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Journal of Applied Mathematics and Stochastic Analysis
Volume 2009 (2009), Article ID 254720, 18 pages
http://dx.doi.org/10.1155/2009/254720
Research Article

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

Institite of Mathematics, University of Gdańsk, Wit Stwosz Street 57, 80-952 Gdańsk, Poland

Received 19 August 2008; Accepted 5 January 2009

Academic Editor: Donal O'Regan

Copyright © 2009 Z. Kamont and K. Kropielnicka. This 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.

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.

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 [58] 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],(1.1)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=𝐸𝐸0𝜕0𝐸.(1.2)For a function 𝑧Ω𝑘, 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 𝜑𝐸0𝜕0𝐸𝑘, 𝜑=(𝜑1,,𝜑𝑘), are given functions. Let us denote by 𝑧=(𝑧1,,𝑧𝑘) an unknown function of the variables (𝑡,𝑥), 𝑥=(𝑥1,,𝑥𝑛). Write𝑓𝔽[𝑧](𝑡,𝑥)=1𝑡,𝑥,𝑧(𝑡,𝑥),𝜕𝑥𝑧1(𝑡,𝑥),,𝑓𝑘𝑡,𝑥,𝑧(𝑡,𝑥),𝜕𝑥𝑧𝑘(𝑡,𝑥)(1.3)and 𝜕𝑥𝑧𝑖=(𝜕𝑥1𝑧𝑖,,𝜕𝑥𝑛𝑧𝑖), 1𝑖𝑘. We consider the system of functional differential equations𝜕𝑡𝑧(𝑡,𝑥)=𝔽[𝑧](𝑡,𝑥)(1.4)with the initial boundary condition𝑧(𝑡,𝑥)=𝜑(𝑡,𝑥)on𝐸0𝜕0𝐸.(1.5)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 𝛼𝐸1+𝑛 satisfies the condition: 𝛼(𝑡,𝑥)(𝑡,𝑥)𝐷 for (𝑡,𝑥)𝐸. For a given 𝑓𝑓=(1𝑓,,𝑘)𝐸×𝑘×𝑘×𝑛𝑘 we put 𝑓𝑓(𝑡,𝑥,𝑤,𝑞)=𝑡,𝑥,𝑤(0,𝜃),𝑤𝛼(𝑡,𝑥)(𝑡,𝑥),𝑞onΣ,(1.6)where 𝜃=(0,,0)𝑛. Then (1.4) is reduced to the system of differential equations with deviated variables𝜕𝑡𝑧𝑖𝑓(𝑡,𝑥)=𝑖𝑡,𝑥,𝑧(𝑡,𝑥),𝑧𝛼(𝑡,𝑥),𝜕𝑥𝑧𝑖(𝑡,𝑥),𝑖=1,,𝑘.(1.7)

Example 1.2. For the above 𝑓 we define 𝑓𝑓(𝑡,𝑥,𝑤,𝑞)=𝑡,𝑥,𝑤(0,𝜃),𝐷𝑤(𝜏,𝑦)𝑑𝑦𝑑𝜏,𝑞onΣ.(1.8)Then (1.4) is equivalent to the system of differential integral equations𝜕𝑡𝑧𝑖𝑓(𝑡,𝑥)=𝑖𝑡,𝑥,𝑧(𝑡,𝑥),𝐷𝑧(𝑡+𝜏,𝑥+𝑦)𝑑𝑦𝑑𝜏,𝜕𝑥𝑧𝑖(𝑡,𝑥),𝑖=1,,𝑘.(1.9)

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 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,(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 condition10𝑛𝑗=11𝑗||𝜕𝑞𝑗𝑓𝑖||(𝑡,𝑥,𝑤,𝑞)0onΣfor𝑖=1,,𝑘,(1.10)where 0 and =(1,,𝑛) 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 0 and 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 [1214]. 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 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 put||𝑥𝑥=1||||𝑥++𝑛||,𝑝||𝑝=max𝑖||1𝑖𝑘.(2.1)We define a mesh on Ω in the following way. Suppose that (0,), =(1,,𝑛), stand for steps of the mesh. For (𝑟,𝑚)1+𝑛 where 𝑚=(𝑚1,,𝑚𝑛), we define nodal points as follows:𝑡(𝑟)=𝑟0,𝑥(𝑚)=𝑥(𝑚1)1,,𝑥(𝑚𝑛)𝑛=𝑚11,,𝑚𝑛𝑛.(2.2)Let us denote by 𝐻 the set of all =(0,) such that there are 𝐾0 and 𝐾=(𝐾1,,𝐾𝑛)𝑛 satisfying the conditions: 𝐾00=𝑑0 and (𝐾11,,𝐾𝑛𝑛)=𝑑. Set1+𝑛=𝑡(𝑟),𝑥(𝑚)(𝑟,𝑚)1+𝑛,𝐷=𝐷1+𝑛,𝐸=𝐸1+𝑛,𝐸0.=𝐸01+𝑛,𝜕0𝐸=𝜕0𝐸1+𝑛,Ω=𝐸𝐸0.𝜕0𝐸.(2.3)Let 𝑁0 be defined by the relations: 𝑁00𝑎<(𝑁0+1)0 and𝐸=𝑡(𝑟),𝑥(𝑚)𝐸0𝑟𝑁01.(2.4)For functions 𝑤𝐷𝑘 and 𝑧Ω𝑘 we write 𝑤(𝑟,𝑚)=𝑤(𝑡(𝑟),𝑥(𝑚)) on 𝐷 and 𝑧(𝑟,𝑚)=𝑧(𝑡(𝑟),𝑥(𝑚)) on Ω. We need a discrete version of the operator (𝑡,𝑥)𝑧(𝑡,𝑥). For a function 𝑧Ω𝑘 and for a point (𝑡(𝑟),𝑥(𝑚))𝐸 we define a function 𝑧[𝑟,𝑚]𝐷𝑘 by𝑧[𝑟,𝑚]𝑡(𝜏,𝑦)=𝑧(𝑟)+𝜏,𝑥(𝑚)+𝑦,(𝜏,𝑦)𝐷.(2.5)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 𝑇𝐹(𝐷,𝑘)𝐶(𝐷,𝑘). We define 𝑇 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.(2.6)Set 𝜐=(𝜐1,,𝜐𝑛)𝑛 and 𝜐𝑖=0 if 𝑑𝑖=0, 𝜐𝑖=1 if 𝑑𝑖>0 where 1𝑖𝑛. WriteΔ+=𝜆𝜆=1,,𝜆𝑛𝜆𝑖𝜗𝑖for1𝑖𝑛.(2.7)Set 𝑒𝑖=(0,,0,1,0,,0)𝑛 with 1 standing on the 𝑖th place.

Let 𝑤𝐹(𝐷,𝑘) and (𝑡,𝑥)𝐷. Suppose that 𝑑0>0. There exists (𝑡(𝑟),𝑥(𝑚))𝐷 such that (𝑡(𝑟+1),𝑥(𝑚+𝜐))𝐷 and 𝑡(𝑟)𝑡𝑡(𝑟+1), 𝑥(𝑚)𝑥𝑥(𝑚+𝜐). Write𝑇[𝑤](𝑡,𝑥)=1𝑡𝑡(𝑟)0𝜆Δ+𝑤(𝑟,𝑚+𝜆)𝑥𝑥(𝑚)𝜆1𝑥𝑥(𝑚)1𝜆+𝑡𝑡(𝑟)0𝜆Δ+𝑤(𝑟+1,𝑚+𝜆)𝑥𝑥(𝑚)𝜆1𝑥𝑥(𝑚)1𝜆,(2.8) where𝑥𝑥(𝑚)𝜆=𝑛𝑖=1𝑥𝑖𝑥(𝑚𝑖)𝑖𝑖𝜆𝑖,1𝑥𝑥(𝑚)1𝜆=𝑛𝑖=1𝑥1𝑖𝑥(𝑚𝑖)𝑖𝑖1𝜆𝑖(2.9) and we take 00=1 in the above formulas. If 𝑑0=0 then we put𝑇[𝑤](𝑡,𝑥)=𝜆Δ+𝑤(𝑟,𝑚+𝜆)𝑥𝑥(𝑚)𝜆1𝑥𝑥(𝑚)1𝜆.(2.10)Then we have defined 𝑇[𝑤] on 𝐷. It is easy to see that 𝑇[𝑤]𝐶(𝐷,𝑘). The above interpolating operator has been first proposed in [10, Chapter 5].

For 𝑤,𝑤𝐹(𝐷,𝑘) we write 𝑤𝑤 if 𝑤(𝑟,𝑚)𝑤(𝑟,𝑚) where (𝑡(𝑟),𝑥(𝑚))𝐷. In a similar way we define the relation 𝑤𝑤 for 𝑤,𝑤𝐶(𝐷,𝑘) and the relation 𝑧𝑧 for 𝑧,𝑧𝐹(Ω,𝑘) and for 𝑧,𝑧𝐶(Ω,𝑘).

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

Assumption (𝐻[𝑓]). The function 𝑓=(𝑓1,,𝑓𝑘)Σ𝑘 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 is ̃𝑥(𝑏,𝑏), ̃𝑥=(𝑥1,,𝑥𝑛), such that(𝑥̃𝑥)𝜕𝑞𝑓𝑖(𝑡,𝑥,𝑤,𝑞)𝜃onΣfor𝑖=1,,𝑘,(2.11)(3)there is 𝜀0>0 such that for 0<0<𝜀0 and 𝑤,𝑤𝐶(𝐷,𝑘), 𝑤𝑤, we have𝑤(0,𝜃)+0𝑓(𝑡,𝑥,𝑤,𝑞)𝑤(0,𝜃)+0𝑓(𝑡,𝑥,𝑤,𝑞),(𝑡,𝑥,𝑞)𝐸×𝑛.(2.12)

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 𝑔𝑖𝑔[𝑧,𝑢](,𝑡,𝑥)=𝑖.1[𝑧,𝑢](,𝑡,𝑥),,𝑔𝑖.𝑛[𝑧,𝑢](,𝑡,𝑥)(2.13)the 𝑖th bicharacteristic of (1.4) corresponding to (𝑧,𝑢). Then 𝑔𝑖[𝑧,𝑢](,𝑡,𝑥) is a solution of the Cauchy problem𝑦(𝜏)=𝜕𝑞𝑓𝑖𝜏,𝑦(𝜏),𝑧(𝜏,𝑦(𝜏)),𝑢𝜏,𝑦(𝜏),𝑦(𝑡)=𝑥.(2.14)Assumption (2.11) states that the bicharacteristics satisfy the following monotonicity conditions: If 𝑥𝑗𝑥𝑗0 the function 𝑔𝑖𝑗[𝑧,𝑢](,𝑡,𝑥) is non increasing. If 𝑥𝑗𝑥𝑗<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 function 𝑓𝑓=(1𝑓,,𝑘)𝐸××𝐶(𝐷,𝑘)×𝑛𝑘 of the variables (𝑡,𝑥,𝑝,𝑤,𝑞). Write 𝑓𝑖𝑓(𝑡,𝑥,𝑤,𝑞)=𝑖(𝑡,𝑥,𝑤𝑖(0,𝜃),𝑤,𝑞), 𝑖=1,,𝑘, on Σ. Then system (1.4) is equivalent to 𝜕𝑡𝑧𝑖𝑓(𝑡,𝑥)=𝑖𝑡,𝑥,𝑧𝑖(𝑡,𝑥),𝑧(𝑡,𝑥),𝜕𝑥𝑧𝑖(𝑡,𝑥),𝑖=1,,𝑘.(2.15)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=(𝜕𝑝𝑓1,,𝜕𝑝𝑓𝑘) and 𝜕𝑝𝑓𝑖(𝑡,𝑥,𝑝,𝑤,𝑞)𝐿 for 𝑖=1,,𝑘 and 1+𝐿00. Then the monotonicity condition (3) of Assumption (𝐻[𝑓])  is satisfied.

Let us denote by 𝐻 the set of all =(0,)𝐻 such that𝑖𝑏<min𝑖𝑥𝑖,𝑥𝑖+𝑏𝑖,𝑖=1,,𝑛.(2.16)Suppose that 𝜔Ω. We apply difference operators 𝛿=(𝛿1,,𝛿𝑛) given byif𝑥𝑗𝑥(𝑚𝑗)𝑗<𝑏𝑗then𝛿𝑗𝜔(𝑟,𝑚)=1𝑗𝜔(𝑟,𝑚+𝑒𝑗)𝜔(𝑟,𝑚),if𝑏𝑗<𝑥(𝑚𝑗)𝑗<𝑥𝑗then𝛿𝑗𝜔(𝑟,𝑚)=1𝑗𝜔(𝑟,𝑚)𝜔(𝑟,𝑚𝑒𝑗),(2.17)and we put 𝑗=1,,𝑛 in (2.17). Let 𝛿0 be defined by𝛿0𝜔(𝑟,𝑚)=10𝜔(𝑟+1,𝑚)𝜔(𝑟,𝑚)(2.18)and 𝛿0𝑧=(𝛿0𝑧1,,𝛿0𝑧𝑘). Write𝔽[𝑧](𝑟,𝑚)=𝑓1𝑡(𝑟),𝑥(𝑚),𝑇𝑧[𝑟,𝑚],𝛿𝑧1(𝑟+1,𝑚),,𝑓𝑘𝑡(𝑟),𝑥(𝑚),𝑇𝑧[𝑟,𝑚],𝛿𝑧𝑘(𝑟+1,𝑚).(2.19) Given 𝜑𝐸0.𝜕0𝐸𝑘, we consider the functional difference equation𝛿0𝑧(𝑟,𝑚)=𝔽[𝑧](𝑟,𝑚)(2.20)with the initial boundary condition𝑧(𝑟,𝑚)=𝜑(𝑟,𝑚)on𝐸0.𝜕0𝐸.(2.21)

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 𝑇𝑧[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, 0<𝜀0 and the functions 𝑢,𝑣Ω𝑘 satisfy the difference functional inequality𝛿0𝑢(𝑟,𝑚)𝔽[𝑢](𝑟,𝑚)𝛿0𝑣(𝑟,𝑚)𝔽[𝑣](𝑟,𝑚)on𝐸,(2.22)(2)the initial boundary estimate 𝑢(𝑟,𝑚)𝑣(𝑟,𝑚) holds on 𝐸0.𝜕0𝐸. Then 𝑢(𝑟,𝑚)𝑣(𝑟,𝑚)on𝐸.(2.23)

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))𝐸. Assume that 𝑢(𝑗,𝑚)𝑣(𝑗,𝑚) for (𝑡(𝑗),𝑥(𝑚))𝐸([0,𝑡(𝑟)]×𝑛). We prove that 𝑢(𝑟+1,𝑚)𝑣(𝑟+1,𝑚) for (𝑡(𝑟+1,𝑚),𝑥(𝑚))𝐸. Write𝑈𝑖(𝑟,𝑚)=𝑢𝑖(𝑟,𝑚)+0𝑓𝑖𝑡(𝑟),𝑥(𝑚),𝑇𝑢[𝑟,𝑚],𝛿𝑢𝑖(𝑟+1,𝑚)𝑣𝑖(𝑟,𝑚)0𝑓𝑖𝑡(𝑟),𝑥(𝑚),𝑇𝑣[𝑟,𝑚],𝛿𝑢𝑖(𝑟+1,𝑚),𝑖=1,,𝑘.(2.24) It follows from (2.22) that𝑢𝑖𝑣𝑖(𝑟+1,𝑚)𝑈𝑖(𝑟,𝑚)+0𝑓𝑖𝑡(𝑟),𝑥(𝑚),𝑇𝑣[𝑟,𝑚],𝛿𝑢𝑖(𝑟+1,𝑚)𝑓𝑖𝑡(𝑟),𝑥(𝑚),𝑇𝑣[𝑟,𝑚],𝛿𝑣𝑖(𝑟+1,𝑚),(2.25) where 𝑖=1,,𝑘. The monotonicity condition (3) of Assumption (𝐻[𝑓])  implies the inequalities 𝑈𝑖(𝑟,𝑚)0 for (𝑡(𝑟),𝑥(𝑚))𝐸, 𝑖=1,,𝑘. Then we have𝑢𝑖𝑣𝑖(𝑟+1,𝑚)0𝑛𝑗=110𝜕𝑞𝑗𝑓𝑖𝑄𝑖(𝑟,𝑚)[𝑣,𝜏]𝑑𝜏𝛿𝑗𝑢𝑖𝑣𝑖(𝑟+1,𝑚),(2.26)where 𝑖=1,,𝑘 and𝑄𝑖(𝑟,𝑚)𝑡[𝑣,𝜏]=(𝑟),𝑥(𝑚),𝑇𝑣[𝑟,𝑚],𝛿𝑣𝑖(𝑟+1,𝑚)𝑢+𝜏𝛿𝑖𝑣𝑖(𝑟+1,𝑚).(2.27)WriteΓ+(𝑚)=𝑗{1,,𝑛}𝑥(𝑚𝑗)𝑗𝑥𝑗,𝑏𝑗,Γ(𝑚)={1,,𝑛}Γ+(𝑚).(2.28)It follows from (2.11), (2.17) that𝑢𝑖𝑣𝑖(𝑟+1,𝑚)1+0𝑛𝑗=11𝑗10||𝜕𝑞𝑗𝑓𝑖𝑄𝑖(𝑟,𝑚)[𝑣,𝜏]|𝑑𝜏0𝑗Γ+(𝑚)1𝑗10𝜕𝑞𝑗𝑓𝑖𝑄𝑖(𝑟,𝑚)𝑢[𝑣,𝜏]𝑑𝜏𝑖𝑣𝑖(𝑟+1,𝑚+𝑒𝑗)0𝑗Γ(𝑚)1𝑗10𝜕𝑞𝑗𝑓𝑖𝑄𝑖(𝑟,𝑚)𝑢[𝑣,𝜏]𝑑𝜏𝑖𝑣𝑖(𝑟+1,𝑚𝑒𝑗),𝑖=1,,𝑘.(2.29) We define 𝑚𝑛 and 𝜇, 1𝜇𝑘, as follows:𝑢𝜇𝑣𝜇(𝑟+1,𝑚)=max1𝑖𝑘𝑢max𝑖𝑣𝑖(𝑟+1,𝑚)𝑡(𝑟+1),𝑥(𝑚)Ω.(2.30)If (𝑡(𝑟+1),𝑥(𝑚))𝜕0𝐸 then assumption (2) implies that (𝑢𝜇𝑣𝜇)(𝑟+1,𝑚)0. Let us consider the case when (𝑡(𝑟+1),𝑥(𝑚))𝐸. Then we have from (2.29) that𝑢𝜇𝑣𝜇(𝑟+1,𝑚)1+0𝑛𝑗=11𝑗10||𝜕𝑞𝑗𝑓𝑖(𝑄𝑖(𝑟,𝑚)||[𝑣,𝜏])𝑑𝜏0𝑢𝜇𝑣𝜇(𝑟+1,𝑚)𝑗Γ+(𝑚)1𝑗10𝜕𝑞𝑗𝑓𝑖𝑄𝑖(𝑟,𝑚)[𝑣,𝜏]𝑑𝜏𝑗Γ(𝑚)1𝑗10𝜕𝑞𝑗𝑓𝑖𝑄(𝑟,𝑚𝑖.[𝑣,𝜏]𝑑𝜏(2.31) 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:𝑁11,,𝑁𝑛𝑛<𝑏1,,𝑏𝑛𝑁1+11𝑁,,𝑛+1𝑛(3.1)and we assume that (𝑁𝑖+1)𝑖=𝑏𝑖 if 𝑑𝑖=0. For 𝑤𝐶(𝐷,𝑘) we write𝑤𝐷=max𝑤(𝑡,𝑥)(𝑡,𝑥)𝐷.(3.2)In a similar way we define the norm in the space 𝐹(𝐷,𝑘) : if 𝑤𝐷𝑘 then𝑤𝐷𝑤=max(𝑟,𝑚)𝑡(𝑟),𝑥(𝑚)𝐷.(3.3)The following properties of the operator 𝑇 are important in our considerations.

Lemma 3.1. Suppose that 𝑤𝐷𝑘 is of class 𝐶1 and 𝑤 is the restriction of 𝑤 to the set 𝐷. Let 𝐶 be such a constant that 𝜕𝑡𝑤𝐷, 𝜕𝑥𝑖𝑤𝐷𝐶 for 1𝑖𝑛. Then 𝑇[𝑤]𝑤𝐷𝐶 where =0+1++𝑛.

Lemma 3.2. Suppose that 𝑤𝐷𝑘 is of class 𝐶2 and 𝑤 is the restriction of 𝑤 to the set 𝐷. Let 𝐶 be such a constant that 𝜕𝑡𝑡𝑤𝐷, 𝜕𝑡𝑥𝑖𝑤𝐷, 𝜕𝑥𝑖𝑥𝑗𝑤𝐷𝐶,   𝑖,𝑗=1,,𝑛. Then 𝑇[𝑤]𝑤𝐷𝐶2.

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 𝜑𝐹(𝐸0.𝜕0𝐸,𝑘) then there exists exactly one solution 𝑢=(𝑢.1,,𝑢.𝑘)Ω𝑘 of difference functional problem (2.20), (2.21).

Proof. Suppose that 0𝑟𝑁01 is fixed and that the solution 𝑢 of problem (2.20), (2.21) is given on the set Ω([𝑑0,𝑡(𝑟)]×𝑛). We prove that the vectors 𝑢(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 equations10𝑧𝑖(𝑟+1,𝑚)𝑢(𝑟,𝑚).𝑖=𝑓𝑖𝑡(𝑟),𝑥(𝑚),𝑇(𝑢)[𝑟,𝑚],𝛿𝑧𝑖(𝑟+1,𝑚),(3.4)where 𝑁𝑚𝑁,𝑖=1,,𝑘, with the initial boundary condition (2.21). There exists 𝑄>0 such that𝑄0𝑗Γ+(𝑚)1𝑗𝜕𝑞𝑗𝑓𝑖𝑡(𝑟),𝑥(𝑚),𝑇𝑢[𝑟,𝑚],𝑞𝑗Γ(𝑚)1𝑗𝜕𝑞𝑗𝑓𝑖𝑡(𝑟),𝑥(𝑚),𝑇𝑢[𝑟,𝑚],𝑞,(3.5) where 𝑁𝑚𝑁,𝑖=1,,𝑘. It is clear that system (3.4) is equivalent to the following one:𝑧𝑖(𝑟+1,𝑚)=1𝑄𝑄+1𝑧𝑖(𝑟+1,𝑚)+𝑢(𝑟,𝑚).𝑖+0𝑓𝑖𝑡(𝑟),𝑥(𝑚),𝑇𝑢[𝑟,𝑚],𝛿𝑧𝑖(𝑟+1,𝑚),𝑁𝑚𝑁,𝑖=1,,𝑘(3.6) Write 𝑆={𝑥(𝑚)𝑥(𝑚)[𝑐,𝑐]}. Elements of the space 𝐹(𝑆,𝑘) are denoted by 𝜉, 𝜉. For 𝜉𝑆𝑘, 𝜉=(𝜉1,,𝜉𝑘), we write 𝜉(𝑚)=𝜉(𝑥(𝑚)) and𝛿𝜉𝑖(𝑚)=𝛿1𝜉𝑖(𝑚),,𝛿𝑛𝜉𝑖(𝑚)𝛿,1𝑖𝑘,𝑗𝜉𝑖(𝑚)=1𝑗𝜉(𝑚+𝑒𝑗𝑖𝑗)𝜉𝑖(𝑚)if𝑥(𝑚𝑗)𝑗𝑥𝑗,𝑏𝑗,𝛿𝑗𝜉𝑖(𝑚)=1𝑗𝜉𝑖(𝑚)𝜉(𝑚𝑒𝑗)𝑖if𝑥(𝑚𝑗)𝑗𝑏𝑗,𝑥𝑗,(3.7)where 𝑗=1,,𝑛. The norm in the space 𝐹(𝑆,𝑘) is defined by𝜉𝜉=max(𝑚)𝑥(𝑚)𝑆.(3.8)Let us consider the set𝑋=𝑆𝜉𝐹,𝑘𝜉(𝑚)=𝜑(𝑟+1,𝑚)for𝑥(𝑚)[𝑐,𝑐](𝑏,𝑏}.(3.9)We consider the operator 𝑊𝑋𝑋,  𝑊=(𝑊.1,,𝑊.𝑛) defined by𝑊.𝑖[𝜉](𝑚)=1𝑄𝑄+1𝜉𝑖(𝑚)+𝑢(𝑟,𝑚).𝑖+0𝑓𝑖𝑡(𝑟),𝑥(𝑚)𝑢,𝑇[𝑟,𝑚],𝛿𝜉𝑖(𝑚),(3.10)where 𝑁𝑚𝑁,  𝑖=1,,𝑘 and𝑊[𝜉](𝑚)=𝜑(𝑟+1,𝑚)for𝑥(𝑚)[𝑐,𝑐](𝑏,𝑏),(3.11)where 𝜉=(𝜉1,,𝜉𝑘)𝐹(𝑆,𝑘). We prove that𝑊[𝜉]𝑊[𝜉]𝑄𝑄+1𝜉𝜉𝑆on𝐹,𝑘.(3.12)
It follows from (3.10) that we have for 𝑁𝑚𝑁:𝑊.𝑖[𝜉](𝑚)𝑊.𝑖[𝜉](𝑚)=1𝑄𝑄+1𝜉𝑖𝜉𝑖(𝑚)0𝑗Γ+(𝑚)1𝑗10𝜕𝑞𝑗𝑓𝑖𝑃𝑖(𝑟,𝑚)[𝑢𝜉,𝜏]𝑑𝜏𝑖𝜉𝑖(𝑚)+𝑗Γ(𝑚)1𝑗10𝜕𝑞𝑗𝑓𝑖𝑃𝑖(𝑟,𝑚)[𝑢,𝜏]𝑑𝜏(𝜉𝑖𝜉𝑖)(𝑚)+0𝑗Γ+(𝑚)1𝑗10𝜕𝑞𝑗𝑓𝑖𝑃𝑖(𝑟,𝑚)[𝑢,𝜏]𝑑𝜏(𝜉𝑖𝜉𝑖)(𝑚+𝑒𝑗)0𝑗Γ(𝑚)1𝑗10𝜕𝑞𝑗𝑓𝑖𝑃𝑖(𝑟,𝑚)[𝑢,𝜏]𝑑𝜏(𝜉𝑖𝜉𝑖)(𝑚𝑒𝑗),(3.13) where 𝑖=1,,𝑘 and𝑃𝑖(𝑟,𝑚)𝑢=𝑡,𝜏(𝑟),𝑥(𝑚),𝑇𝑢[𝑟,𝑚],𝛿𝜉𝑖(𝑚)𝜉+𝜏𝛿𝑖𝜉𝑖(𝑚).(3.14)It follows from the above relations and from (3.5) that|||𝑊.𝑖[𝜉](𝑚)𝑊.𝑖[𝜉](𝑚)|||𝑄𝑄+1𝜉𝜉for𝑁𝑚𝑁,𝑖=1,,𝑘.(3.15)According to (3.12) we have𝑊.𝑖[𝜉](𝑚)𝑊.𝑖[𝜉](𝑚)=0for𝑥(𝑚)[𝑐,𝑐](𝑏,𝑏),𝑖=1,,𝑘.(3.16)This completes the proof of (3.12).
It follows from the Banach fixed point theorem that there exists exactly one solution 𝜉𝑆𝑘 of the equation 𝜉=𝑊[𝜉] and consequently, there exists exactly one solution of (3.6), (2.21). Then the vectors 𝑢(𝑟+1,𝑚), 𝑁𝑚𝑁, exist and they are unique. Then the proof is completed by induction with respect to 𝑟, 0𝑟𝑁0.

Assumption (𝐻[𝜎]). The function 𝜎[0,𝑎]×++ 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 problem𝜂(𝑡)=𝜎(𝑡,𝜂(𝑡)),𝜂(0)=0,(3.17)is ̃𝜂(𝑡)=0 for 𝑡[0,𝑎].

Assumption (𝐻[𝑓,𝜎]). 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)𝜎𝑡,𝑤𝑤𝐷,𝑖=1,,𝑘,(3.18)where (𝑡,𝑥,𝑞)𝐸×𝑛.

Theorem 3.4. Suppose that Assumptions (𝐻[𝑓])  and   (𝐻[𝑓,𝜎])  are satisfied and
(1)𝑣Ω is a solution of (1.4), (1.5) and 𝑣 is of class 𝐶1 on Ω,(2)𝐻, 0<𝜀 and 𝜑𝐸0.𝜕0𝐸𝑘 and there is 𝛼0𝐻+ such that𝜑(𝑟,𝑚)𝜑(𝑟,𝑚)𝛼0()on𝐸0.𝜕0𝐸,lim0𝛼0()=0.(3.19) Under these assumptions there is a solution 𝑢Ω𝑘 of (2.20), (2.21) and there is 𝛼𝐻+ such that 𝑢𝑣(𝑟,𝑚)𝛼()on𝐸,lim0𝛼()=0,(3.20)where 𝑣 is the restriction of 𝑣 to the set Ω.

Proof. The existence of 𝑢 follows from Theorem 3.3. Let Γ𝐸𝑘, Γ0.𝐸0.𝜕0𝐸𝑘 be defined by the relations𝛿0𝑣(𝑟,𝑚)=𝔽𝑣(𝑟,𝑚)+Γ(𝑟,𝑚)on𝐸,𝑣(3.21)(𝑟,𝑚)=𝜑(𝑟+1,𝑚)+Γ(𝑟,𝑚)0.𝑡for(𝑟),𝑥(𝑚)𝐸0.𝜕0𝐸.(3.22)From Lemma 3.1 and from assumption (1) of the theorem it follows that there are 𝛾,𝛾0𝐻+ such thatΓ(𝑟,𝑚)𝛾()on𝐸,Γ(𝑟,𝑚)0.𝛾0()on𝐸0.𝜕0𝐸(3.23)and lim0𝛾()=0, lim0𝛾0()=0. Write 𝐽=[0,𝑎] and 𝐽={𝑡(𝑟)0𝑟𝑁0}. For 𝛽𝐽 we put 𝛽(𝑟)=𝛽(𝑡(𝑟)). Let 𝛽𝐽+ be a solution of the difference problem𝛽(𝑟+1)=𝛽(𝑟)+0𝜎𝑡(𝑟),𝛽(𝑟)+0𝛾(),0𝑟𝑁01,𝛽(0)=𝛼0().(3.24)We prove that𝑢𝑣(𝑟,𝑚)𝛽(𝑟)on𝐸.(3.25)Let 𝑣𝑣=(.1𝑣,,.𝑘)Ω𝑘 be defined by𝑣(𝑟,𝑚).𝑖=𝑣(𝑟,𝑚).𝑖+𝛽(0)on𝐸0.,𝑣(𝑟,𝑚).𝑖=𝑣(𝑟,𝑚).𝑖+𝛽(𝑖)on𝐸𝜕0𝐸,(3.26) where 𝑖=1,,𝑘. We prove that the difference functional inequality𝛿0𝑣𝔽𝑣(𝑟,𝑚),𝑡(𝑟),𝑥(𝑚)𝐸,(3.27)is satisfied. It follows from Assumption (𝐻[𝑓,𝜎]) and from (3.21) that𝛿0𝑣(𝑟,𝑚).𝑖=𝛿0𝑣(𝑟,𝑚).𝑖+10𝛽(𝑟+1)𝛽(𝑟)=𝑓𝑖𝑡(𝑟),𝑥(𝑚),𝑇𝑣[𝑟,𝑚]𝑣,𝛿(𝑟+1,𝑚).𝑖+10𝛽(𝑟+1)𝛽(𝑟)+𝑓𝑖𝑡(𝑟),𝑥(𝑚),𝑇𝑣[𝑟,𝑚],𝛿𝑣(𝑟+1,𝑚).𝑖𝑓𝑖𝑡(𝑟),𝑥(𝑚),𝑇𝑣[𝑟,𝑚],𝛿𝑣(𝑟+1,𝑚).𝑖𝑓𝑖𝑡(𝑟),𝑥(𝑚),𝑇𝑣[𝑟,𝑚]𝑣,𝛿(𝑟+1,𝑚).𝑖𝑡𝜎(𝑟),𝛽(𝑟)+10𝛽(𝑟+1)𝛽(𝑟)=𝑓𝑖(𝑡(𝑟),𝑥(𝑚),𝑇(𝑣)[𝑟,𝑚]𝑣,𝛿(𝑟+1,𝑚).𝑖),𝑖=1,,𝑘.(3.28) This completes the proof of (3.27).
Since 𝑣(𝑟,𝑚)𝑣(𝑟,𝑚) on 𝐸0.𝜕0𝐸, it follows from Theorem 2.3 that 𝑢(𝑟,𝑚)𝑣(𝑟,𝑚)+𝛽(𝑟) on 𝐸. In a similar way we prove that 𝑣(𝑟,𝑚)𝛽(𝑟)𝑢(𝑟,𝑚) on 𝐸. The above estimates imply (3.25). Consider the Cauchy problem𝜂(𝑡)=𝜎(𝑡,𝜂(𝑡))+𝛾(),𝜂(0)=𝛼0().(3.29)It follows from Assumption (𝐻[𝜎]) that there is ̃𝜀>0 such that for ̃𝜀  the maximal solution 𝜂(,) of (3.29) is defined on [0,𝑎] andlim0𝜂(𝑡,)=0uniformlyon[0,𝑎].(3.30)Since 𝜂(,) is convex function then we have the difference inequality𝜂𝑡(𝑟+1)𝑡,𝜂(𝑟),+0𝜎𝑡(𝑟)𝑡,𝜂(𝑟),+0𝛾(),(3.31)where 0𝑟𝑁01. Since 𝛽 satisfies (3.24), the above relations imply the estimate𝛽(𝑟)𝑡𝜂(𝑟),𝜂(𝑎,),0𝑟𝑁0.(3.32)It follows from (3.30) that condition (3.20) is satisfied with 𝛼()=𝜂(𝑎,). This completes the proof.

Lemma 3.5. Suppose that Assumption (𝐻[𝑓]) is satisfied and
(1)𝑣Ω is a solution of (1.4), (1.5) and 𝑣 is of class 𝐶2 on Ω,(2)𝐻, 0<𝜀 and 𝜑𝐸0.𝜕0𝐸𝑘 and there is 𝛼0𝐻+ such that𝜑(𝑟,𝑚)𝜑(𝑟,𝑚)𝛼0()on𝐸0.𝜕0𝐸,lim0𝛼0()=0.(3.33)(3)there exists 𝐿+ such that estimates𝑓𝑖(𝑡,𝑥,𝑤,𝑞)𝑓𝑖(𝑡,𝑥,𝑤,𝑞)𝐿𝑤𝑤𝐷,𝑖=1,,𝑘,(3.34)are satisfied for (𝑡,𝑥,𝑞)𝐸×𝑛, 𝑤,𝑤𝐶(𝐷,𝑘) and 𝑤𝑤,(4)there is 𝐶+ such that𝜕𝑞𝑓𝑖(𝑡,𝑥,𝑤,𝑞)𝐶onΣfor𝑖=1,,𝑘.(3.35) Under these assumptions there is a solution 𝑢Ω𝑘 of (2.20), (2.21), and 𝑢𝑣(𝑟,𝑚)𝛼()on𝐸,(3.36)where 𝛼()=𝛼0()𝑒𝐿𝑎𝑒+̃𝛾()𝐿𝑎1𝐿if𝐿>0,𝛼()=𝛼0()+𝑎̃𝛾()if𝐿=0,̃𝛾()=0.5𝐶0(1+𝐶)+𝐿𝐶2+0.5𝐶𝐶(3.37)and 𝐶+ is such that 𝜕𝑡𝑡𝑣(𝑡,𝑥),𝜕𝑡𝑥𝑖𝑣(𝑡,𝑥),𝜕𝑥𝑖𝑥𝑗𝑣(𝑡,𝑥)𝐶(3.38)on Ω for 𝑖,𝑖=1,,𝑛.

Proof. It follows that the solution 𝛽𝐽+ of the difference problem𝛽(𝑟+1)=1+𝐿0𝛽(𝑟)+0𝛾(),0𝑟𝑁0𝛽1,(0)=𝛼0()(3.39) satisfies the condition: 𝛽(𝑟)𝛼() for 0𝑟𝑁0. Moreover we haveΓ(𝑟,𝑚)̃𝛾()on𝐸,(3.40)where Γ 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 Examples

Example 4.1. For 𝑛=2 we put 𝐸=[0,0.5]×[1,1]×[1,1],𝐸0={0}×[1,1]×[1,1].(4.1)Consider the differential integral equation𝜕𝑡𝑧(𝑡,𝑥,𝑦)=arctan2𝑥𝜕𝑥𝑧(𝑡,𝑥,𝑦)+2𝑦𝜕𝑦𝑧(𝑡,𝑥,𝑦)𝑡2𝑥2𝑦2𝑥2𝑦2𝑧(𝑡,𝑥,𝑦)+𝑡1𝑦2𝑥1𝑠𝑧(𝑡,𝑠,𝑦)𝑑𝑠+𝑡1𝑥2𝑦1𝑥𝑠𝑧(𝑡,𝑥,𝑠)𝑑𝑠+𝑧(𝑡,𝑥,𝑦)4+0.252𝑦1214(4.2) 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].(4.3)

The function 𝑣(𝑡,𝑥,𝑦)=exp[0.25𝑡(𝑥21)(𝑦21)] is the solution of the above problem. Let us denote by 𝑧 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) and𝜀(𝑟)=12𝑁112𝑁21𝑚Π||𝑧(𝑟,𝑚)𝑣(𝑟,𝑚)||,0𝑟𝑁0,(4.4)where𝑚Π=𝑚=1,𝑚22𝑁1+1𝑚1𝑁11,𝑁1+1𝑚2𝑁21(4.5)and 𝑁11=1, 𝑁22=1, 𝑁00=0.5. The numbers 𝜀(𝑟) can be called average errors of the difference method for fixed 𝑡(𝑟). We put 0=1=2=0.005 and we have the values of the above defined errors which are shown in Table 1.

tab1
Table 1: Table of errors.

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].(4.6)Consider the differential equation with deviated variables𝜕𝑡𝑧(𝑡,𝑥,𝑦)=2𝑥𝜕𝑥𝑧(𝑡,𝑥,𝑦)+2𝑦𝜕𝑦𝑧(𝑡,𝑥,𝑦)+cos2𝑥𝜕𝑥𝑧(𝑡,𝑥,𝑦)2𝑦𝜕𝑦𝑥𝑧(𝑡,𝑥,𝑦)𝑡2𝑦2+𝑧(𝑡,𝑥,𝑦)𝑧𝑡2,𝑥,𝑦+𝑓(𝑡,𝑥,𝑦)𝑧(𝑡,𝑥,𝑦)1,(4.7) 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],(4.8) where𝑥𝑓(𝑡,𝑥,𝑦)=20.250.25𝑦2+𝑡8𝑥2𝑦2𝑥2𝑦2exp0.5𝑡2𝑥𝑡20.250.25𝑦2.(4.9)

The function 𝑣(𝑡,𝑥,𝑦)=exp[[𝑡(𝑥20.25)(0.25𝑦2)] is the solution of the above problem. Let us denote by 𝑧 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 𝜀 be defined by (4.4) with 𝑁11=0.5, 𝑁22=0.5, 𝑁00=0.5. We put 0=1=2=0.005 and we have the values of the above defined errors which are shown in Table 2.

tab2
Table 2: Table of errors.

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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