JAMSAJournal of Applied Mathematics and Stochastic Analysis1687-21771048-9533Hindawi Publishing Corporation25472010.1155/2009/254720254720Research ArticleImplicit Difference Inequalities Corresponding to First-Order Partial Differential Functional EquationsKamont Z.1KropielnickaK.1O'ReganDonal1Institite of MathematicsUniversity of GdańskWit Stwosz Street 5780-952 GdańskPolandug.gda.pl200927012009200919082008050120092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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  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 spaces X 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. WriteE=[0,a]×(b,b),D=[d0,0]×[d,d],where a>0, b=(b1,,bn)n, bi>0 for 1in and d=(d1,,dn)+n, d0+, +=[0,+). Let c=b+d andE0=[d0,0]×[c,c],0E=[0,a]×([c,c](b,b)),Ω=EE00E.For a function z:Ωk, z=(z1,,zk), and for a point (t,x)E¯ where E¯ is the closure of E, we define a function z(t,x):Dk 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 [td0,t]×[xd,x+d] and this restriction is shifted to the set D. Write Σ=E×C(D,k)×n and suppose that f=(f1,,fk):Σk and φ:E00Ek, φ=(φ1,,φk), are given functions. Let us denote by z=(z1,,zk) an unknown function of the variables (t,x), x=(x1,,xn). Write𝔽[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), 1ik. We consider the system of functional differential equationstz(t,x)=𝔽[z](t,x)with the initial boundary conditionz(t,x)=φ(t,x)onE00E.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 function α:E1+n satisfies the condition: α(t,x)(t,x)D for (t,x)E. For a given f˜=(f˜1,,f˜k):E×k×k×nk we put f(t,x,w,q)=f˜(t,x,w(0,θ),w(α(t,x)(t,x)),q)onΣ,where θ=(0,,0)n. Then (1.4) is reduced to the system of differential equations with deviated variablestzi(t,x)=f˜i(t,x,z(t,x),z(α(t,x)),xzi(t,x)),i=1,,k.

Example 1.2.

For the above f˜ we define 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 equationstzi(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

the function f of the variables (t,x,w,q), q=(q1,,qn), is of class C1 with respect to q and the functions qfi=(q1fi,,qnfi), 1ik, are bounded,

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 condition1h0j=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 Section 2. 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 . For further bibliographic informations concerning differential and functional differential inequalities and applications see the survey paper  and the monographs [16, 17].

2. Functional Difference Inequalities

For any two sets U and W we denote by F(U,W) the class of all functions defined on U and taking values in W. Let and be the sets of natural numbers and integers, respectively. For x=(x1,,xn)n, p=(p1,,pk)k we putx=|x1|++|xn|,p=max{|pi|:1ik}.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)1+n where m=(m1,,mn), we define nodal points as follows: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 and K=(K1,,Kn)n satisfying the conditions: K0h0=d0 and (K1h1,,Knhn)=d. Seth1+n={(t(r),x(m)):(r,m)1+n},Dh=Dh1+n,Eh=Eh1+n,E0.h=E0h1+n,0Eh=0Eh1+n,Ωh=EhE0.h0Eh.Let N0 be defined by the relations: N0h0a<(N0+1)h0 andEh={(t(r),x(m))Eh:0rN01}.For functions w:Dhk and z:Ωhk 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:Ωhk and for a point (t(r),x(m))Eh we define a function z[r,m]:Dhk byz[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,k). Then we need an interpolating operator Th:F(Dh,k)C(D,k). We define Th in the following way. Let us denote by (ϑ1,,ϑn) the family of sets defined byϑi={0,1}ifdi>0,ϑi={0}ifdi=0,1in.Set υ=(υ1,,υn)n and υi=0 if di=0, υi=1 if di>0 where 1in. WriteΔ+={λ=(λ1,,λn):λiϑifor1in}.Set ei=(0,,0,1,0,,0)n with 1 standing on the ith place.

Let wF(Dh,k) 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)tt(r+1), x(m)xx(m+υ). WriteTh[w](t,x)=(1tt(r)h0)λΔ+w(r,m+λ)(xx(m)h)λ(1xx(m)h)1λ+tt(r)h0λΔ+w(r+1,m+λ)(xx(m)h)λ(1xx(m)h)1λ, where(xx(m)h)λ=i=1n(xixi(mi)hi)λi,(1xx(m)h)1λ=i=1n(1xixi(mi)hi)1λi and we take 00=1 in the above formulas. If d0=0 then we putTh[w](t,x)=λΔ+w(r,m+λ)(xx(m)h)λ(1xx(m)h)1λ.Then we have defined Th[w] on D. It is easy to see that Th[w]C(D,k). The above interpolating operator has been first proposed in [10, Chapter 5].

For w,w¯F(Dh,k) we write ww¯ if w(r,m)w¯(r,m) where (t(r),x(m))Dh. In a similar way we define the relation ww¯ for w,w¯C(D,k) and the relation zz¯ for z,z¯F(Ωh,k) and for z,z¯C(Ω,k).

We formulate an implicit difference scheme for (1.4), (1.5). For x,yn we write xy=(x1y1,,xnyn)n.

Assumption (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M147"><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></inline-formula>).

The function f=(f1,,fk):Σk of the variables (t,x,w,q), q=(q1,,qn), is continuous and

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,

there is x˜(b,b), x˜=(x˜1,,x˜n), such that(xx˜)qfi(t,x,w,q)θonΣfori=1,,k,

there is ε0>0 such that for 0<h0<ε0 and w,w¯C(D,k), ww¯, we havew(0,θ)+h0f(t,x,w,q)w¯(0,θ)+h0f(t,x,w¯,q),(t,x,q)E×n.

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 zC(Ω,k), uC(Ω,n). Let us denote by 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 problemy(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 xjx˜j0 the function gij[z,u](,t,x) is non increasing. If xjx˜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 . 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 function f˜=(f˜1,,f˜k):E××C(D,k)×nk 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 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

f˜ is nondecreasing with respect to the functional variable,

there exists the derivative pf˜=(pf˜1,,pf˜k) and pf˜i(t,x,p,w,q)L for i=1,,k and 1+Lh00.

Then the monotonicity condition (3) of Assumption (H[f])  is satisfied.

Let us denote by H the set of all h=(h0,h)H such thathi<min{bix˜i,x˜i+bi},i=1,,n.Suppose that ω:Ωh. We apply difference operators δ=(δ1,,δn) given byifx˜jxj(mj)<bjthenδjω(r,m)=1hj[ω(r,m+ej)ω(r,m)],ifbj<xj(mj)<x˜jthenδjω(r,m)=1hj[ω(r,m)ω(r,mej)],and we put j=1,,n in (2.17). Let δ0 be defined byδ0ω(r,m)=1h0[ω(r+1,m)ω(r,m)]and δ0z=(δ0z1,,δ0zk). Write𝔽h[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.h0Ehk, we consider the functional difference equationδ0z(r,m)=𝔽h[z](r,m)with the initial boundary conditionz(r,m)=φh(r,m)onE0.h0Eh.

The above problem is considered as an implicit difference method for (1.4), (1.5). It is important that the difference expressions (δ1zi,,δnzi), 1ik, 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  are not applicable to (2.20), (2.21).

Theorem 2.3.

Suppose that Assumption (H[f])  is satisfied and

hH, h0<ε0 and the functions u,v:Ωhk satisfy the difference functional inequalityδ0u(r,m)𝔽h[u](r,m)δ0v(r,m)𝔽h[v](r,m)onEh,

the initial boundary estimate u(r,m)v(r,m) holds on E0.h0Eh.

Thenu(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)]×n). We prove that u(r+1,m)v(r+1,m) for (t(r+1,m),x(m))Eh. WriteUi(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(uivi)(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(uivi)(r+1,m)h0j=1n01qjfi(Qi(r,m)[v,τ])dτδj(uivi)(r+1,m),where i=1,,k andQi(r,m)[v,τ]=(t(r),x(m),Thv[r,m],δvi(r+1,m)+τδ(uivi)(r+1,m)).WriteΓ+(m)={j{1,,n}:xj(mj)[x˜j,bj)},Γ(m)={1,,n}Γ+(m).It follows from (2.11), (2.17) that(uivi)(r+1,m)[1+h0j=1n1hj01|qjfi(Qi(r,m)[v,τ])|dτ]h0jΓ+(m)1hj01qjfi(Qi(r,m)[v,τ])dτ(uivi)(r+1,m+ej)h0jΓ(m)1hj01qjfi(Qi(r,m)[v,τ])dτ(uivi)(r+1,mej),i=1,,k. We define m˜n and μ, 1μk, as follows:(uμvμ)(r+1,m˜)=max1ikmax{(uivi)(r+1,m):(t(r+1),x(m))Ωh}.If (t(r+1),x(m˜))0Eh then assumption (2) implies that (uμvμ)(r+1,m˜)0. Let us consider the case when (t(r+1),x(m˜))Eh. Then we have from (2.29) that(uμvμ)(r+1,m˜)[1+h0j=1n1hj01|qjfi(Qi(r,m˜)[v,τ])|dτ]h0(uμvμ)(r+1,m˜)[jΓ+(m˜)1hj01qjfi(Qi(r,m˜)[v,τ])dτjΓ(m˜)1hj01qjfi(Qi(r,m˜)[v,τ])dτ]. It follows that (uμvμ)(r+1,m˜)0. The the proof of (2.23) is completed by induction.

3. Implicit Difference Schemes

We define N=(N1,,Nn)Nn by the relations:(N1h1,,Nnhn)<(b1,,bn)((N1+1)h1,,(Nn+1)hn)and we assume that (Ni+1)hi=bi if di=0. For wC(D,k) we writewD=max{w(t,x):(t,x)D}.In a similar way we define the norm in the space F(Dh,k) : if w:Dhk thenwDh=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 that w:Dk is of class C1 and wh is the restriction of w to the set Dh. Let C˜ be such a constant that twD, xiwDC˜ for 1in. Then Th[wh]wDC˜h where h=h0+h1++hn.

Lemma 3.2.

Suppose that w:Dk is of class C2 and wh is the restriction of w to the set Dh. Let C˜ be such a constant that ttwD, txiwD, xixjwDC˜,   i,j=1,,n. Then Th[wh]wDC˜h2.

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 φhF(E0.h0Eh,k) then there exists exactly one solution uh=(uh.1,,uh.k):Ωhk of difference functional problem (2.20), (2.21).

Proof.

Suppose that 0rN01 is fixed and that the solution uh of problem (2.20), (2.21) is given on the set Ωh([d0,t(r)]×n). We prove that the vectors uh(r+1,m),  NmN, exist and that they are unique. It is sufficient to show that there exists exactly one solution of the system of equations1h0(zi(r+1,m)uh.i(r,m))=fi(t(r),x(m),T(uh)[r,m],δzi(r+1,m)),where NmN,i=1,,k, with the initial boundary condition (2.21). There exists Qh>0 such thatQhh0[jΓ+(m)1hjqjfi(t(r),x(m),Th(uh)[r,m],q)jΓ(m)1hjqjfi(t(r),x(m),Th(uh)[r,m],q)], where NmN,i=1,,k. It is clear that system (3.4) is equivalent to the following one: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))],NmN,i=1,,k Write Sh={x(m):x(m)[c,c]}. Elements of the space F(Sh,k) are denoted by ξ, ξ¯. For ξ:Shk, ξ=(ξ1,,ξk), we write ξ(m)=ξ(x(m)) andδξi(m)=(δ1ξi(m),,δnξi(m)),1ik,δjξi(m)=1hj[ξi(m+ejj)ξi(m)]ifxj(mj)[x˜j,bj),δjξi(m)=1hj[ξi(m)ξi(mej)]ifxj(mj)(bj,x˜j),where j=1,,n. The norm in the space F(Sh,k) is defined byξ=max{ξ(m):x(m)Sh}.Let us consider the setXh={ξF(Sh,k):ξ(m)=φ(r+1,m)forx(m)[c,c](b,b)}.We consider the operator Wh:XhXh,  Wh=(Wh.1,,Wh.n) defined byWh.i[ξ](m)=1Qh+1[Qhξi(m)+uh.i(r,m)+h0fi(t(r),x(m),T(uh)[r,m],δξi(m))],where NmN,  i=1,,k andWh[ξ](m)=φh(r+1,m)forx(m)[c,c](b,b),where ξ=(ξ1,,ξk)F(Sh,k). We prove thatWh[ξ]Wh[ξ¯]QhQh+1ξξ¯onF(Sh,k).

It follows from (3.10) that we have for NmN:Wh.i[ξ](m)Wh.i[ξ¯](m)=1Qh+1[Qh(ξiξ¯i)(m)h0jΓ+(m)1hj01qjfi(Pi(r,m)[uh,τ])dτ(ξiξ¯i)(m)+jΓ(m)1hj01qjfi(Pi(r,m)[uh,τ])dτ(ξiξ¯i)(m)+h0jΓ+(m)1hj01qjfi(Pi(r,m)[uh,τ])dτ(ξiξ¯i)(m+ej)h0jΓ(m)1hj01qjfi(Pi(r,m)[uh,τ])dτ(ξiξ¯i)(mej)], where i=1,,k andPi(r,m)[uh,τ]=(t(r),x(m),Th(uh)[r,m],δξ¯i(m)+τδ(ξiξ¯i)(m)).It follows from the above relations and from (3.5) that|Wh.i[ξ](m)Wh.i[ξ¯](m)|QhQh+1ξξ¯forNmN,i=1,,k.According to (3.12) we haveWh.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 solution ξ¯:Shk 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), NmN, exist and they are unique. Then the proof is completed by induction with respect to r, 0rN0.

Assumption (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M322"><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></inline-formula>).

The function σ:[0,a]×++ satisfies the conditions:

σ is continuous and it is nondecreasing with respect to the both variables,

σ(t,0)=0 for t[0,a] and the maximal solution of the Cauchy problemη(t)=σ(t,η(t)),η(0)=0,is η˜(t)=0 for t[0,a].

Assumption (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M330"><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>f</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></inline-formula>).

There is σ:[0,a]×++ such that Assumption (H[σ])  is satisfied and for w,w¯(D,k), ww¯, we havefi(t,x,w,q)fi(t,x,w¯,q)σ(t,ww¯D),i=1,,k,where (t,x,q)E×n.

Theorem 3.4.

Suppose that Assumptions (H[f])  and   (H[f,σ])  are satisfied and

v:Ω is a solution of (1.4), (1.5) and v is of class C1 on Ω,

hH*, h0<ε and φh:E0.h0Ehk and there is α0:H*+ such thatφ(r,m)φh(r,m)α0(h)onE0.h0Eh,limh0α0(h)=0.

Under these assumptions there is a solution uh:Ωhk of (2.20), (2.21) and there is α:H*+ such that(uhvh)(r,m)α(h)on  Eh,limh0α(h)=0,where vh is the restriction of v to the set Ωh.

Proof.

The existence of uh follows from Theorem 3.3. Let Γh:Ehk, Γ0.h:E0.h0Ehk be defined by the relationsδ0vh(r,m)=𝔽h[vh](r,m)+Γh(r,m)onEh,vh(r,m)=φh(r+1,m)+Γ0.h(r,m)for(t(r),x(m))E0.h0Eh.From Lemma 3.1 and from assumption (1) of the theorem it follows that there are γ,γ0:H*+ such thatΓh(r,m)γ(h)onEh,Γ0.h(r,m)γ0(h)onE0.h0Ehand limh0γ(h)=0, limh0γ0(h)=0. Write J=[0,a] and Jh={t(r):0rN0}. For β:Jh we put β(r)=β(t(r)). Let βh:Jh+ be a solution of the difference problemβ(r+1)=β(r)+h0σ(t(r),β(r))+h0γ(h),0rN01,β(0)=α0(h).We prove that(uhvh)(r,m)βh(r)onEh.Let v˜h=(v˜h.1,,v˜h.k):Ωhk be defined byv˜h.i(r,m)=vh.i(r,m)+βh(0)onE0.h,v˜h.i(r,m)=vh.i(r,m)+βh(i)onEh0Eh, where i=1,,k. We prove that the difference functional inequalityδ0v˜h𝔽h[v˜h](r,m),(t(r),x(m))Eh,is satisfied. It follows from Assumption (H[f,σ]) and from (3.21) thatδ0v˜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).

Since vh(r,m)v˜h(r,m) on E0.h0Eh, 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 problemη(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 hε˜  the maximal solution η(,h) of (3.29) is defined on [0,a] andlimh0η(t,h)=0uniformlyon[0,a].Since η(,h) is convex function then we have the difference inequalityη(t(r+1),h)η(t(r),h)+h0σ(t(r),η(t(r),h))+h0γ(h),where 0rN01. Since βh satisfies (3.24), the above relations imply the estimateβh(r)η(t(r),h)η(a,h),0rN0.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

v:Ω is a solution of (1.4), (1.5) and v is of class C2 on Ω,

hH*, h0<ε and φh:E0.h0Ehk and there is α0:H*+ such thatφ(r,m)φh(r,m)α0(h)onE0.h0Eh,  limh0α0(h)=0.

there exists L+ such that estimatesfi(t,x,w,q)fi(t,x,w˜,q)Lww˜D,i=1,,k,are satisfied for (t,x,q)E×n, w,w˜C(D,k) and ww˜,

there is C¯+ such thatqfi(t,x,w,q)C¯onΣfori=1,,k.

Under these assumptions there is a solution uh:Ωhk of (2.20), (2.21), and(uhvh)(r,m)α˜(h)onEh,whereα˜(h)=α0(h)eLa+γ˜(h)eLa1LifL>0,α˜(h)=α0(h)+aγ˜(h)ifL=0,γ˜(h)=0.5C˜h0(1+C¯)+LC˜h2+0.5C¯C˜hand C˜+ is such thatttv(t,x),txiv(t,x),xixjv(t,x)C˜on Ω for i,i=1,,n.

Proof.

It follows that the solution βh:Jh+ of the difference problemβ(r+1)=(1+Lh0)β(r)+h0γ(h),0rN01,β(0)=α0(h) satisfies the condition: βh(r)α˜(h) for 0rN0. Moreover we haveΓh(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 solution v of problem (1.4), (1.5). One may obtain them by the method of differential inequalities, see [10, Chapter 5].

4. Numerical ExamplesExample 4.1.

For n=2 we put E=[0,0.5]×[1,1]×[1,1],E0={0}×[1,1]×[1,1].Consider the differential integral equationtz(t,x,y)=arctan[2xxz(t,x,y)+2yyz(t,x,y)t(2x2y2x2y2)z(t,x,y)]+t(1y2)1xsz(t,s,y)ds+t(1x2)1ysz(t,x,s)ds+z(t,x,y)[4+0.25(x21)(y21)]4 with the initial boundary conditionz(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 function v(t,x,y)=exp[0.25t(x21)(y21)] 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. Write m=(m1,m2) andεh(r)=1(2N11)(2N21)mΠ|zh(r,m)v(r,m)|,0rN0,whereΠ={m=(m1,m2):2:N1+1m1N11,N1+1m2N21}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 of errors.

 t(r) 0.25 0.3 0.35 0.4 0.45 0.5 εh(r) 0.0006 0.0007 0.0009 0.001 0.0012 0.0014

Note 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.

Let n=2 and 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 variablestz(t,x,y)=2xxz(t,x,y)+2yyz(t,x,y)+cos[2xxz(t,x,y)2yyz(t,x,y)t(x2y2)z(t,x,y)]+z(t2,x,y)+f(t,x,y)z(t,x,y)1, with the initial boundary conditionsz(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], wheref(t,x,y)=(x20.25)(0.25y2)+t[8x2y2x2y2]exp{(0.5t2t)(x20.25)(0.25y2)}.

The function v(t,x,y)=exp[[t(x20.25)(0.25y2)] 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.

Let εh 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 of errors.

 t(r) 0.25 0.3 0.35 0.4 0.45 0.5 εh(r) 0.0002 0.0003 0.0004 0.0004 0.0005 0.0006

Note 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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