On Second Order of Accuracy Difference Scheme of the Approximate Solution of Nonlocal Elliptic-Parabolic Problems

Abstract

A second order of accuracy difference scheme for the approximate solution of the abstract nonlocal boundary value problem −𝑑²𝑢(𝑡)/𝑑𝑡²+𝐴𝑢(𝑡)=𝑔(𝑡), (0≤𝑡≤1), 𝑑𝑢(𝑡)/𝑑𝑡−𝐴𝑢(𝑡)=𝑓(𝑡), (−1≤𝑡≤0), 𝑢(1)=𝑢(−1)+𝜇 for differential equations in a Hilbert space 𝐻 with a self-adjoint positive definite operator 𝐴 is considered. The well posedness of this difference scheme in Hölder spaces is established. In applications, coercivity inequalities for the solution of a difference scheme for elliptic-parabolic equations are obtained and a numerical example is presented.

1. Introduction

The role played by coercive inequalities in the study of boundary value problems for elliptic and parabolic partial differential equations is well known (see [1–4]).

Nonlocal problems are widely used for mathematical modeling of various processes of physics, biology, chemistry, ecology, engineering, and industry when it is impossible to determine the boundary or initial values of the unknown function. Theory and numerical methods of solutions of the nonlocal boundary value problems for partial differential equations of variable type have been studied extensively by many researchers (see, e.g., [5–34] and the references therein).

In paper [35], the nonlocal boundary value problem−𝑑²𝑢(𝑡)𝑑𝑡²+𝐴𝑢(𝑡)=𝑔(𝑡)(0≤𝑡≤1),𝑑𝑢(𝑡)𝑑𝑡−𝐴𝑢(𝑡)=𝑓(𝑡)(−1≤𝑡≤0),𝑢(1)=𝑢(−1)+𝜇 (1.1) for the differential equation in a Hilbert space 𝐻 with the self-adjoint positive definite operator 𝐴 was considered. The well posedness of problem (1.1) in Hölder spaces was established. The first order of accuracy difference scheme for approximate solutions of nonlocal boundary value problem (1.1) was presented. In applications, the coercivity inequalities for solutions of difference schemes for elliptic-parabolic equations were obtained.

In the present paper, the second order of accuracy difference scheme generated by Crank-Nicholson difference scheme−𝜏⁻²(𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1)+𝐴𝑢_𝑘=𝑔_𝑘,𝑔_𝑘=𝑔(𝑡_𝑘),𝑡_𝑘𝜏=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,₋₁(𝑢_𝑘−𝑢_𝑘−11)−2(𝐴𝑢_𝑘−1+𝐴𝑢_𝑘)=𝑓_𝑘,𝑓_𝑘=𝑓(𝑡_𝑘−1/2𝑡),_𝑘−1/21=(𝑘−2𝑢)𝜏,−(𝑁−1)≤𝑘≤0,_𝑁=𝑢_−𝑁+𝜇,𝑢₂−4𝑢₁+3𝑢₀=−3𝑢₀+4𝑢₋₁−𝑢₋₂ (1.2) for the approximate solution of problem (1.1) is presented. The well posedness of difference scheme (1.2) in Hölder spaces is established. As an application, coercivity inequalities for solutions of difference schemes for elliptic-parabolic equations are obtained. A numerical example is given.

2. The Formula for the Solution of Problem (1.2)

The following operators: 𝑃=(𝐼−𝜏𝐴2)𝐺,𝐺=(𝐼+𝜏𝐴2)⁻¹,𝑅=(𝐼+𝜏𝐵)⁻¹,𝑇_𝜏=(𝐼+𝐵⁻¹𝜏𝐴(𝐼+𝜏𝐴+2𝐺⁻²)𝐾(𝐼−𝑅^2𝑁−1)𝜏+𝐾(𝐼−²𝐴2)𝐺⁻²𝑅^2𝑁−1𝜏−𝐾(𝐼−²𝐴2)𝐺⁻²(2𝐼+𝜏𝐵)𝑅^𝑁𝑃^𝑁)⁻¹ (2.1) exist and are bounded for a self-adjoint positive operator 𝐴. Here 1𝐵=2(𝜏𝐴+𝐴(4+𝜏²5𝐴)),𝐾=(𝐼+2𝜏𝐴+4(𝜏𝐴)²)⁻¹. (2.2)

Theorem 2.1. For any 𝑔_𝑘,1≤𝑘≤𝑁−1, and 𝑓_𝑘,−𝑁+1≤𝑘≤0, the solution of problem (1.2) exists and the following formula holds: 𝑢_𝑘=(𝐼−𝑅^2𝑁)⁻¹{[𝑅^𝑘−𝑅^2𝑁−𝑘]𝑢₀+[𝑅^𝑁−𝑘−𝑅^𝑁+𝑘][𝑃^𝑁𝑢₀−𝜏⁰_{𝑠=−𝑁+1}𝑃^{𝑠+𝑁−1}𝐺𝑓_𝑠+𝜇]−[𝑅^𝑁−𝑘−𝑅^𝑁+𝑘](𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵^−1𝑁−1_𝑠=1[𝑅^𝑁−𝑠−𝑅^𝑁+𝑠]𝑔_𝑠𝜏}+(𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵^−1𝑁−1_𝑠=1[𝑅^{|𝑘−𝑠|}−𝑅^𝑘+𝑠]𝑔_𝑠𝑢𝜏,1≤𝑘≤𝑁,_𝑘=𝑃^−𝑘𝑢₀−𝜏⁰_𝑠=𝑘+1𝑃^{𝑠−𝑘−1}𝐺𝑓_𝑠𝑢,−𝑁≤𝑘≤−1,₀=12𝑇_𝜏𝐾𝐺⁻²×{(2𝐼−𝜏²𝐴){(2+𝜏𝐵)𝑅^𝑁[−𝜏⁰_{𝑠=−𝑁+1}𝑃^{𝑠+𝑁−1}𝐺𝑓_𝑠+𝜇]−𝑅^𝑁−1𝐵^−1𝑁−1_𝑠=1[𝑅^𝑁−𝑠−𝑅^𝑁+𝑠]𝑔_𝑠𝜏+(𝐼−𝑅^2𝑁)𝐵^−1𝑁−1_𝑠=1𝑅^𝑠−1𝑔_𝑠𝜏}+(𝐼−𝑅^2𝑁)(𝐼+𝜏𝐵)(𝜏𝐵⁻¹𝑔₁−4𝐺𝐵⁻¹𝑓₀+𝑃𝐺𝐵⁻¹𝑓₀+𝐺𝐵⁻¹𝑓₋₁𝑇)},_𝜏=(𝐼+𝐵⁻¹𝜏𝐴(𝐼+𝜏𝐴+2𝐺⁻²)𝐾(𝐼−𝑅^2𝑁−1)𝜏+𝐾(𝐼−²𝐴2)𝐺⁻²𝑅^2𝑁−1𝜏−𝐾(𝐼−²𝐴2)𝐺⁻²(2𝐼+𝜏𝐵)𝑅^𝑁𝑃^𝑁)⁻¹. (2.3)(2.4)(2.5)

Proof. For any {𝑓_𝑘}⁻¹_𝑘=−𝑁 and 𝜉, the solution of the auxiliary inverse Cauchy difference problem 𝜏⁻¹(𝑢_𝑘−𝑢_𝑘−11)−2(𝐴𝑢_𝑘−1+𝐴𝑢_𝑘)=𝑓_𝑘,−(𝑁−1)≤𝑘≤0,𝑢₀=𝜉 (2.6) exists and the following formula holds [36] 𝑢_𝑘=𝑃^−𝑘𝜉−𝜏⁰_𝑠=𝑘+1𝑃^{𝑠−𝑘−1}𝐺𝑓_𝑠,−𝑁≤𝑘≤−1. (2.7)
Putting 𝜉=𝑢₀, we get (2.4).
Now, we consider the following auxiliary difference problem −𝜏⁻²(𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1)+𝐴𝑢_𝑘=𝑔_𝑘,𝑔_𝑘=𝑔(𝑡_𝑘),𝑡_𝑘𝑢=𝑘𝜏,1≤𝑘≤𝑁−1,₀=𝜉,𝑢_𝑁=𝜓. (2.8)
It is well known that for the solution of (2.8) the following formula holds [37, 38]: 𝑢_𝑘=(𝐼−𝑅^2𝑁)⁻¹{[𝑅^𝑘−𝑅^2𝑁−𝑘]𝜉+[𝑅^𝑁−𝑘−𝑅^𝑁+𝑘]𝜓−[𝑅^𝑁−𝑘−𝑅^𝑁+𝑘](𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵^−1𝑁−1_𝑠=1[𝑅^𝑁−𝑠−𝑅^𝑁+𝑠]𝑔_𝑠𝜏}+(𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵^−1𝑁−1_𝑠=1[𝑅^{|𝑘−𝑠|}−𝑅^𝑘+𝑠]𝑔_𝑠𝜏,1≤𝑘≤𝑁. (2.9)
Applying (2.7) and putting 𝜉=𝑢₀,𝜓=𝑃^𝑁𝑢₀−𝜏∑⁰_{𝑠=−𝑁+1}𝑃^{𝑁+𝑠−1}𝐺𝑓_𝑠+𝜇, in (2.9), we get (2.3).
For 𝑢₀, using (2.3), (2.4), and the condition 𝑢₂−4𝑢₁+3𝑢₀=−3𝑢₀+4𝑢₋₁−𝑢₋₂, (2.10) we obtain the operator equation (2𝐼−𝜏²𝐴){(𝐼−𝑅^2𝑁)⁻¹{[𝑅−𝑅^2𝑁−1]𝑢₀+[𝑅^𝑁−1−𝑅^𝑁+1][𝑃^𝑁𝑢₀−𝜏⁰_{𝑠=−𝑁+1}𝑃^{𝑠+𝑁−1}𝐺𝑓_𝑠+𝜇]−[𝑅^𝑁−1−𝑅^𝑁+1](𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵∑^−1𝑁−1_𝑠=1[𝑅^𝑁−𝑠−𝑅^𝑁+𝑠]𝑔_𝑠𝜏}+(𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵^−1𝑁−1_𝑠=1[𝑅^𝑠−1−𝑅^1+𝑠]𝑔_𝑠𝜏}=−𝜏²𝑔₁+𝐺²5(2𝐼+4𝜏𝐴+2(𝜏𝐴)²)𝑢₀+4𝐺𝜏𝑓₀−𝑃𝐺𝜏𝑓₀−𝐺𝜏𝑓₋₁. (2.11)
The operator 𝐼+𝐵⁻¹𝜏𝐴(𝐼+𝜏𝐴+2𝐺⁻²)𝐾(𝐼−𝑅^2𝑁−1)𝜏+𝐾(𝐼−²𝐴2)𝐺⁻²𝑅^2𝑁−1𝜏−𝐾(𝐼−²𝐴2)𝐺⁻²(2𝐼+𝜏𝐵)𝑅^𝑁𝑃^𝑁 (2.12) has an inverse 𝑇_𝜏=(𝐼+𝐵⁻¹𝜏𝐴(𝐼+𝜏𝐴+2𝐺⁻²)𝐾(𝐼−𝑅^2𝑁−1)𝜏+𝐾(𝐼−²𝐴2)𝐺⁻²𝑅^2𝑁−1𝜏−𝐾(𝐼−²𝐴2)𝐺⁻²(2𝐼+𝜏𝐵)𝑅^𝑁𝑃^𝑁)⁻¹. (2.13)
Hence, we obtain that 𝑢₀=12𝑇_𝜏𝐾𝐺⁻²×{(2𝐼−𝜏²𝐴){(2+𝜏𝐵)𝑅^𝑁[−𝜏⁰_{𝑠=−𝑁+1}𝑃^{𝑠+𝑁−1}𝐺𝑓_𝑠+𝜇]−𝑅^𝑁−1𝐵^−1𝑁−1_𝑠=1[𝑅^𝑁−𝑠−𝑅^𝑁+𝑠]𝑔_𝑠𝜏+(𝐼−𝑅^2𝑁)𝐵^−1𝑁−1_𝑠=1𝑅^𝑠−1𝑔_𝑠𝜏}+(𝐼−𝑅^2𝑁)(𝐼+𝜏𝐵)(𝜏𝐵⁻¹𝑔₁−4𝐺𝐵⁻¹𝑓₀+𝑃𝐺𝐵⁻¹𝑓₀+𝐺𝐵⁻¹𝑓₋₁)}. (2.14)
This concludes the proof of Theorem 2.1.

3. Main Theorems

Here, we study well posedness of problem (1.2). First, we give some necessary estimates for 𝑃^𝑘,𝑅^𝑘, and 𝑇_𝜏. For a self-adjoint positive operator 𝐴, the following estimates are satisfied [36, 38, 39]:‖𝑃^𝑘‖_𝐻→𝐻≤1,‖𝐺‖_𝐻→𝐻≤1,𝑘𝜏‖𝐴𝑃^𝑘𝐺²‖_𝐻→𝐻≤𝑀,𝑘≥1,𝛿>0,‖𝑅^𝑘‖_𝐻→𝐻≤𝑀(1+𝛿𝜏)^−𝑘,𝑘𝜏‖𝐵𝑅^𝑘‖_𝐻→𝐻≤𝑀,𝑘≥1,𝛿>0, (3.1)(3.2) where 𝑀 is independent of 𝜏. From these estimates, it follows that‖(𝐼+𝐵⁻¹𝜏𝐴(𝐼+𝜏𝐴+2𝐺⁻²)𝐾(𝐼−𝑅^2𝑁−1𝜏)+𝐾(𝐼−²𝐴2)𝐺⁻²𝑅^2𝑁−1𝜏−𝐾(𝐼−²𝐴2)𝐺⁻²(2𝐼+𝜏𝐵)𝑅^𝑁𝑃^𝑁)⁻¹)⁻¹‖_𝐻→𝐻≤𝑀. (3.3)

Let 𝐹_𝜏(𝐻)=𝐹([𝑎,𝑏]_𝜏,𝐻) be the linear space of mesh functions 𝜑^𝜏={𝜑_𝑘}^𝑁𝑏_𝑁𝑎 defined on [𝑎,𝑏]_𝜏={𝑡_𝑘=𝑘ℎ,𝑁_𝑎≤𝑘≤𝑁_𝑏,𝑁_𝑎𝜏=𝑎,𝑁_𝑏𝜏=𝑏} with values in the Hilbert space 𝐻. Next on 𝐹_𝜏(𝐻) we denote 𝐶([𝑎,𝑏]_𝜏,𝐻) and 𝐶^𝛼_0,1([−1,1]_𝜏,𝐻),𝐶^𝛼_0,1([−1,0]_𝜏,𝐻),𝐶^𝛼₀([0,1]_𝜏,𝐻)(0<𝛼<1) Banach spaces with the norms ‖𝜑^𝜏‖_{𝐶([𝑎,𝑏]𝜏,𝐻)}=max_{𝑁𝑎≤𝑘≤𝑁𝑏}‖𝜑_𝑘‖_𝐻,‖𝜑^𝜏‖_{𝐶𝛼0,1([−1,1]𝜏,𝐻)}=‖𝜑^𝜏‖_{𝐶([−1,1]𝜏,𝐻)}+sup_{−𝑁≤𝑘<𝑘+𝑟≤0}‖𝜑_𝑘+𝑟−𝜑_𝑘‖_𝐸(−𝑘)^𝛼𝑟^𝛼+sup_{1≤𝑘<𝑘+𝑟≤𝑁−1}‖𝜑_𝑘+𝑟−𝜑_𝑘‖_𝐸((𝑘+𝑟)𝜏)^𝛼(𝑁−𝑘)^𝛼𝑟^𝛼,‖𝜑^𝜏‖_{𝐶𝛼0([−1,0]𝜏,𝐻)}=‖𝜑^𝜏‖_{𝐶([−1,0]𝜏,𝐻)}+sup_{−𝑁≤𝑘<𝑘+𝑟≤0}‖𝜑_𝑘+𝑟−𝜑_𝑘‖_𝐸(−𝑘)^𝛼𝑟^𝛼,‖𝜑^𝜏‖_{𝐶𝛼0,1([0,1]𝜏,𝐻)}=‖𝜑^𝜏‖_{𝐶([0,1]𝜏,𝐻)}+sup_{1≤𝑘<𝑘+𝑟≤𝑁−1}‖𝜑_𝑘+𝑟−𝜑_𝑘‖_𝐸((𝑘+𝑟)𝜏)^𝛼(𝑁−𝑘)^𝛼𝑟^𝛼. (3.4)

Nonlocal boundary value problem (1.2) is said to be stable in 𝐹([−1,1]_𝜏,𝐻) if we have the inequality ‖𝑢^𝜏‖_{𝐹([−1,1]𝜏,𝐻)}≤𝑀[‖𝑓^𝜏‖_{𝐹([−1,0]𝜏,𝐻)}+‖𝑔^𝜏‖_{𝐹([0,1]𝜏,𝐻)}+‖𝜇‖_𝐻], (3.5) where 𝑀 is independent of not only 𝑓^𝜏,𝑔^𝜏,and𝜇 but also 𝜏.

Theorem 3.1. Nonlocal boundary value problem (1.2) is stable in 𝐶([−1,1]_𝜏,𝐻) norm.

Proof. By [38], we have ‖{𝑢_𝑘}₁^𝑁−1‖_{𝐶([0,1]𝜏,𝐻)}≤𝑀[‖𝑔^𝜏‖_{𝐶([0,1]𝜏,𝐻)}+‖𝑢₀‖_𝐻+‖𝑢_𝑁‖_𝐻] (3.6) for the solution of boundary value problem (2.8).
By [36], we get ‖{𝑢_𝑘}⁰_−𝑁‖_{𝐶([−1,0]𝜏,𝐻)}≤𝑀[‖𝑓^𝜏‖_{𝐶([−1,0]𝜏,𝐻)}+‖𝑢₀‖_𝐻] (3.7) for the solution of an inverse Cauchy difference problem (2.6). Then, the proof of Theorem 3.1 is based on the stability inequalities (3.6), (3.7), and on the estimates ‖𝑢₀‖_𝐻≤𝑀[‖𝑓^𝜏‖_{𝐶([−1,0]𝜏,𝐻)}+‖𝑔^𝜏‖_{𝐶([1,0]𝜏,𝐻)}+‖𝜇‖_𝐻],‖𝑢_𝑁‖_𝐻≤𝑀[‖𝑓^𝜏‖_{𝐶([−1,0]𝜏,𝐻)}+‖𝑔^𝜏‖_{𝐶([1,0]𝜏,𝐻)}+‖𝜇‖_𝐻], (3.8)(3.9) for the solution of the boundary value problem (1.2). Estimates (3.8) and (3.9) follow from formula (2.5) and estimates (3.1), (3.2), and (3.3) which conclude the proof of Theorem 3.1.

Theorem 3.2. Assume that 𝜇∈𝐷(𝐴) and 𝑓₀,𝑓₋₁,𝑔₁∈𝐷(𝐼+𝜏𝐵). Then, for the solution of difference problem (1.2), we have the following almost coercivity inequality: ‖{𝜏⁻²(𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1)}₁^𝑁−1‖_{𝐶([0,1]𝜏,𝐻)}+‖{𝜏⁻¹(𝑢_𝑘−𝑢_𝑘−1)}⁰_−𝑁+1‖_{𝐶([−1,0]𝜏,𝐻)}+‖{𝐴𝑢_𝑘}₁^𝑁−1‖_{𝐶([0,1]𝜏,𝐻)}1+‖{2(𝐴𝑢_𝑘+𝐴𝑢_𝑘−1)}⁰_−𝑁+1‖_{𝐶([−1,0]𝜏,𝐻)}1≤𝑀[min{ln𝜏,1+|ln‖𝐴‖_𝐻→𝐻|}[‖𝑓^𝜏‖_{𝐶([−1,0]𝜏,𝐻)}+‖𝑔^𝜏‖_{𝐶([0,1]𝜏,𝐻)}]+‖𝐴𝜇‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₀‖_𝐻+‖(𝐼+𝜏𝐵)𝑔₁‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₋₁‖_𝐻], (3.10) where 𝑀 does not dependent on not only 𝑓^𝜏,𝑔^𝜏,and𝜇 but also 𝜏.

Proof. By [40], we have ‖{𝜏⁻¹(𝑢_𝑘−𝑢_𝑘−1)}⁰_−𝑁+1‖_{𝐶([−1,0]𝜏,𝐻)}1+‖{2(𝐴𝑢_𝑘+𝐴𝑢_𝑘−1)}⁰_−𝑁+1‖_{𝐶([−1,0]𝜏,𝐻)}1≤𝑀[min{ln𝜏,1+|ln‖𝐴‖_𝐻→𝐻|}‖𝑓^𝜏‖_{𝐶([−1,0]𝜏)}+‖𝐴𝑢₀‖_𝐻] (3.11) for the solution of an inverse Cauchy difference problem (2.6).
By [38], we get ‖{𝜏⁻²(𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1)}₁^𝑁−1‖_{𝐶([0,1]𝜏,𝐻)}+‖{𝐴𝑢_𝑘}₁^𝑁−1‖_{𝐶([0,1]𝜏,𝐻)}1≤𝑀[min{ln𝜏,1+|ln‖𝐴‖_𝐻→𝐻|}‖𝑔^𝜏‖_{𝐶([0,1]𝜏,𝐻)}+‖𝐴𝑢₀‖_𝐻+‖𝐴𝑢_𝑁‖_𝐻] (3.12) for the solution of boundary value problem (2.8).
Then, the proof of Theorem 3.2 is based on almost coercivity inequalities (3.11), (3.12), and on the estimates ‖𝐴𝑢₀‖_𝐻≤𝑀[‖𝐴𝜇‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₀‖_𝐻1+min{ln𝜏,1+|ln‖𝐴‖_𝐻→𝐻|}[‖𝑓^𝜏‖_{𝐶([−1,0]𝜏,𝐻)}+‖𝑔^𝜏‖_{𝐶([0,1]𝜏,𝐻)}]],‖𝐴𝑢_𝑁‖_𝐻≤𝑀[[‖𝐴𝜇‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₀‖_𝐻]1+min{ln𝜏,1+|ln‖𝐴‖_𝐻→𝐻|}[‖𝑓^𝜏‖_{𝐶([−1,0]𝜏,𝐻)}+‖𝑔^𝜏‖_{𝐶([0,1]𝜏,𝐻)}]] (3.13) for the solution of boundary value problem (1.2). The proof of these estimates follows the scheme of the papers [38, 40] and relies on both the formula (2.5) and the estimates (3.1), (3.2), and (3.3).
This concludes the proof of Theorem 3.2.

Let 𝐶^𝛼_0,1([−1,1]_𝜏,𝐻),𝐶^𝛼₀([−1,0]_𝜏,𝐻),0<𝛼<1 be the Banach spaces with the norms ‖𝜑^𝜏‖𝐶^𝛼_{0,1([−1,1]𝜏,𝐻)}=‖𝜑^𝜏‖_{𝐶([−1,1]𝜏,𝐻)}+sup_{−𝑁≤𝑘<𝑘+2𝑟≤0}‖𝜑_𝑘+2𝑟−𝜑_𝑘‖_𝐸(−𝑘)^𝛼(2𝑟)^𝛼+sup_{1≤𝑘<𝑘+𝑟≤𝑁−1}‖𝜑_𝑘+2𝑟−𝜑_𝑘‖_𝐸((𝑘+𝑟)𝜏)^𝛼(𝑁−𝑘)^𝛼𝑟^𝛼,‖𝜑^𝜏‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}=‖𝜑^𝜏‖_{𝐶([−1,0]𝜏,𝐻)}+sup_{−𝑁≤𝑘<𝑘+2𝑟≤0}‖𝜑_𝑘+2𝑟−𝜑_𝑘‖_𝐸(−𝑘)^𝛼(2𝑟)^𝛼. (3.14)

Theorem 3.3. Let the assumptions of Theorem 3.2 be satisfied. Then, boundary value problem (1.2) is well posed in Hölder spaces 𝐶^𝛼_0,1([−1,1]_𝜏,𝐻) and 𝐶^𝛼_0,1([−1,1]_𝜏,𝐻), and the following coercivity inequalities hold: ‖{𝜏⁻²(𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1)}₁^𝑁−1‖_{𝐶𝛼0,1([0,1]𝜏,𝐻)}+‖{𝜏⁻¹(𝑢_𝑘−𝑢_𝑘−1)}⁰_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}+‖{𝐴𝑢_𝑘}₁^𝑁−1‖_{𝐶𝛼0,1([0,1]𝜏,𝐻)}1+‖{2(𝐴𝑢_𝑘+𝐴𝑢_𝑘−1)}⁰_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}1≤𝑀[𝛼(1−𝛼)[‖𝑓^𝜏‖_{𝐶𝛼0([−1,0]𝜏,𝐻)}+‖𝑔^𝜏‖_{𝐶𝛼0,1([0,1]𝜏,𝐻)}]+‖𝐴𝜇‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₀‖_𝐻+‖(𝐼+𝜏𝐵)𝑔₁‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₋₁‖_𝐻],‖{𝜏⁻²(𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1)}₁^𝑁−1‖_{𝐶𝛼0,1([0,1]𝜏,𝐻)}+‖{𝜏⁻¹(𝑢_𝑘−𝑢_𝑘−1)}⁰_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}+‖{𝐴𝑢_𝑘}₁^𝑁−1‖_{𝐶𝛼0,1([0,1]𝜏,𝐻)}1+‖{2(𝐴𝑢_𝑘+𝐴𝑢_𝑘−1)}⁰_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}1≤𝑀[𝛼(1−𝛼)[‖𝑓^𝜏‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}+‖𝑔^𝜏‖_{𝐶𝛼0,1([0,1]𝜏,𝐻)}]+‖𝐴𝜇‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₀‖_𝐻+‖(𝐼+𝜏𝐵)𝑔₁‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₋₁‖_𝐻], (3.15) where 𝑀 is independent of not only 𝑓^𝜏,𝑔^𝜏, and 𝜇 but also 𝜏 and 𝛼.

Proof. By [39, 40], ‖{𝜏⁻¹(𝑢_𝑘−𝑢_𝑘−1)}⁰_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}1+‖{2(𝐴𝑢_𝑘+𝐴𝑢_𝑘−1)}⁰_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}1≤𝑀[𝛼(1−𝛼)‖𝑓^𝜏‖_{𝐶𝛼0([−1,0]𝜏,𝐻)}+‖𝐴𝑢₀‖_𝐻],‖{𝜏⁻¹(𝑢_𝑘−𝑢_𝑘−1)}⁰_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}1+‖{2(𝐴𝑢_𝑘+𝐴𝑢_𝑘−1)}⁰_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}1≤𝑀[𝛼(1−𝛼)‖𝑓^𝜏‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}+‖𝐴𝑢₀‖_𝐻] (3.16)(3.17) for the solution of an inverse Cauchy difference problem (2.6) can be written.
By [37, 38], we get ‖{𝜏⁻²(𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1)}₁^𝑁−1‖_{𝐶𝛼0,1([0,1]𝜏,𝐻)}+‖{𝐴𝑢_𝑘}₁^𝑁−1‖_{𝐶𝛼0,1([0,1]𝜏,𝐻)}1≤𝑀[𝛼(1−𝛼)‖𝑔^𝜏‖_{𝐶𝛼0,1([0,1]𝜏,𝐻)}+‖𝐴𝑢₀‖_𝐻+‖𝐴𝑢_𝑁‖_𝐻] (3.18) for the solution of boundary value problem (2.8).
Then, the proof of Theorem 3.3 is based on coercivity inequalities (3.16)–(3.18), and the estimates ‖𝐴𝑢₀‖_𝐻1≤𝑀[𝛼(1−𝛼)[‖𝑓^𝜏‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}+‖𝑔^𝜏‖_{𝐶𝛼0,1([0,1]𝜏,𝐻)}]+‖𝐴𝑢‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₀‖_𝐻+‖(𝐼+𝜏𝐵)𝑔₁‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₋₁‖_𝐻],‖𝐴𝑢_𝑁‖_𝐻1≤𝑀[𝛼(1−𝛼)[‖𝑓^𝜏‖𝐶^𝛼₀^([−1,0]_𝜏^,𝐻)+‖𝑔^𝜏‖_{𝐶𝛼0,1([0,1]𝜏,𝐻)}]+‖𝐴𝜇‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₀‖_𝐻+‖(𝐼+𝜏𝐵)𝑔₁‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₋₁‖_𝐻] (3.19)(3.20) for the solution of boundary value problem (1.2).
Estimates (3.19) and (3.20) follow from the formulas 𝐴𝑢₀=12𝑇_𝜏𝐾𝐺⁻²×{(2𝐼−𝜏²𝐴){(2+𝜏𝐵)𝑅^𝑁[−𝜏⁰_{𝑠=−𝑁+1}𝐴𝑃^{𝑠+𝑁−1}𝐺(𝑓_𝑠−𝑓_−𝑁+1)+𝐴𝜇]−𝑅^𝑁−1𝐴𝐵^−1𝑁−1_𝑠=1𝑅^𝑁−𝑠(𝑔_𝑠−𝑔_𝑁−1)𝜏+𝑅^𝑁−1𝐴𝐵^−1𝑁−1_𝑠=1𝑅^𝑁+𝑠(𝑔_𝑠−𝑔₁)𝜏+(𝐼−𝑅^2𝑁)𝐴𝐵^−1𝑁−1_𝑠=1𝑅^𝑠−1(𝑔_𝑠−𝑔₁)𝜏}+(𝐼−𝑅^2𝑁)(𝐼+𝜏𝐵)(𝜏𝐵⁻¹𝐴𝑔₁−4𝐺𝐵⁻¹𝐴𝑓₀+𝑃𝐺𝐵⁻¹𝐴𝑓₀+𝐺𝐵⁻¹𝐴𝑓₋₁)+(2𝐼−𝜏²𝐴)(2+𝜏𝐵)𝑅^𝑁(𝑃^𝑁−𝐼)𝑓_−𝑁+1+𝐴𝐵⁻²(𝑅^𝑁−1−𝐼){𝑅^𝑁−1𝑔_𝑁−1+(𝑅^2𝑁−𝑅^2𝑁−1−𝐼)𝑔₁}},𝐴𝑢_𝑁=12𝑃^𝑁𝑇_𝜏𝐾𝐺⁻²×{(2𝐼−𝜏²𝐴){(2+𝜏𝐵)𝑅^𝑁[−𝜏⁰_{𝑠=−𝑁+1}𝐴𝑃^{𝑠+𝑁−1}𝐺(𝑓_𝑠−𝑓_−𝑁+1)+𝐴𝜇]−𝑅^𝑁−1𝐴𝐵^−1𝑁−1_𝑠=1𝑅^𝑁−𝑠(𝑔_𝑠−𝑔_𝑁−1)𝜏+𝑅^𝑁−1𝐴𝐵^−1𝑁−1_𝑠=1𝑅^𝑁+𝑠(𝑔_𝑠−𝑔₁)𝜏+(𝐼−𝑅^2𝑁)𝐴𝐵^−1𝑁−1_𝑠=1𝐵𝑅^𝑠−1(𝑔_𝑠−𝑔₁)𝜏}+(𝐼−𝑅^2𝑁)(𝐼+𝜏𝐵)(𝜏𝐵⁻¹𝐴𝑔₁−4𝐺𝐵⁻¹𝐴𝑓₀+𝑃𝐺𝐵⁻¹𝐴𝑓₀+𝐺𝐵⁻¹𝐴𝑓₋₁)+(2𝐼−𝜏²𝐴)(2+𝜏𝐵)𝑅^𝑁(𝑃^𝑁−𝐼)𝑓_−𝑁+1+𝐴𝐵⁻²(𝑅^𝑁−1−𝐼){𝑅^𝑁−1𝑔_𝑁−1+(𝑅^2𝑁−𝑅^2𝑁−1−𝐼)𝑔₁}}−𝜏⁰_{𝑠=−𝑁+1}𝐴𝑃^{𝑠+𝑁−1}𝐺(𝑓_𝑠−𝑓_−𝑁+1)+𝐴𝜇+(𝑃^𝑁−𝐼)𝑓_−𝑁+1 (3.21)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) for the solution of problem (1.2) and estimates (3.1), (3.2), and (3.3).
This finalizes the proof of Theorem 3.3.

4. Applications

In this section, we indicate applications of Theorems 3.1, 3.2, and 3.3 to obtain the stability, the almost coercive stability, and the coercive stability estimates for the solutions of these difference schemes for the approximate solution of nonlocal mixed problems. First, let Ω be the unit open cube in the 𝑛-dimensional Euclidean space ℝ^𝑛(0<𝑥_𝑘<1,1≤𝑘≤𝑛) with boundary 𝑆,Ω=Ω∪𝑆. In [−1,1]×Ω, the boundary value problem for the multidimensional elliptic-parabolic equation−𝑢_𝑡𝑡−^𝑛_𝑟=1(𝑎_𝑟(𝑥)𝑢_𝑥𝑟)_𝑥𝑟𝑢=𝑔(𝑡,𝑥),0<𝑡<1,𝑥∈Ω,_𝑡+^𝑛_𝑟=1(𝑎_𝑟(𝑥)𝑢_𝑥𝑟)_𝑥𝑟=𝑓(𝑡,𝑥),−1<𝑡<0,𝑥∈Ω,𝑢(𝑡,𝑥)=0,𝑥∈𝑆,−1≤𝑡≤1,𝑢(1,𝑥)=𝑢(−1,𝑥)+𝜇(𝑥),𝑥∈Ω,𝑢(0+,𝑥)=𝑢(0−,𝑥),𝑢_𝑡(0+,𝑥)=𝑢_𝑡(0−,𝑥),𝑥∈Ω (4.1) is considered. Problem (4.1) has a unique smooth solution 𝑢(𝑡,𝑥) for 𝑓(𝑡,𝑥)(𝑡∈(−1,0),𝑥∈Ω),𝑔(𝑡,𝑥)(𝑡∈(0,1),𝑥∈Ω) the smooth functions, and 𝑎_𝑟(𝑥)⩾𝑎>0(𝑥∈Ω).

The discretization of problem (4.1) is carried out in two steps. In the first step, the grid setsΩ_ℎ={𝑥=𝑥_𝑚=(ℎ₁𝑚₁,…,ℎ_𝑛𝑚_𝑛),𝑚=(𝑚₁,…,𝑚_𝑛),0≤𝑚_𝑟≤𝑁_𝑟,ℎ_𝑟𝑁_𝑟Ω=1,𝑟=1,…,𝑛},_ℎ=Ω_ℎ∩Ω,𝑆_ℎ=Ω_ℎ∩𝑆 (4.2) are defined. To the differential operator 𝐴 generated by problem (4.1), we assign the difference operator 𝐴^𝑥_ℎ by the formula𝐴^𝑥_ℎ𝑢^ℎ_𝑥=−^𝑛_𝑟=1(𝑎_𝑟(𝑥)𝑢^ℎ_𝑥𝑟)_{𝑥𝑟,𝑚𝑟} (4.3) acting in the space of grid functions 𝑢^ℎ(𝑥), satisfying the conditions 𝑢^ℎ(𝑥)=0 for all 𝑥∈𝑆_ℎ. With the help of 𝐴^𝑥_ℎ, we arrive at the nonlocal boundary value problem−𝑑²𝑢^ℎ(𝑡,𝑥)𝑑𝑡²+𝐴^𝑥_ℎ𝑢^ℎ(𝑡,𝑥)=𝑔^ℎ(𝑡,𝑥),0<𝑡<1,𝑥∈Ω_ℎ,𝑑𝑢^ℎ(𝑡,𝑥)𝑑𝑡−𝐴^𝑥_ℎ𝑢^ℎ(𝑡,𝑥)=𝑓^ℎ(𝑡,𝑥),−1<𝑡<0,𝑥∈Ω_ℎ,𝑢^ℎ(1,𝑥)=𝑢^ℎ(−1,𝑥)+𝜇^ℎ(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ(0+,𝑥)=𝑢^ℎ(0−,𝑥),𝑑𝑢^ℎ(0+,𝑥)=𝑑𝑡𝑑𝑢^ℎ(0−,𝑥)Ω𝑑𝑡,𝑥∈_ℎ (4.4) for an infinite system of ordinary differential equations.

Replacing problem (4.4) by the difference scheme (1.2), one can obtain the second order of accuracy difference scheme−𝑢^ℎ_𝑘+1(𝑥)−2𝑢^ℎ_𝑘(𝑥)+𝑢^ℎ_𝑘−1(𝑥)𝜏²+𝐴^𝑥_ℎ𝑢^ℎ_𝑘(𝑥)=𝑔^ℎ_𝑘𝑔(𝑥),_ℎ𝑘(𝑥)=𝑔(𝑡_𝑘,𝑥_𝑛),𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,𝑥∈Ω_ℎ,𝑢^ℎ_𝑘(𝑥)−𝑢^ℎ_𝑘−1(𝑥)𝜏−𝐴^𝑥_ℎ2(𝑢^ℎ_𝑘(𝑥)+𝑢^ℎ_𝑘−1(𝑥))=𝑓^ℎ_𝑘𝑓(𝑥),_ℎ𝑘(𝑥)=𝑓(𝑡_𝑘−1/2,𝑥_𝑛),𝑡_𝑘−1/21=(𝑘−2)𝜏,−𝑁+1≤𝑘≤0,𝑥∈Ω_ℎ,𝑢^ℎ_𝑁(𝑥)=𝑢^ℎ_−𝑁(𝑥)+𝜇^ℎ(𝑥),𝑥∈Ω_ℎ,−𝑢^ℎ₂(𝑥)+4𝑢^ℎ₁(𝑥)−3𝑢^ℎ₀(𝑥)=3𝑢^ℎ₀(𝑥)−4𝑢^ℎ₋₁(𝑥)+𝑢^ℎ₋₂(𝑥),𝑥∈Ω_ℎ. (4.5)

Let us give a corollary of Theorems 3.1 and 3.2.

Theorem 4.1. Let 𝜏 and |ℎ|=ℎ²₁+⋅⋅⋅+ℎ²_𝑛 be sufficiently small positive numbers. Then, solutions of difference scheme (4.5) satisfy the following stability and almost coercivity estimates: ‖{𝑢^ℎ_𝑘}^𝑁−1_−𝑁‖_{𝐶([−1,1]𝜏,𝐿2ℎ)}≤𝑀[‖{𝑓^ℎ_𝑘}⁻¹_−𝑁+1‖_{𝐶([−1,0]𝜏,𝐿2ℎ)}+‖{𝑔^ℎ_𝑘}₁^𝑁−1‖_{𝐶([0,1]𝜏,𝐿2ℎ)}+‖𝜇^ℎ‖_𝐿2ℎ],‖{𝜏⁻²(𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^ℎ_𝑘−1)}₁^𝑁−1‖_{𝐶([0,1]𝜏,𝐿2ℎ)}+‖{𝑢^ℎ_𝑘}₁^𝑁−1‖_{𝐶([0,1]𝜏,𝑊22ℎ)}+‖{𝜏⁻¹(𝑢^ℎ_𝑘−𝑢^ℎ_𝑘−1)}⁰_−𝑁+1‖_{𝐶([−1,0]𝜏,𝐿2ℎ)}𝑢+‖{^ℎ𝑘+𝑢^ℎ_𝑘−12}⁰_−𝑁+1‖_{𝐶([−1,0]𝜏,𝑊22ℎ)}≤𝑀[‖𝑓^ℎ₀‖_𝐿2ℎ+‖𝑓^ℎ₋₁‖_𝐿2ℎ+‖𝑔^ℎ₁‖_𝐿2ℎ+‖𝜇^ℎ‖_𝑊22ℎ+𝜏‖𝑓^ℎ₀‖_𝑊12ℎ+𝜏‖𝑓^ℎ₋₁‖_𝑊12ℎ+𝜏‖𝑔^ℎ₁‖_𝑊12ℎ1+ln𝜏+|ℎ|[‖{𝑓^ℎ_𝑘}⁻¹_−𝑁+1‖_{𝐶([−1,0]𝜏,𝐿2ℎ)}+‖{𝑔^ℎ_𝑘}₁^𝑁−1‖_{𝐶([0,1]𝜏,𝐿2ℎ)}]]. (4.6)(15)(16)(17)(18)(19)(20) Here, 𝑀 is independent of not only 𝜏,ℎ,𝜇^ℎ(𝑥) but also 𝑔^ℎ_𝑘(𝑥),1≤𝑘≤𝑁−1, and 𝑓^ℎ_𝑘,−𝑁+1≤𝑘≤0.

The proof of Theorem 4.1 is based on Theorems 3.1 and 3.2, the estimate1min{ln𝜏,1+|ln‖𝐴^𝑥_ℎ‖_{𝐿2ℎ→𝐿2ℎ}1|}≤𝑀ln,𝜏+|ℎ| (4.7) the symmetry properties of the difference operator 𝐴^𝑥_ℎ defined by formula (4.3) in 𝐿_2ℎ, and the following theorem.

Theorem 4.2. For the solution of the elliptic difference problem 𝐴^𝑥_ℎ𝑢^ℎ(𝑥)=𝜔^ℎ(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ(𝑥)=0,𝑥∈𝑆_ℎ (4.8) the following coercivity inequality holds [41]: ^𝑛_𝑟=1‖(𝑢^ℎ)_{𝑥𝑟𝑥𝑟,𝑚𝑟}‖_𝐿2ℎ≤𝑀‖𝜔^ℎ‖_𝐿2ℎ. (4.9)

Let us give a corollary of Theorem 3.3.

Theorem 4.3. Let 𝜏 and |ℎ| be sufficiently small positive numbers. Then, solutions of difference scheme (4.5) satisfy the following coercivity stability estimates: ‖{𝜏⁻²(𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^ℎ_𝑘−1)}₁^𝑁−1‖_{𝐶𝛼0,1([0,1]𝜏,𝐿2ℎ)}+‖{𝜏⁻¹(𝑢^ℎ_𝑘−𝑢^ℎ_𝑘−1)}⁰_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝐿2ℎ)}+‖{𝑢^ℎ_𝑘}₁^𝑁−1‖_{𝐶𝛼0,1([0,1]𝜏,𝑊22ℎ)}𝑢+‖{^ℎ𝑘+𝑢^ℎ_𝑘−12}⁰_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝑊}²_2ℎ)≤𝑀[‖𝜇^ℎ‖_𝑊22ℎ+𝜏‖𝑓^ℎ₀‖_𝑊12ℎ+𝜏‖𝑓^ℎ₋₁‖_𝑊12ℎ+𝜏‖𝑔^ℎ₁‖_𝑊12ℎ+1𝛼(1−𝛼)[‖{𝑓^ℎ_𝑘}⁻¹_−𝑁+1‖_{𝐶𝛼0([−1,0]𝜏,𝐿2ℎ)}+‖{𝑔^ℎ_𝑘}₁^𝑁−1‖_{𝐶𝛼0,1([0,1]𝜏,𝐿2ℎ)}]],‖{𝜏⁻²(𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^ℎ_𝑘−1)}₁^𝑁−1‖_{𝐶𝛼0,1([0,1]𝜏,𝐿2ℎ)}𝑢+‖{^ℎ𝑘+𝑢^ℎ_𝑘−12}⁰_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝑊}²_2ℎ)+‖{𝜏⁻¹(𝑢^ℎ_𝑘−𝑢^ℎ_𝑘−1)}⁰_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝐿2ℎ)}+‖{𝑢^ℎ_𝑘}₁^𝑁−1‖_{𝐶𝛼0,1([0,1]𝜏,𝑊22ℎ)}≤𝑀[‖𝜇^ℎ‖_𝑊22ℎ+𝜏‖𝑓^ℎ₀‖_𝑊12ℎ+𝜏‖𝑓^ℎ₋₁‖_𝑊12ℎ+𝜏‖𝑔^ℎ₁‖_𝑊12ℎ+1𝛼(1−𝛼)[‖{𝑓^ℎ_𝑘}⁻¹_−𝑁+1‖𝐶^𝛼_{0([−1,0]𝜏,𝐿2ℎ)}+‖{𝑔^ℎ_𝑘}₁^𝑁−1‖_{𝐶𝛼0,1([0,1]𝜏,𝐿2ℎ)}]], (4.10)(21)(22)(23)(24)(25)(26)(27)(28)(29) where 𝑀 is independent of not only 𝜏,ℎ,and𝜇^ℎ(𝑥) but also 𝑔^ℎ_𝑘(𝑥),1≤𝑘≤𝑁−1 and𝑓^ℎ_𝑘,−𝑁+1≤𝑘≤0.

The proof of Theorem 4.3 is based on the abstract Theorems 3.3 and 4.2, and the symmetry properties of the difference operator 𝐴^𝑥_ℎ defined by the formula (4.3).

Second, the mixed boundary value problem for the elliptic-parabolic equation−𝑢_𝑡𝑡−(𝑎(𝑥)𝑢_𝑥)_𝑥𝑢+𝛿𝑢=𝑔(𝑡,𝑥),0<𝑡<1,0<𝑥<1,_𝑡+(𝑎(𝑥)𝑢_𝑥)_𝑥−𝛿𝑢=𝑓(𝑡,𝑥),−1<𝑡<0,0<𝑥<1,𝑢(𝑡,0)=𝑢(𝑡,1),𝑢_𝑥(𝑡,0)=𝑢_𝑥(𝑡,1),−1≤𝑡≤1,𝑢(1,𝑥)=𝑢(−1,𝑥)+𝜇(𝑥),0≤𝑥≤1,𝑢(0+,𝑥)=𝑢(0−,𝑥),𝑢_𝑡(0+,𝑥)=𝑢_𝑡(0−,𝑥),0≤𝑥≤1 (4.11) is considered. Problem (4.11) has a unique smooth solution 𝑢(𝑡,𝑥) for 𝑓(𝑡,𝑥)(𝑡∈[−1,0],𝑥∈[0,1]),𝑔(𝑡,𝑥)(𝑡∈[0,1],𝑥∈[0,1]), the smooth functions, and 𝑎(𝑥)⩾𝑎>0(𝑥∈(0,1)),𝛿=const>0.

Note that in a similar manner one can construct the difference schemes of the second order of accuracy with respect to one variable for approximate solutions of the boundary value problem (4.11). Abstract theorems given above permit us to obtain the stability, the almost stability and the coercive stability estimates for the solutions of these difference schemes.

5. Numerical Results

We consider the nonlocal boundary value problem𝜕𝑢+𝜕𝜕𝑡²𝑢𝜕𝑥²𝜕=sin𝑥,−1<𝑡≤0,0<𝑥<𝜋,₂𝑢𝜕𝑡²+𝜕²𝑢𝜕𝑥²=sin𝑥,0<𝑡<1,0<𝑥<𝜋,𝑢(1,𝑥)=𝑢(−1,𝑥)+2sinh1sin𝑥,0≤𝑥≤𝜋,𝑢(𝑡,0)=𝑢(𝑡,𝜋)=0,−1≤𝑡≤1 (5.1) for the elliptic-parabolic equation.

The exact solution of this problem is 𝑢(𝑡,𝑥)=(𝑒^𝑡−1)sin𝑥.

Now, we give the results of the numerical analysis. The errors computed by 𝐸^𝑁_𝑀=max_{−𝑁≤𝑘≤𝑁,1≤𝑛≤𝑀−1}|𝑢(𝑡_𝑘,𝑥_𝑛)−𝑢^𝑘_𝑛| (5.2) of the numerical solutions are given inTable 1.

Table 1 

Comparison of the errors.

Thus, the second order of accuracy difference scheme is more accurate than the first order of accuracy difference scheme.

Acknowledgments

The authors would like to thank the referees and Professor P. E. Sobolevskii (Jerusalem, Israel) for helpful suggestions to the improvement of this paper.