi(Qi(r,m)[v,τ])|dτ]≤h0∑j∈Γ+(m)1hj∫01∂qjfi(Qi(r,m)[v,τ])dτ(ui−vi)(r+1,m+ej)−h0∑j∈Γ−(m)1hj∫01∂qjfi(Qi(r,m)[v,τ])dτ(ui−vi)(r+1,m−ej),i=1,…,k. We define m˜∈ℤn and μ∈ℕ, 1≤μ≤k, as follows:(2.30)(uμ−vμ)(r+1,m˜)=max1≤i≤kmax{(ui−vi)(r+1,m):(t(r+1),x(m))∈Ωh}.If (t(r+1),x(m˜))∈∂0Eh then assumption (2) implies that (uμ−vμ)(r+1,m˜)≤0. Let us consider the case when (t(r+1),x(m˜))∈Eh. Then we have from (2.29) that(2.31)(uμ−vμ)(r+1,m˜)[1+h0∑j=1n1hj∫01|∂qjfi(Qi(r,m˜)[v,τ])|dτ]≤h0(uμ−vμ)(r+1,m˜)[∑j∈Γ+(m˜)1hj∫01∂qjfi(Qi(r,m˜)[v,τ])dτ−∑j∈Γ−(m˜)1hj∫01∂qjfi(Qi(r,m˜)[v,τ])dτ]. It follows that (uμ−vμ)(r+1,m˜)≤0.
The the proof of (2.23) is completed by induction.


## 3. Implicit Difference Schemes

We defineN=(N1,…,Nn)∈Nn by the relations:(3.1)(N1h1,…,Nnhn)<(b1,…,bn)≤((N1+1)h1,…,(Nn+1)hn)and we assume that (Ni+1)hi=bi if di=0.
For w∈C(D,ℝk) we write(3.2)∥w∥D=max{∥w(t,x)∥∞:(t,x)∈D}.In a similar way we define the
norm in the space F(Dh,ℝk) :
if w:Dh→ℝk then(3.3)∥w∥Dh=max{∥w(r,m)∥∞:(t(r),x(m))∈Dh}.The following properties of the
operator Th are important in our considerations.Lemma 3.1.
Suppose
thatw:D→ℝk is of class C1 and wh is the restriction of w to the set Dh.
Let C˜ be such a constant that ∥∂tw∥D, ∥∂xiw∥D≤C˜ for 1≤i≤n. Then ∥Th[wh]−w∥D≤C˜∥h∥ where ∥h∥=h0+h1+⋯+hn.Lemma 3.2.
Suppose
thatw:D→ℝk is of class C2 and wh is the restriction of w to the set Dh.
Let C˜ be such a constant that ∥∂ttw∥D, ∥∂txiw∥D, ∥∂xixjw∥D≤C˜,   i,j=1,…,n. Then ∥Th[wh]−w∥D≤C˜∥h∥2.The above lemmas
are consequences of  [10, Lemma 3.19 and Theorem 5.27].We first prove a theorem on the existence and
uniqueness of solutions to (2.20), (2.21).Theorem 3.3.
If
Assumption(H[f])  is satisfied and φh∈F(E0.h∪∂0Eh,ℝk) then there exists exactly one solution uh=(uh.1,…,uh.k):Ωh→ℝk of difference functional problem (2.20), (2.21).Proof.
Suppose
that0≤r≤N0−1 is fixed and that the solution uh of problem (2.20), (2.21) is given on the set Ωh∩([−d0,t(r)]×ℝn).
We prove that the vectors uh(r+1,m),  −N≤m≤N,
exist and that they are unique. It is sufficient to show that there exists
exactly one solution of the system of equations(3.4)1h0(zi(r+1,m)−uh.i(r,m))=fi(t(r),x(m),T(uh)[r,m],δzi(r+1,m)),where −N≤m≤N,i=1,…,k, with the initial boundary condition (2.21). There
exists Qh>0 such that(3.5)Qh≥h0[∑j∈Γ+(m)1hj∂qjfi(t(r),x(m),Th(uh)[r,m],q)−∑j∈Γ−(m)1hj∂qjfi(t(r),x(m),Th(uh)[r,m],q)], where −N≤m≤N,i=1,…,k. It is clear that system (3.4) is equivalent to
the following one:(3.6)zi(r+1,m)=1Qh+1[Qhzi(r+1,m)+uh.i(r,m)+h0fi(t(r),x(m),Th(uh)[r,m],δzi(r+1,m))],−N≤m≤N,i=1,…,k Write Sh={x(m):x(m)∈[−c,c]}. Elements of the space F(Sh,ℝk) are denoted by ξ, ξ¯.
For ξ:Sh→ℝk, ξ=(ξ1,…,ξk),
we write ξ(m)=ξ(x(m)) and(3.7)δξi(m)=(δ1ξi(m),…,δnξi(m)),1≤i≤k,δjξi(m)=1hj[ξi(m+ejj)−ξi(m)]ifxj(mj)∈[x˜j,bj),δjξi(m)=1hj[ξi(m)−ξi(m−ej)]ifxj(mj)∈(bj,x˜j),where j=1,…,n. The norm in the space F(Sh,ℝk) is defined by(3.8)∥ξ∥⋆=max{∥ξ(m)∥∞:x(m)∈Sh}.Let us consider the set(3.9)Xh={ξ∈F(Sh,ℝk):ξ(m)=φ(r+1,m)forx(m)∈[−c,c]∖(−b,b)}.We consider the operator Wh:Xh→Xh,  Wh=(Wh.1,…,Wh.n) defined by(3.10)Wh.i[ξ](m)=1Qh+1[Qhξi(m)+uh.i(r,m)+h0fi(t(r),x(m),T(uh)[r,m],δξi(m))],where −N≤m≤N,  i=1,…,k and(3.11)Wh[ξ](m)=φh(r+1,m)forx(m)∈[−c,c]∖(−b,b),where ξ=(ξ1,…,ξk)∈F(Sh,ℝk).
We prove that(3.12)∥Wh[ξ]−Wh[ξ¯]∥⋆≤QhQh+1∥ξ−ξ¯∥⋆onF(Sh,ℝk).
It follows from (3.10) that we have for −N≤m≤N:(3.13)Wh.i[ξ](m)−Wh.i[ξ¯](m)=1Qh+1[Qh(ξi−ξ¯i)(m)−h0∑j∈Γ+(m)1hj∫01∂qjfi(Pi(r,m)[uh,τ])dτ(ξi−ξ¯i)(m)+∑j∈Γ−(m)1hj∫01∂qjfi(Pi(r,m)[uh,τ])dτ(ξi−ξ¯i)(m)+h0∑j∈Γ+(m)1hj∫01∂qjfi(Pi(r,m)[uh,τ])dτ(ξi−ξ¯i)(m+ej)−h0∑j∈Γ−(m)1hj∫01∂qjfi(Pi(r,m)[uh,τ])dτ(ξi−ξ¯i)(m−ej)], where i=1,…,k and(3.14)Pi(r,m)[uh,τ]=(t(r),x(m),Th(uh)[r,m],δξ¯i(m)+τδ(ξi−ξ¯i)(m)).It follows from the above
relations and from (3.5) that(3.15)|Wh.i[ξ](m)−Wh.i[ξ¯](m)|≤QhQh+1∥ξ−ξ¯∥⋆for−N≤m≤N,i=1,…,k.According to (3.12) we have(3.16)Wh.i[ξ](m)−Wh.i[ξ¯](m)=0forx(m)∈[−c,c]∖(−b,b),i=1,…,k.This completes the proof of (3.12).
It follows from the Banach fixed point theorem that
there exists exactly one solutionξ¯:Sh→ℝk of the equation ξ=Wh[ξ] and consequently, there exists exactly one
solution of (3.6), (2.21). Then the vectors uh(r+1,m), −N≤m≤N,
exist and they are unique. Then the proof is completed by induction with
respect to r, 0≤r≤N0.Assumption (H[σ]).
The functionσ:[0,a]×ℝ+→ℝ+ satisfies the conditions:
(1)
σ is continuous and it is nondecreasing with
respect to the both variables,(2)
σ(t,0)=0 for t∈[0,a] and the maximal solution of the Cauchy problem(3.17)η′(t)=σ(t,η(t)),η(0)=0,is η˜(t)=0  for t∈[0,a].Assumption (H[f,σ]).
There isσ:[0,a]×ℝ+→ℝ+ such that Assumption (H[σ])  is satisfied and for w,w¯∈ℂ(D,ℝk), w≥w¯,
we have(3.18)fi(t,x,w,q)−fi(t,x,w¯,q)≤σ(t,∥w−w¯∥D),i=1,…,k,where (t,x,q)∈E×ℝn.Theorem 3.4.
Suppose that Assumptions(H[f])  and     (H[f,σ])  are satisfied and
(1)
v:Ω→ℝ is a solution of  (1.4),  (1.5) and v  is of class C1  on Ω,(2)
h∈H*, h0<ε  and φh:E0.h∪∂0Eh→ℝk  and there is α0:H*→ℝ+  such that(3.19)∥φ(r,m)−φh(r,m)∥∞≤α0(h)onE0.h∪∂0Eh,limh→0α0(h)=0.    Under these
assumptions there is a solution uh:Ωh→ℝk of (2.20), (2.21) and there is α:H*→ℝ+ such that(3.20)∥(uh−vh)(r,m)∥∞≤α(h)onEh,limh→0α(h)=0,where vh is the restriction of v to the set Ωh.Proof.
The
existence ofuh follows from Theorem 3.3. Let Γh:Eh′→ℝk, Γ0.h:E0.h∪∂0Eh→ℝk be defined by the relations(3.21)δ0vh(r,m)=𝔽h[vh](r,m)+Γh(r,m)onEh′,(3.22)vh(r,m)=φh(r+1,m)+Γ0.h(r,m)for(t(r),x(m))∈E0.h∪∂0Eh.From Lemma 3.1 and from
assumption (1) of the theorem it follows that there are γ,γ0:H*→ℝ+ such that(3.23)∥Γh(r,m)∥∞≤γ(h)onEh′,∥Γ0.h(r,m)∥∞≤γ0(h)onE0.h∪∂0Ehand limh→0γ(h)=0, limh→0γ0(h)=0. Write J=[0,a] and Jh={t(r):0≤r≤N0}. For β:Jh→ℝ we put β(r)=β(t(r)). Let βh:Jh→ℝ+ be a solution of the difference
problem(3.24)β(r+1)=β(r)+h0σ(t(r),β(r))+h0γ(h),0≤r≤N0−1,β(0)=α0(h).We prove that(3.25)∥(uh−vh)(r,m)∥∞≤βh(r)onEh.Let v˜h=(v˜h.1,…,v˜h.k):Ωh→ℝk be defined by(3.26)v˜h.i(r,m)=vh.i(r,m)+βh(0)onE0.h,v˜h.i(r,m)=vh.i(r,m)+βh(i)onEh∪∂0Eh, where i=1,…,k. We prove that the difference functional
inequality(3.27)δ0v˜h≥𝔽h[v˜h](r,m),(t(r),x(m))∈Eh′,is satisfied. It follows from
Assumption (H[f,σ])   and from (3.21) that(3.28)δ0v˜h.i(r,m)=δ0vh.i(r,m)+1h0(βh(r+1)−βh(r))=fi(t(r),x(m),Th(v˜h)[r,m],δv˜h.i(r+1,m))+1h0(βh(r+1)−βh(r))+[fi(t(r),x(m),Th(vh)[r,m],δvh.i(r+1,m))−fi(t(r),x(m),Th(v˜h)[r,m],δvh.i(r+1,m))]≥fi(t(r),x(m),Th(v˜h)[r,m],δv˜h.i(r+1,m))−σ(t(r),βh(r))+1h0(βh(r+1)−βh(r))=fi(t(r),x(m),Th(v˜h)[r,m],δv˜h.i(r+1,m)),i=1,…,k. This completes the proof of (3.27).
Sincevh(r,m)≤v˜h(r,m) on E0.h∪∂0Eh,
it follows from Theorem 2.3
that uh(r,m)≤vh(r,m)+βh(r) on Eh. In a similar way we prove that vh(r,m)−βh(r)≤uh(r,m) on Eh. The above estimates imply (3.25). Consider the
Cauchy problem(3.29)η′(t)=σ(t,η(t))+γ(h),η(0)=α0(h).It follows from Assumption (H[σ])  that there is ε˜>0 such that for ∥h∥≤ε˜  the maximal solution η(⋅,h) of (3.29) is defined on [0,a] and(3.30)limh→0η(t,h)=0uniformlyon[0,a].Since η(⋅,h) is convex function then we have the difference
inequality(3.31)η(t(r+1),h)≥η(t(r),h)+h0σ(t(r),η(t(r),h))+h0γ(h),where 0≤r≤N0−1.
Since βh satisfies (3.24), the above relations imply the
estimate(3.32)βh(r)≤η(t(r),h)≤η(a,h),0≤r≤N0.It follows from (3.30) that
condition (3.20) is satisfied with α(h)=η(a,h).
This completes the proof.Lemma 3.5.
Suppose  that  Assumption(H[f])  is satisfied
and
(1)
v:Ω→ℝ is a solution of (1.4), (1.5)  and v  is of class C2 on Ω,(2)
h∈H*, h0<ε and φh:E0.h∪∂0Eh→ℝk and there is α0:H*→ℝ+ such that(3.33)∥φ(r,m)−φh(r,m)∥∞≤α0(h)onE0.h∪∂0Eh,limh→0α0(h)=0.(3)
there existsL∈ℝ+ such that estimates(3.34)fi(t,x,w,q)−fi(t,x,w˜,q)≤L∥w−w˜∥D,i=1,…,k,are satisfied for (t,x,q)∈E×ℝn, w,w˜∈C(D,ℝk) and w≥w˜,(4)
there isC¯∈ℝ+  such that(3.35)∥∂qfi(t,x,w,q)∥≤C¯onΣfori=1,…,k.           Under these
assumptions there is a solution uh:Ωh→ℝk of (2.20), (2.21), and(3.36)∥(uh−vh)(r,m)∥∞≤α˜(h)onEh,where(3.37)α˜(h)=α0(h)eLa+γ˜(h)eLa−1LifL>0,α˜(h)=α0(h)+aγ˜(h)ifL=0,γ˜(h)=0.5C˜h0(1+C¯)+LC˜∥h′∥2+0.5C¯C˜∥h∥and C˜∈ℝ+  is such that(3.38)∥∂ttv(t,x)∥∞,∥∂txiv(t,x)∥∞,∥∂xixjv(t,x)∥∞≤C˜on Ω for i,i=1,…,n.Proof.
It
follows that the solutionβh:Jh→ℝ+ of the difference problem(3.39)β(r+1)=(1+Lh0)β(r)+h0γ(h),0≤r≤N0−1,β(0)=α0(h) satisfies the condition: βh(r)≤α˜(h) for 0≤r≤N0.
Moreover we have(3.40)∥Γh(r,m)∥∞≤γ˜(h)onEh′,where Γh is given by (3.21). Then we obtain the assertion
from Lemma 3.2
and Theorem 3.4.Remark 3.6.
In the result on error estimates we need
estimates for the derivatives of the solutionv of problem (1.4), (1.5). One may obtain them by
the method of differential inequalities, see [10, Chapter 5].


## 4. Numerical Examples

Example 4.1.
Forn=2 we put (4.1)E=[0,0.5]×[−1,1]×[−1,1],E0={0}×[−1,1]×[−1,1].Consider the differential
integral equation(4.2)∂tz(t,x,y)=arctan[2x∂xz(t,x,y)+2y∂yz(t,x,y)−t(2x2y2−x2−y2)z(t,x,y)]+t(1−y2)∫−1xsz(t,s,y)ds+t(1−x2)∫−1ysz(t,x,s)ds+z(t,x,y)[4+0.25(x2−1)(y2−1)]−4 with the initial boundary
condition(4.3)z(0,x,y)=1,(x,y)∈[−1,1]×[−1,1],z(t,−1,y)=z(t,1,y)=1,(t,y)∈[0,0.5]×[−1,1],z(t,x,−1)=z(t,x,1)=1,(t,x)∈[0,0.5]×[−1,1].The functionv(t,x,y)=exp[0.25t(x2−1)(y2−1)] is the solution of the above problem. Let us
denote by zh an approximate solution which is obtained by
using the implicit difference scheme.The Newton method is used for solving nonlinear
systems generated by the implicit difference scheme. Writem=(m1,m2) and(4.4)εh(r)=1(2N1−1)(2N2−1)∑m∈Π|zh(r,m)−v(r,m)|,0≤r≤N0,where(4.5)Π={m=(m1,m2):∈ℤ2:−N1+1≤m1≤N1−1,−N1+1≤m2≤N2−1}and N1h1=1, N2h2=1, N0h0=0.5. The numbers εh(r) can be called average  errors of the difference
method for fixed t(r). We put h0=h1=h2=0.005  and  we have   the  values of the above
defined errors which are shown in Table 1.Table 1
Table of errors.
t(r)0.250.300.350.400.450.50εh(r)0.00060.00070.00090.00100.00120.0014Note that our equation and the
steps of the mesh do not satisfy condition (1.10) which is necessary for the
explicit difference method to be convergent. In our numerical example the
average errors for the explicit difference method exceeded 102.Example 4.2.
Letn=2 and (4.6)E=[0,0.5]×[−0.5,0.5]×[−0.5,0.5],E0={0}×[−0.5,0.5]×[−0.5,0.5].Consider the differential
equation with deviated variables(4.7)∂tz(t,x,y)=2x∂xz(t,x,y)+2y∂yz(t,x,y)+cos[2x∂xz(t,x,y)−2y∂yz(t,x,y)−t(x2−y2)z(t,x,y)]+z(t2,x,y)+f(t,x,y)z(t,x,y)−1, with the   initial boundary
conditions(4.8)z(0,x,y)=1,(x,y)∈[−0.5,0.5]×[−0.5,0.5]z(t,−0.5,y)=z(t,0.5,y)=1,(t,y)∈[0,0.5]×[−0.5,0.5],z(t,x,−0.5)=z(t,x,0,5)=1,(t,x)∈[0,0.5]×[−0.5,0.5], where(4.9)f(t,x,y)=(x2−0.25)(0.25−y2)+t[8x2y2−x2−y2]−exp{(0.5t2−t)(x2−0.25)(0.25−y2)}.The functionv(t,x,y)=exp[[t(x2−0.25)(0.25−y2)] is the solution of the above problem. Let us
denote by zh an approximate solution which is obtained by
using the implicit difference scheme.The Newton method is used for solving nonlinear
systems generated by the implicit difference scheme.Letεh be defined by (4.4) with N1h1=0.5, N2h2=0.5, N0h0=0.5.
We put h0=h1=h2=0.005  and we have the  values of the above
defined errors which are shown in Table 2.Table 2
Table of errors.
t(r)0.250.300.350.400.450.5εh(r)0.00020.00030.00040.00040.00050.0006Note that our equation and the
steps of the mesh do not satisfy condition (1.10) which is necessary for the
explicit difference method to be convergent. In our numerical example the
average errors for the explicit difference method exceeded 102.The above examples show that there are implicit
difference schemes which are convergent, and the corresponding classical method
is not convergent. This is due to the fact that we need assumption (1.10) for
explicit difference methods. We do not need this condition in our implicit
methods.Our results show that implicit difference schemes are
convergent on all meshes.



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