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

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 [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 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,…,𝑏_𝑛)∈ℝ^𝑛, 𝑏_𝑖>0 for 1≤𝑖≤𝑛 and 𝑑=(𝑑₁,…,𝑑_𝑛)∈ℝ^𝑛+, 𝑑₀∈ℝ₊, ℝ₊=[0,+∞). Let 𝑐=𝑏+𝑑 and𝜕𝐸₀=−𝑑₀,0×[−𝑐,𝑐],₀𝐸=Ω[0,=𝑎]𝐸×∪([𝐸−₀𝑐,𝑐]⧵(−𝑏,𝑏)),∪𝜕₀𝐸.(1.2)For a function 𝑧∶Ω→ℝ^𝑘, 𝑧=(𝑧₁,…,𝑧_𝑘), and for a point (𝑡,𝑥)∈𝐸 where 𝐸 is the closure of 𝐸, we define a function 𝑧_(𝑡,𝑥)∶𝐷→ℝ^𝑘 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,ℝ^𝑘)×ℝ^𝑛 and suppose that 𝑓=(𝑓₁,…,𝑓_𝑘)∶Σ→ℝ^𝑘 and 𝜑∶𝐸₀∪𝜕₀𝐸→ℝ^𝑘, 𝜑=(𝜑₁,…,𝜑_𝑘), are given functions. Let us denote by 𝑧=(𝑧₁,…,𝑧_𝑘) an unknown function of the variables (𝑡,𝑥), 𝑥=(𝑥₁,…,𝑥_𝑛). Write𝔽[𝑧](𝑡,𝑥)=𝑓₁𝑡,𝑥,𝑧_(𝑡,𝑥),𝜕_𝑥𝑧₁(𝑡,𝑥),…,𝑓_𝑘𝑡,𝑥,𝑧_(𝑡,𝑥),𝜕_𝑥𝑧_𝑘(𝑡,𝑥)(1.3)and 𝜕_𝑥𝑧_𝑖=(𝜕_𝑥1𝑧_𝑖,…,𝜕_𝑥𝑛𝑧_𝑖), 1≤𝑖≤𝑘. 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 𝛼∶𝐸→ℝ^1+𝑛 satisfies the condition: 𝛼(𝑡,𝑥)−(𝑡,𝑥)∈𝐷 for (𝑡,𝑥)∈𝐸. For a given 𝑓=(𝑓₁,…,𝑓_𝑘)∶𝐸×ℝ^𝑘×ℝ^𝑘×ℝ^𝑛→ℝ^𝑘 we put 𝑓(𝑡,𝑥,𝑤,𝑞)=𝑓𝑡,𝑥,𝑤(0,𝜃),𝑤𝛼(𝑡,𝑥)−(𝑡,𝑥),𝑞onΣ,(1.6)where 𝜃=(0,…,0)∈ℝ^𝑛. Then (1.4) is reduced to the system of differential equations with deviated variables𝜕_𝑡𝑧_𝑖(𝑡,𝑥)=𝑓_𝑖𝑡,𝑥,𝑧(𝑡,𝑥),𝑧𝛼(𝑡,𝑥),𝜕_𝑥𝑧_𝑖(𝑡,𝑥),𝑖=1,…,𝑘.(1.7)

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.(1.8)Then (1.4) is equivalent to the system of differential integral equations𝜕_𝑡𝑧_𝑖(𝑡,𝑥)=𝑓_𝑖𝑡,𝑥,𝑧(𝑡,𝑥),_𝐷𝑧(𝑡+𝜏,𝑥+𝑦)𝑑𝑦𝑑𝜏,𝜕_𝑥𝑧_𝑖(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,…,𝑞_𝑛), is of class 𝐶¹ with respect to 𝑞 and the functions 𝜕_𝑞𝑓_𝑖=(𝜕_𝑞1𝑓_𝑖,…,𝜕_𝑞𝑛𝑓_𝑖), 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 condition1−ℎ_0𝑛_𝑗=11ℎ_𝑗||𝜕_𝑞𝑗𝑓_𝑖(𝑡,𝑥,𝑤,𝑞)||≥0onΣfor𝑖=1,…,𝑘,(1.10)where ℎ₀ and ℎ′=(ℎ₁,…,ℎ_𝑛) are steps of the mesh with respect to 𝑡 and (𝑥₁,…,𝑥_𝑛) 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 [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 sets 𝑈 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,…,𝑥_𝑛)∈ℝ^𝑛, 𝑝=(𝑝₁,…,𝑝_𝑘)∈ℝ^𝑘 we put‖𝑥‖=||𝑥₁||+⋯+||𝑥_𝑛||,‖𝑝‖_∞=max||𝑝_𝑖||∶1≤𝑖≤𝑘.(2.1)We define a mesh on Ω in the following way. Suppose that (ℎ₀,ℎ^), ℎ′=(ℎ₁,…,ℎ_𝑛), stand for steps of the mesh. For (𝑟,𝑚)∈ℤ^1+𝑛 where 𝑚=(𝑚₁,…,𝑚_𝑛), we define nodal points as follows:𝑡^(𝑟)=𝑟ℎ₀,𝑥^(𝑚)=𝑥^(𝑚1)1,…,𝑥^{(𝑚𝑛)𝑛}=𝑚₁ℎ₁,…,𝑚_𝑛ℎ_𝑛.(2.2)Let us denote by 𝐻 the set of all ℎ=(ℎ₀,ℎ^) such that there are 𝐾₀∈ℤ and 𝐾=(𝐾₁,…,𝐾_𝑛)∈ℤ^𝑛 satisfying the conditions: 𝐾₀ℎ₀=𝑑₀ and (𝐾₁ℎ₁,…,𝐾_𝑛ℎ_𝑛)=𝑑. Set𝐷_ℎℝ^ℎ1+𝑛=𝑡^(𝑟),𝑥^(𝑚)∶(𝑟,𝑚)∈ℤ^1+𝑛,=𝜕𝐷₀∩ℝ^ℎ1+𝑛,𝐸_ℎ=𝐸∩ℝ^ℎ1+𝑛,𝐸_0.ℎ=𝐸₀∩ℝ^ℎ1+𝑛,𝐸_ℎ=𝜕₀𝐸∩ℝ^ℎ1+𝑛,Ω_ℎ=𝐸_ℎ∪𝐸_0.ℎ∪𝜕₀𝐸_ℎ.(2.3)Let 𝑁₀∈ℕ be defined by the relations: 𝑁₀ℎ₀≤𝑎<(𝑁₀+1)ℎ₀ and𝐸^ℎ=𝑡^(𝑟),𝑥^(𝑚)∈𝐸_ℎ∶0≤𝑟≤𝑁₀−1.(2.4)For functions 𝑤∶𝐷_ℎ→ℝ^𝑘 and 𝑧∶Ω_ℎ→ℝ^𝑘 we write 𝑤^(𝑟,𝑚)=𝑤(𝑡^(𝑟),𝑥^(𝑚)) on 𝐷_ℎ and 𝑧^(𝑟,𝑚)=𝑧(𝑡^(𝑟),𝑥^(𝑚)) on Ω_ℎ. We need a discrete version of the operator (𝑡,𝑥)→𝑧_(𝑡,𝑥). For a function 𝑧∶Ω_ℎ→ℝ^𝑘 and for a point (𝑡^(𝑟),𝑥^(𝑚))∈𝐸_ℎ we define a function 𝑧_[𝑟,𝑚]∶𝐷_ℎ→ℝ^𝑘 by𝑧_[𝑟,𝑚](𝜏,𝑦)=𝑧𝑡^(𝑟)+𝜏,𝑥^(𝑚)+𝑦,(𝜏,𝑦)∈𝐷_ℎ.(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,ℝ^𝑘). Then we need an interpolating operator 𝑇_ℎ∶𝐹(𝐷_ℎ,ℝ^𝑘)→𝐶(𝐷,ℝ^𝑘). We define 𝑇_ℎ in the following way. Let us denote by (𝜗₁,…,𝜗_𝑛) the family of sets defined by𝜗_𝑖={0,1}if𝑑_𝑖>0,𝜗_𝑖={0}if𝑑_𝑖=0,1≤𝑖≤𝑛.(2.6)Set 𝜐=(𝜐₁,…,𝜐_𝑛)∈ℤ^𝑛 and 𝜐_𝑖=0 if 𝑑_𝑖=0, 𝜐_𝑖=1 if 𝑑_𝑖>0 where 1≤𝑖≤𝑛. WriteΔ₊=𝜆=𝜆₁,…,𝜆_𝑛∶𝜆_𝑖∈𝜗_𝑖for1≤𝑖≤𝑛.(2.7)Set 𝑒_𝑖=(0,…,0,1,0,…,0)∈ℝ^𝑛 with 1 standing on the 𝑖th place.

Let 𝑤∈𝐹(𝐷_ℎ,ℝ^𝑘) and (𝑡,𝑥)∈𝐷. Suppose that 𝑑₀>0. There exists (𝑡^(𝑟),𝑥^(𝑚))∈𝐷_ℎ such that (𝑡^(𝑟+1),𝑥^(𝑚+𝜐))∈𝐷_ℎ and 𝑡^(𝑟)≤𝑡≤𝑡^(𝑟+1), 𝑥^(𝑚)≤𝑥≤𝑥^(𝑚+𝜐). Write𝑇_ℎ[𝑤](𝑡,𝑥)=1−𝑡−𝑡^(𝑟)ℎ₀_𝜆∈Δ+𝑤^{(𝑟,𝑚+𝜆)}𝑥−𝑥^(𝑚)ℎ^^𝜆1−𝑥−ℎ𝑥^(𝑚)^1−𝜆+𝑡−𝑡^(𝑟)ℎ₀_𝜆∈Δ+𝑤^{(𝑟+1,𝑚+𝜆)}𝑥−𝑥^(𝑚)ℎ^^𝜆1−𝑥−ℎ𝑥^(𝑚)^1−𝜆,(2.8) where𝑥−ℎ𝑥^(𝑚)^𝜆=^𝑛_𝑖=1𝑥^𝑖−𝑥^{(𝑚𝑖)𝑖}ℎ_𝑖^𝜆𝑖1−𝑥−𝑥_(𝑚),ℎ^^1−𝜆=^𝑛_𝑖=11−𝑥^𝑖−𝑥^{(𝑚𝑖)𝑖}ℎ_𝑖^1−𝜆𝑖(2.9) and we take 0⁰=1 in the above formulas. If 𝑑₀=0 then we put𝑇_ℎ[𝑤](𝑡,𝑥)=_𝜆∈Δ+𝑤^{(𝑟,𝑚+𝜆)}𝑥−𝑥^(𝑚)ℎ^^𝜆1−𝑥−ℎ𝑥^(𝑚)^1−𝜆.(2.10)Then we have defined 𝑇_ℎ[𝑤] on 𝐷. It is easy to see that 𝑇_ℎ[𝑤]∈𝐶(𝐷,ℝ^𝑘). The above interpolating operator has been first proposed in [10, Chapter 5].

For 𝑤,𝑤∈𝐹(𝐷_ℎ,ℝ^𝑘) we write 𝑤≤𝑤 if 𝑤^(𝑟,𝑚)≤𝑤^(𝑟,𝑚) where (𝑡^(𝑟),𝑥^(𝑚))∈𝐷_ℎ. In a similar way we define the relation 𝑤≤𝑤 for 𝑤,𝑤∈𝐶(𝐷,ℝ^𝑘) and the relation 𝑧≤𝑧 for 𝑧,𝑧∈𝐹(Ω_ℎ,ℝ^𝑘) and for 𝑧,𝑧∈𝐶(Ω,ℝ^𝑘).

We formulate an implicit difference scheme for (1.4), (1.5). For 𝑥,𝑦∈ℝ^𝑛 we write 𝑥⋄𝑦=(𝑥₁𝑦₁,…,𝑥_𝑛𝑦_𝑛)∈ℝ^𝑛.

Assumption (𝐻[𝑓]). The function 𝑓=(𝑓₁,…,𝑓_𝑘)∶Σ→ℝ^𝑘 of the variables (𝑡,𝑥,𝑤,𝑞), 𝑞=(𝑞₁,…,𝑞_𝑛), is continuous and
(1)the partial derivatives (𝜕_𝑞1𝑓_𝑖,…,𝜕_𝑞𝑛𝑓_𝑖)=𝜕_𝑞𝑓_𝑖, 𝑖=1,…,𝑘, exist on Σ and the functions 𝜕_𝑞𝑓_𝑖, 𝑖=1,…,𝑘, are continuous and bounded on Σ,(2)there is 𝑥̃∈(−𝑏,𝑏), 𝑥̃=(𝑥₁,…,𝑥_𝑛), such that(𝑥−𝑥̃)⋄𝜕_𝑞𝑓_𝑖(𝑡,𝑥,𝑤,𝑞)≥𝜃onΣfor𝑖=1,…,𝑘,(2.11)(3)there is 𝜀₀>0 such that for 0<ℎ₀<𝜀₀ and 𝑤,𝑤∈𝐶(𝐷,ℝ^𝑘), 𝑤≤𝑤, we have𝑤(0,𝜃)+ℎ₀𝑓(𝑡,𝑥,𝑤,𝑞)≤𝑤(0,𝜃)+ℎ₀𝑓(𝑡,𝑥,𝑤,𝑞),(𝑡,𝑥,𝑞)∈𝐸×ℝ^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 𝑧∈𝐶(Ω,ℝ^𝑘), 𝑢∈𝐶(Ω,ℝ^𝑛). Let us denote by 𝑔_𝑖[𝑧,𝑢](⋅,𝑡,𝑥)=𝑔_𝑖.1[𝑧,𝑢](⋅,𝑡,𝑥),…,𝑔_𝑖.𝑛[𝑧,𝑢](⋅,𝑡,𝑥)(2.13)the 𝑖th bicharacteristic of (1.4) corresponding to (𝑧,𝑢). Then 𝑔_𝑖[𝑧,𝑢](⋅,𝑡,𝑥) is a solution of the Cauchy problem𝑦^(𝜏)=−𝜕_𝑞𝑓_𝑖𝜏,𝑦(𝜏),𝑧_{(𝜏,𝑦(𝜏))},𝑢𝜏,𝑦(𝜏),𝑦(𝑡)=𝑥.(2.14)Assumption (2.11) states that the bicharacteristics satisfy the following monotonicity conditions: If 𝑥_𝑗−𝑥_𝑗≥0 the function 𝑔_𝑖𝑗[𝑧,𝑢](⋅,𝑡,𝑥) is non increasing. If 𝑥_𝑗−𝑥_𝑗<0 then 𝑔_𝑖𝑗[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 𝑓=(𝑓₁,…,𝑓_𝑘)∶𝐸×ℝ×𝐶(𝐷,ℝ^𝑘)×ℝ^𝑛→ℝ^𝑘 of the variables (𝑡,𝑥,𝑝,𝑤,𝑞). Write 𝑓_𝑖(𝑡,𝑥,𝑤,𝑞)=𝑓_𝑖(𝑡,𝑥,𝑤_𝑖(0,𝜃),𝑤,𝑞), 𝑖=1,…,𝑘, on Σ. Then system (1.4) is equivalent to 𝜕_𝑡𝑧_𝑖(𝑡,𝑥)=𝑓_𝑖𝑡,𝑥,𝑧_𝑖(𝑡,𝑥),𝑧_(𝑡,𝑥),𝜕_𝑥𝑧_𝑖(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=(𝜕_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 by 𝐻^⋆ the set of all ℎ=(ℎ₀,ℎ^)∈𝐻 such thatℎ_𝑖<min𝑏_𝑖−𝑥_𝑖,𝑥_𝑖+𝑏_𝑖,𝑖=1,…,𝑛.(2.16)Suppose that 𝜔∶Ω_ℎ→ℝ. We apply difference operators 𝛿=(𝛿₁,…,𝛿_𝑛) given byif𝑥_𝑗≤𝑥^{(𝑚𝑗)𝑗}<𝑏_𝑗then𝛿_𝑗𝜔^(𝑟,𝑚)=1ℎ_𝑗𝜔^{(𝑟,𝑚+𝑒𝑗)}if−𝑏_𝑗−𝜔^(𝑟,𝑚),<𝑥^{(𝑚𝑗)𝑗}<𝑥_𝑗then𝛿_𝑗𝜔^(𝑟,𝑚)=1ℎ_𝑗𝜔^(𝑟,𝑚)−𝜔^{(𝑟,𝑚−𝑒𝑗)},(2.17)and we put 𝑗=1,…,𝑛 in (2.17). Let 𝛿₀ be defined by𝛿₀𝜔^(𝑟,𝑚)=1ℎ₀𝜔^{(𝑟+1,𝑚)}−𝜔^(𝑟,𝑚)(2.18)and 𝛿₀𝑧=(𝛿₀𝑧₁,…,𝛿₀𝑧_𝑘). Write𝔽_ℎ[𝑧]^(𝑟,𝑚)=𝑓₁𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ𝑧_[𝑟,𝑚],𝛿𝑧^{1(𝑟+1,𝑚)},…,𝑓_𝑘𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ𝑧_[𝑟,𝑚],𝛿𝑧^{𝑘(𝑟+1,𝑚)}.(2.19) Given 𝜑_ℎ∶𝐸_0.ℎ∪𝜕₀𝐸_ℎ→ℝ^𝑘, we consider the functional difference equation𝛿₀𝑧^(𝑟,𝑚)=𝔽_ℎ[𝑧]^(𝑟,𝑚)(2.20)with the initial boundary condition𝑧^(𝑟,𝑚)=𝜑^{ℎ(𝑟,𝑚)}on𝐸_0.ℎ∪𝜕₀𝐸_ℎ.(2.21)

The above problem is considered as an implicit difference method for (1.4), (1.5). It is important that the difference expressions (𝛿₁𝑧_𝑖,…,𝛿_𝑛𝑧_𝑖), 1≤𝑖≤𝑘, are calculated at the point (𝑡^(𝑟+1),𝑥^(𝑚)) and the functional variable 𝑇_ℎ𝑧_[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∶Ω_ℎ→ℝ^𝑘 satisfy the difference functional inequality𝛿₀𝑢^(𝑟,𝑚)−𝔽_ℎ[𝑢]^(𝑟,𝑚)≤𝛿₀𝑣^(𝑟,𝑚)−𝔽_ℎ[𝑣]^(𝑟,𝑚)on𝐸^ℎ,(2.22)(2)the initial boundary estimate 𝑢^(𝑟,𝑚)≤𝑣^(𝑟,𝑚) holds on 𝐸_0.ℎ∪𝜕₀𝐸_ℎ. Then 𝑢^(𝑟,𝑚)≤𝑣^(𝑟,𝑚)on𝐸_ℎ.(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_ℎ. Assume that 𝑢^(𝑗,𝑚)≤𝑣^(𝑗,𝑚) for (𝑡^(𝑗),𝑥^(𝑚))∈𝐸_ℎ∩([0,𝑡^(𝑟)]×ℝ^𝑛). We prove that 𝑢^{(𝑟+1,𝑚)}≤𝑣^{(𝑟+1,𝑚)} for (𝑡^{(𝑟+1,𝑚)},𝑥^(𝑚))∈𝐸_ℎ. Write𝑈^{𝑖(𝑟,𝑚)}=−𝑢𝑣^{𝑖(𝑟,𝑚)}+_{𝑖(𝑟,𝑚)}ℎ₀𝑓_𝑖𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ𝑢_[𝑟,𝑚],𝛿𝑢^{𝑖(𝑟+1,𝑚)}−ℎ₀𝑓_𝑖𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ𝑣_[𝑟,𝑚],𝛿𝑢^{𝑖(𝑟+1,𝑚)},𝑖=1,…,𝑘.(2.24) It follows from (2.22) that𝑢_𝑖−𝑣_𝑖^{(𝑟+1,𝑚)}≤𝑈^{𝑖(𝑟,𝑚)}+ℎ₀𝑓−_𝑖𝑓𝑡^(𝑟)_𝑖,𝑥^(𝑚),𝑇_ℎ𝑣_[𝑟,𝑚],𝛿𝑢^{𝑖(𝑟+1,𝑚)}𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ𝑣_[𝑟,𝑚],𝛿𝑣^{𝑖(𝑟+1,𝑚)},(2.25) where 𝑖=1,…,𝑘. The monotonicity condition (3) of Assumption (𝐻[𝑓])  implies the inequalities 𝑈^{𝑖(𝑟,𝑚)}≤0 for (𝑡^(𝑟),𝑥^(𝑚))∈𝐸_ℎ, 𝑖=1,…,𝑘. Then we have𝑢_𝑖−𝑣_𝑖^{(𝑟+1,𝑚)}≤ℎ^0𝑛_𝑗=1¹⁰𝜕_𝑞𝑗𝑓_𝑖𝑄^{𝑖(𝑟,𝑚)}[𝑣,𝜏]𝑑𝜏𝛿_𝑗𝑢_𝑖−𝑣_𝑖^{(𝑟+1,𝑚)},(2.26)where 𝑖=1,…,𝑘 and𝑄^{𝑖(𝑟,𝑚)}[𝑣,𝜏]=𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ𝑣_[𝑟,𝑚],𝛿𝑣^{𝑖(𝑟+1,𝑚)}+𝜏𝛿𝑢_𝑖−𝑣_𝑖^{(𝑟+1,𝑚)}.(2.27)WriteΓ^+(𝑚)=𝑗∈{1,…,𝑛}∶𝑥^{(𝑚𝑗)𝑗}∈𝑥_𝑗,𝑏_𝑗,Γ^−(𝑚)={1,…,𝑛}⧵Γ^+(𝑚).(2.28)It follows from (2.11), (2.17) that𝑢_𝑖−𝑣_𝑖^{(𝑟+1,𝑚)}1+ℎ^0𝑛_𝑗=11ℎ_𝑗¹⁰≤ℎ₀||𝜕_𝑞𝑗𝑓_𝑖𝑄^{𝑖(𝑟,𝑚)}[𝑣,𝜏]|𝑑𝜏_𝑗∈_Γ+(𝑚)1ℎ_𝑗¹⁰𝜕_𝑞𝑗𝑓_𝑖𝑄^{𝑖(𝑟,𝑚)}[𝑣,𝜏]𝑑𝜏𝑢_𝑖−𝑣_𝑖^{(𝑟+1,𝑚+𝑒𝑗)}−ℎ_0𝑗∈_Γ−(𝑚)1ℎ_𝑗¹⁰𝜕_𝑞𝑗𝑓_𝑖𝑄^{𝑖(𝑟,𝑚)}[𝑣,𝜏]𝑑𝜏𝑢_𝑖−𝑣_𝑖^{(𝑟+1,𝑚−𝑒𝑗)},𝑖=1,…,𝑘.(2.29) We define 𝑚∈ℤ^𝑛 and 𝜇∈ℕ, 1≤𝜇≤𝑘, as follows:𝑢_𝜇−𝑣_𝜇^{(𝑟+1,𝑚)}=max_{1≤𝑖≤𝑘}max𝑢_𝑖−𝑣_𝑖^{(𝑟+1,𝑚)}∶𝑡^(𝑟+1),𝑥^(𝑚)∈Ω_ℎ.(2.30)If (𝑡^(𝑟+1),𝑥^(𝑚))∈𝜕₀𝐸_ℎ then assumption (2) implies that (𝑢_𝜇−𝑣_𝜇)^{(𝑟+1,𝑚)}≤0. Let us consider the case when (𝑡^(𝑟+1),𝑥^(𝑚))∈𝐸_ℎ. Then we have from (2.29) that𝑢_𝜇−𝑣_𝜇^{(𝑟+1,𝑚)}1+ℎ^0𝑛_𝑗=11ℎ_𝑗¹⁰≤ℎ₀||𝜕_𝑞𝑗𝑓_𝑖(𝑄^{𝑖(𝑟,𝑚)}[𝑣,𝜏])||𝑑𝜏𝑢_𝜇−𝑣_𝜇^{(𝑟+1,𝑚)}_{𝑗∈Γ+(𝑚)}1ℎ_𝑗¹⁰−𝜕_𝑞𝑗𝑓_𝑖𝑄^{𝑖(𝑟,𝑚)}[𝑣,𝜏]𝑑𝜏_{𝑗∈Γ−(𝑚)}1ℎ_𝑗¹⁰𝜕_𝑞𝑗𝑓_𝑖𝑄^{(𝑟,𝑚}_𝑖[𝑣,𝜏]𝑑𝜏.(2.31) It follows that (𝑢_𝜇−𝑣_𝜇)^{(𝑟+1,𝑚)}≤0. The the proof of (2.23) is completed by induction.

3. Implicit Difference Schemes

We define 𝑁=(𝑁₁,…,𝑁_𝑛)∈𝑁^𝑛 by the relations:𝑁₁ℎ₁,…,𝑁_𝑛ℎ_𝑛<𝑏₁,…,𝑏_𝑛≤𝑁₁+1ℎ₁,…,𝑁_𝑛+1ℎ_𝑛(3.1)and we assume that (𝑁_𝑖+1)ℎ_𝑖=𝑏_𝑖 if 𝑑_𝑖=0. For 𝑤∈𝐶(𝐷,ℝ^𝑘) we write‖𝑤‖_𝐷=max‖‖𝑤(𝑡,𝑥)‖‖_∞∶(𝑡,𝑥)∈𝐷.(3.2)In a similar way we define the norm in the space 𝐹(𝐷_ℎ,ℝ^𝑘) : if 𝑤∶𝐷_ℎ→ℝ^𝑘 then‖𝑤‖_𝐷ℎ=max‖‖𝑤^(𝑟,𝑚)‖‖_∞∶𝑡^(𝑟),𝑥^(𝑚)∈𝐷_ℎ.(3.3)The following properties of the operator 𝑇_ℎ are important in our considerations.

Lemma 3.1. Suppose that 𝑤∶𝐷→ℝ^𝑘 is of class 𝐶¹ and 𝑤_ℎ is the restriction of 𝑤 to the set 𝐷_ℎ. Let 𝐶 be such a constant that ‖𝜕_𝑡𝑤‖_𝐷, ‖𝜕_𝑥𝑖𝑤‖_𝐷≤𝐶 for 1≤𝑖≤𝑛. Then ‖𝑇_ℎ[𝑤_ℎ]−𝑤‖_𝐷≤𝐶‖ℎ‖ where ‖ℎ‖=ℎ₀+ℎ₁+⋯+ℎ_𝑛.

Lemma 3.2. Suppose that 𝑤∶𝐷→ℝ^𝑘 is of class 𝐶² and 𝑤_ℎ is the restriction of 𝑤 to the set 𝐷_ℎ. Let 𝐶 be such a constant that ‖𝜕_𝑡𝑡𝑤‖_𝐷, ‖𝜕_𝑡𝑥𝑖𝑤‖_𝐷, ‖𝜕_{𝑥𝑖𝑥𝑗}𝑤‖_𝐷≤𝐶,   𝑖,𝑗=1,…,𝑛. Then ‖𝑇_ℎ[𝑤_ℎ]−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 𝜑_ℎ∈𝐹(𝐸_0.ℎ∪𝜕₀𝐸_ℎ,ℝ^𝑘) then there exists exactly one solution 𝑢_ℎ=(𝑢_ℎ.1,…,𝑢_ℎ.𝑘)∶Ω_ℎ→ℝ^𝑘 of difference functional problem (2.20), (2.21).

Proof. Suppose that 0≤𝑟≤𝑁₀−1 is fixed and that the solution 𝑢_ℎ of problem (2.20), (2.21) is given on the set Ω_ℎ∩([−𝑑₀,𝑡^(𝑟)]×ℝ^𝑛). We prove that the vectors 𝑢^{ℎ(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 equations1ℎ₀𝑧^{𝑖(𝑟+1,𝑚)}−𝑢^{(𝑟,𝑚)ℎ.𝑖}=𝑓_𝑖𝑡^(𝑟),𝑥^(𝑚),𝑇(𝑢_ℎ)_[𝑟,𝑚],𝛿𝑧^{𝑖(𝑟+1,𝑚)},(3.4)where −𝑁≤𝑚≤𝑁,𝑖=1,…,𝑘, with the initial boundary condition (2.21). There exists 𝑄_ℎ>0 such that𝑄_ℎ≥ℎ₀_{𝑗∈Γ+(𝑚)}1ℎ_𝑗𝜕_𝑞𝑗𝑓_𝑖𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ𝑢_ℎ_[𝑟,𝑚],𝑞−_{𝑗∈Γ−(𝑚)}1ℎ_𝑗𝜕_𝑞𝑗𝑓_𝑖𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ𝑢_ℎ_[𝑟,𝑚],𝑞,(3.5) where −𝑁≤𝑚≤𝑁,𝑖=1,…,𝑘. It is clear that system (3.4) is equivalent to the following one:𝑧^{𝑖(𝑟+1,𝑚)}=1𝑄_ℎ+1𝑄^ℎ𝑧^{𝑖(𝑟+1,𝑚)}+𝑢^{(𝑟,𝑚)ℎ.𝑖}+ℎ₀𝑓_𝑖𝑡^(𝑟),𝑥^(𝑚)−,𝑇𝑁^ℎ≤𝑢_ℎ𝑚≤_[𝑟,𝑚],𝑁𝛿,𝑧𝑖=1^{𝑖(𝑟+1,𝑚)},…,,𝑘(3.6) Write 𝑆_ℎ={𝑥^(𝑚)∶𝑥^(𝑚)∈[−𝑐,𝑐]}. Elements of the space 𝐹(𝑆_ℎ,ℝ^𝑘) are denoted by 𝜉, 𝜉. For 𝜉∶𝑆_ℎ→ℝ^𝑘, 𝜉=(𝜉₁,…,𝜉_𝑘), we write 𝜉^(𝑚)=𝜉(𝑥^(𝑚)) and𝛿𝛿𝜉^𝑖(𝑚)=𝛿₁𝜉^𝑖(𝑚),…,𝛿_𝑛𝜉^𝑖(𝑚),1≤𝑖≤𝑘,_𝑗𝜉^𝑖(𝑚)=1ℎ_𝑗𝜉^{(𝑚+𝑒𝑗𝑖𝑗)}𝛿_𝑗−𝜉^𝑖(𝑚)if𝑥^{(𝑚𝑗)𝑗}∈𝑥_𝑗,𝑏_𝑗,𝜉^𝑖(𝑚)=1ℎ_𝑗𝜉^𝑖(𝑚)−𝜉^{(𝑚−𝑒𝑗)𝑖}if𝑥^{(𝑚𝑗)𝑗}∈𝑏_𝑗,𝑥_𝑗,(3.7)where 𝑗=1,…,𝑛. The norm in the space 𝐹(𝑆_ℎ,ℝ^𝑘) is defined by‖𝜉‖_⋆=max‖‖𝜉^(𝑚)‖‖_∞∶𝑥^(𝑚)∈𝑆_ℎ.(3.8)Let us consider the set𝑋_ℎ=𝜉∈𝐹𝑆_ℎ,ℝ^𝑘∶𝜉^(𝑚)=𝜑^{(𝑟+1,𝑚)}for𝑥^(𝑚)∈[−𝑐,𝑐]⧵(−𝑏,𝑏}.(3.9)We consider the operator 𝑊_ℎ∶𝑋_ℎ→𝑋_ℎ,  𝑊_ℎ=(𝑊_ℎ.1,…,𝑊_ℎ.𝑛) defined by𝑊_ℎ.𝑖[𝜉]^(𝑚)=1𝑄_ℎ+1𝑄^ℎ𝜉^𝑖(𝑚)+𝑢^{(𝑟,𝑚)ℎ.𝑖}+ℎ₀𝑓_𝑖𝑡^(𝑟),𝑥^(𝑚),𝑇𝑢_ℎ_[𝑟,𝑚],𝛿𝜉^𝑖(𝑚),(3.10)where −𝑁≤𝑚≤𝑁,  𝑖=1,…,𝑘 and𝑊_ℎ[𝜉]^(𝑚)=𝜑^{ℎ(𝑟+1,𝑚)}for𝑥^(𝑚)∈[−𝑐,𝑐]⧵(−𝑏,𝑏),(3.11)where 𝜉=(𝜉₁,…,𝜉_𝑘)∈𝐹(𝑆_ℎ,ℝ^𝑘). We prove that‖‖𝑊_ℎ[𝜉]−𝑊_ℎ[𝜉]‖‖_⋆≤𝑄^ℎ𝑄_ℎ+1‖𝜉−𝜉‖_⋆on𝐹𝑆_ℎ,ℝ^𝑘.(3.12)
It follows from (3.10) that we have for −𝑁≤𝑚≤𝑁:𝑊_ℎ.𝑖[𝜉]^(𝑚)−𝑊_ℎ.𝑖[𝜉]^(𝑚)=1𝑄_ℎ+1𝑄^ℎ𝜉_𝑖−𝜉_𝑖^(𝑚)−ℎ_0𝑗∈_Γ+(𝑚)1ℎ_𝑗¹⁰𝜕_𝑞𝑗𝑓_𝑖𝑃^{𝑖(𝑟,𝑚)}[𝑢_ℎ,𝜏]𝑑𝜏𝜉_𝑖−𝜉_𝑖^(𝑚)+_{𝑗∈Γ−(𝑚)}1ℎ_𝑗¹⁰𝜕_𝑞𝑗𝑓_𝑖𝑃^{𝑖(𝑟,𝑚)}[𝑢_ℎ,𝜏]𝑑𝜏(𝜉_𝑖−𝜉_𝑖)^(𝑚)+ℎ_0𝑗∈_Γ+(𝑚)1ℎ_𝑗¹⁰𝜕_𝑞𝑗𝑓_𝑖𝑃^{𝑖(𝑟,𝑚)}[𝑢_ℎ,𝜏]𝑑𝜏(𝜉_𝑖−𝜉_𝑖)^{(𝑚+𝑒𝑗)}−ℎ_0𝑗∈_Γ−(𝑚)1ℎ_𝑗¹⁰𝜕_𝑞𝑗𝑓_𝑖𝑃^{𝑖(𝑟,𝑚)}[𝑢_ℎ,𝜏]𝑑𝜏(𝜉_𝑖−𝜉_𝑖)^{(𝑚−𝑒𝑗)},(3.13) where 𝑖=1,…,𝑘 and𝑃^{𝑖(𝑟,𝑚)}𝑢_ℎ,𝜏=𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ𝑢_ℎ_[𝑟,𝑚],𝛿𝜉^𝑖(𝑚)+𝜏𝛿𝜉_𝑖−𝜉_𝑖^(𝑚).(3.14)It follows from the above relations and from (3.5) that|||𝑊_ℎ.𝑖[𝜉]^(𝑚)−𝑊_ℎ.𝑖[𝜉]^(𝑚)|||≤𝑄_ℎ𝑄_ℎ+1‖𝜉−𝜉‖_⋆for−𝑁≤𝑚≤𝑁,𝑖=1,…,𝑘.(3.15)According to (3.12) we have𝑊_ℎ.𝑖[𝜉]^(𝑚)−𝑊_ℎ.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 𝜉∶𝑆_ℎ→ℝ^𝑘 of the equation 𝜉=𝑊_ℎ[𝜉] and consequently, there exists exactly one solution of (3.6), (2.21). Then the vectors 𝑢^{ℎ(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 function 𝜎∶[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 problem𝜂^(𝑡)=𝜎(𝑡,𝜂(𝑡)),𝜂(0)=0,(3.17)is 𝜂̃(𝑡)=0 for 𝑡∈[0,𝑎].

Assumption (𝐻[𝑓,𝜎]). There is 𝜎∶[0,𝑎]×ℝ₊→ℝ₊ such that Assumption (𝐻[𝜎])  is satisfied and for 𝑤,𝑤∈ℂ(𝐷,ℝ^𝑘), 𝑤≥𝑤, we have𝑓_𝑖(𝑡,𝑥,𝑤,𝑞)−𝑓_𝑖(𝑡,𝑥,𝑤,𝑞)≤𝜎𝑡,‖𝑤−𝑤‖_𝐷,𝑖=1,…,𝑘,(3.18)where (𝑡,𝑥,𝑞)∈𝐸×ℝ^𝑛.

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 𝜑_ℎ∶𝐸_0.ℎ∪𝜕₀𝐸_ℎ→ℝ^𝑘 and there is 𝛼₀∶𝐻^∗→ℝ₊ such that‖‖𝜑^(𝑟,𝑚)−𝜑^{ℎ(𝑟,𝑚)}‖‖_∞≤𝛼₀(ℎ)on𝐸_0.ℎ∪𝜕₀𝐸_ℎ,lim_ℎ→0𝛼₀(ℎ)=0.(3.19) Under these assumptions there is a solution 𝑢_ℎ∶Ω_ℎ→ℝ^𝑘 of (2.20), (2.21) and there is 𝛼∶𝐻^∗→ℝ₊ such that ‖‖𝑢_ℎ−𝑣_ℎ^(𝑟,𝑚)‖‖_∞≤𝛼(ℎ)on𝐸_ℎ,lim_ℎ→0𝛼(ℎ)=0,(3.20)where 𝑣_ℎ is the restriction of 𝑣 to the set Ω_ℎ.

Proof. The existence of 𝑢_ℎ follows from Theorem 3.3. Let Γ_ℎ∶𝐸^ℎ→ℝ^𝑘, Γ_0.ℎ∶𝐸_0.ℎ∪𝜕₀𝐸_ℎ→ℝ^𝑘 be defined by the relations𝑣𝛿₀𝑣^{ℎ(𝑟,𝑚)}=𝔽_ℎ𝑣_ℎ^(𝑟,𝑚)+Γ^{ℎ(𝑟,𝑚)}on𝐸^ℎ,(3.21)_{ℎ(𝑟,𝑚)}=𝜑^{ℎ(𝑟+1,𝑚)}+Γ^{(𝑟,𝑚)0.ℎ}for𝑡^(𝑟),𝑥^(𝑚)∈𝐸_0.ℎ∪𝜕₀𝐸_ℎ.(3.22)From Lemma 3.1 and from assumption (1) of the theorem it follows that there are 𝛾,𝛾₀∶𝐻^∗→ℝ₊ such that‖‖Γ^{ℎ(𝑟,𝑚)}‖‖_∞≤𝛾(ℎ)on𝐸^ℎ,‖‖Γ^{(𝑟,𝑚)0.ℎ}‖‖_∞≤𝛾₀(ℎ)on𝐸_0.ℎ∪𝜕₀𝐸_ℎ(3.23)and lim_ℎ→0𝛾(ℎ)=0, lim_ℎ→0𝛾₀(ℎ)=0. Write 𝐽=[0,𝑎] and 𝐽_ℎ={𝑡^(𝑟)∶0≤𝑟≤𝑁₀}. For 𝛽∶𝐽_ℎ→ℝ we put 𝛽^(𝑟)=𝛽(𝑡^(𝑟)). Let 𝛽_ℎ∶𝐽_ℎ→ℝ₊ be a solution of the difference problem𝛽^(𝑟+1)=𝛽^(𝑟)+ℎ₀𝜎𝑡^(𝑟),𝛽^(𝑟)+ℎ₀𝛾(ℎ),0≤𝑟≤𝑁₀−1,𝛽⁽⁰⁾=𝛼₀(ℎ).(3.24)We prove that‖‖𝑢_ℎ−𝑣_ℎ^(𝑟,𝑚)‖‖_∞≤𝛽^ℎ(𝑟)on𝐸_ℎ.(3.25)Let 𝑣_ℎ=(𝑣_ℎ.1,…,𝑣_ℎ.𝑘)∶Ω_ℎ→ℝ^𝑘 be defined by𝑣_{(𝑟,𝑚)ℎ.𝑖}𝑣^{(𝑟,𝑚)ℎ.𝑖}=𝑣^{(𝑟,𝑚)ℎ.𝑖}+𝛽^ℎ(0)on𝐸_0.ℎ,=𝑣^{(𝑟,𝑚)ℎ.𝑖}+𝛽^ℎ(𝑖)on𝐸_ℎ∪𝜕₀𝐸_ℎ,(3.26) where 𝑖=1,…,𝑘. We prove that the difference functional inequality𝛿₀𝑣_ℎ≥𝔽_ℎ𝑣_ℎ^(𝑟,𝑚),𝑡^(𝑟),𝑥^(𝑚)∈𝐸^ℎ,(3.27)is satisfied. It follows from Assumption (𝐻[𝑓,𝜎]) and from (3.21) that𝛿₀𝑣^{(𝑟,𝑚)ℎ.𝑖}=𝛿₀𝑣^{(𝑟,𝑚)ℎ.𝑖}+1ℎ₀𝛽^ℎ(𝑟+1)=𝑓_𝑖−𝛽^ℎ(𝑟)+𝑓𝑡^(𝑟)_𝑖,𝑥^(𝑚),𝑇_ℎ𝑣_ℎ_[𝑟,𝑚],𝛿𝑣^{(𝑟+1,𝑚)ℎ.𝑖}+ℎ1₀𝛽^ℎ(𝑟+1)−𝛽^ℎ(𝑟)≥𝑓_𝑖𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ𝑣_ℎ_[𝑟,𝑚],𝛿𝑣^{(𝑟+1,𝑚)ℎ.𝑖}−𝑓_𝑖𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ𝑣_ℎ_[𝑟,𝑚],𝛿𝑣^{(𝑟+1,𝑚)ℎ.𝑖}=𝑓_𝑖𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ𝑣_ℎ_[𝑟,𝑚],𝛿𝑣^{(𝑟+1,𝑚)ℎ.𝑖}−𝜎𝑡^(𝑟),𝛽^ℎ(𝑟)+ℎ1₀𝛽^ℎ(𝑟+1)−𝛽^ℎ(𝑟)(𝑡^(𝑟),𝑥^(𝑚),𝑇_ℎ(𝑣_ℎ)_[𝑟,𝑚],𝛿𝑣^{(𝑟+1,𝑚)ℎ.𝑖}),𝑖=1,…,𝑘.(3.28) This completes the proof of (3.27).
Since 𝑣^{ℎ(𝑟,𝑚)}≤𝑣^{ℎ(𝑟,𝑚)} on 𝐸_0.ℎ∪𝜕₀𝐸_ℎ, it follows from Theorem 2.3 that 𝑢^{ℎ(𝑟,𝑚)}≤𝑣^{ℎ(𝑟,𝑚)}+𝛽^ℎ(𝑟) on 𝐸_ℎ. In a similar way we prove that 𝑣^{ℎ(𝑟,𝑚)}−𝛽^ℎ(𝑟)≤𝑢^{ℎ(𝑟,𝑚)} on 𝐸_ℎ. The above estimates imply (3.25). Consider the Cauchy problem𝜂^(𝑡)=𝜎(𝑡,𝜂(𝑡))+𝛾(ℎ),𝜂(0)=𝛼₀(ℎ).(3.29)It follows from Assumption (𝐻[𝜎]) that there is 𝜀̃>0 such that for ‖ℎ‖≤𝜀̃  the maximal solution 𝜂(⋅,ℎ) of (3.29) is defined on [0,𝑎] andlim_ℎ→0𝜂(𝑡,ℎ)=0uniformlyon[0,𝑎].(3.30)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),(3.31)where 0≤𝑟≤𝑁₀−1. Since 𝛽_ℎ satisfies (3.24), the above relations imply the estimate𝛽^ℎ(𝑟)≤𝜂𝑡^(𝑟),ℎ≤𝜂(𝑎,ℎ),0≤𝑟≤𝑁₀.(3.32)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 𝜑_ℎ∶𝐸_0.ℎ∪𝜕₀𝐸_ℎ→ℝ^𝑘 and there is 𝛼₀∶𝐻^∗→ℝ₊ such that‖‖𝜑^(𝑟,𝑚)−𝜑^{ℎ(𝑟,𝑚)}‖‖_∞≤𝛼₀(ℎ)on𝐸_0.ℎ∪𝜕₀𝐸_ℎ,lim_ℎ→0𝛼₀(ℎ)=0.(3.33)(3)there exists 𝐿∈ℝ₊ such that estimates𝑓_𝑖(𝑡,𝑥,𝑤,𝑞)−𝑓_𝑖(𝑡,𝑥,𝑤,𝑞)≤𝐿‖𝑤−𝑤‖_𝐷,𝑖=1,…,𝑘,(3.34)are satisfied for (𝑡,𝑥,𝑞)∈𝐸×ℝ^𝑛, 𝑤,𝑤∈𝐶(𝐷,ℝ^𝑘) and 𝑤≥𝑤,(4)there is 𝐶∈ℝ₊ such that‖‖𝜕_𝑞𝑓_𝑖(𝑡,𝑥,𝑤,𝑞)‖‖≤𝐶onΣfor𝑖=1,…,𝑘.(3.35) Under these assumptions there is a solution 𝑢_ℎ∶Ω_ℎ→ℝ^𝑘 of (2.20), (2.21), and ‖‖𝑢_ℎ−𝑣_ℎ^(𝑟,𝑚)‖‖_∞≤𝛼(ℎ)on𝐸_ℎ,(3.36)where 𝛼(ℎ)=𝛼₀(ℎ)𝑒^𝐿𝑎+𝛾̃(ℎ)𝑒^𝐿𝑎−1𝛼(ℎ)=𝛼₀𝐿if𝐿>0,𝛾̃(ℎ)=0.5𝐶ℎ₀(ℎ)+𝑎𝛾̃(ℎ)if𝐿=0,(1+𝐶)+𝐿𝐶‖ℎ^‖²+0.5𝐶𝐶‖ℎ‖(3.37)and 𝐶∈ℝ₊ is such that ‖‖𝜕_𝑡𝑡𝑣(𝑡,𝑥)‖‖_∞,‖‖𝜕_𝑡𝑥𝑖𝑣(𝑡,𝑥)‖‖_∞,‖‖𝜕_{𝑥𝑖𝑥𝑗}𝑣(𝑡,𝑥)‖‖_∞≤𝐶(3.38)on Ω for 𝑖,𝑖=1,…,𝑛.

Proof. It follows that the solution 𝛽_ℎ∶𝐽_ℎ→ℝ₊ of the difference problem𝛽^(𝑟+1)=1+𝐿ℎ₀𝛽𝛽^(𝑟)+ℎ₀𝛾(ℎ),0≤𝑟≤𝑁₀−1,₍₀₎=𝛼₀(ℎ)(3.39) satisfies the condition: 𝛽^ℎ(𝑟)≤𝛼(ℎ) for 0≤𝑟≤𝑁₀. Moreover we have‖‖Γ^{ℎ(𝑟,𝑚)}‖‖_∞≤𝛾̃(ℎ)on𝐸^ℎ,(3.40)where Γ_ℎ 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)=+arc𝑡ta1n−2𝑦𝑥₂𝜕_𝑥𝑧(𝑡,𝑥,𝑦)+2𝑦𝜕_𝑦𝑧(𝑡,𝑥,𝑦)−𝑡2𝑥²𝑦²−𝑥²−𝑦²𝑧(𝑡,𝑥,𝑦)+𝑧(𝑡,𝑥,𝑦)4+^𝑥−10𝑠𝑧.2(5𝑡,𝑠𝑥,₂𝑦)𝑑𝑠+𝑡1−𝑥²^𝑦−1𝑠𝑧(𝑡,𝑥,𝑠)𝑑𝑠−1𝑦²−1−4(4.2) with the initial boundary condition𝑧𝑧((𝑡𝑡,,𝑥−,1𝑧−,(1𝑦0)),𝑥==,𝑦𝑧𝑧)((𝑡𝑡=,,𝑥11,,,𝑦1))==(𝑥11,,,𝑦)∈((𝑡𝑡,,[𝑦𝑥−))1∈∈,1[[]00×,,00[..−55]1],××1[][−,−11,,11]],.(4.3)

The function 𝑣(𝑡,𝑥,𝑦)=exp[0.25𝑡(𝑥²−1)(𝑦²−1)] is the solution of the above problem. Let us denote by 𝑧_ℎ 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^ℎ(𝑟)=12𝑁₁−12𝑁₂−1_𝑚∈Π||𝑧^{ℎ(𝑟,𝑚)}−𝑣^(𝑟,𝑚)||,0≤𝑟≤𝑁₀,(4.4)whereΠ=𝑚=𝑚₁,𝑚₂∶∈ℤ²∶−𝑁₁+1≤𝑚₁≤𝑁₁−1,−𝑁₁+1≤𝑚₂≤𝑁₂−1(4.5)and 𝑁₁ℎ₁=1, 𝑁₂ℎ₂=1, 𝑁₀ℎ₀=0.5. The numbers 𝜀^ℎ(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.

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)=+2𝑥c𝜕o_𝑥s𝑧(2𝑡,𝑥𝑥𝜕,_𝑥𝑦)+2𝑦𝜕_𝑦𝑧(𝑡,𝑥,𝑦)+𝑧(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,(4.7) with the initial boundary conditions𝑧𝑧((𝑡𝑡,,𝑥−,0−𝑧.5(00,.5,𝑦𝑥)),==𝑦)𝑧𝑧=((𝑡𝑡,,1𝑥0,,.50,,𝑦(5𝑥)),==𝑦)11∈,,[−((𝑡0𝑡,,.𝑦𝑥5),)0∈∈.5[[]00×,,00[..−55]0]×.×5,[[−0−.050.].55,,00..55]],,(4.8) where𝑓(𝑡,𝑥,𝑦)=−𝑥e²x−p0.205.5𝑡₂0.25−𝑦²+𝑡8𝑥²𝑦²−𝑥²−𝑦²−𝑡𝑥²−0.250.25−𝑦².(4.9)

The function 𝑣(𝑡,𝑥,𝑦)=exp[[𝑡(𝑥²−0.25)(0.25−𝑦²)] is the solution of the above problem. Let us denote by 𝑧_ℎ 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_ℎ 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.

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.