# A Note on the Second Order of Accuracy Stable Difference Schemes for the Nonlocal Boundary Value Hyperbolic Problem

Authors: Allaberen Ashyralyev; Ozgur Yildirim
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The second order of accuracy absolutely stable difference schemes are presented for the nonlocal boundary value hyperbolic problem for the differential equations in a Hilbert spaceH with the self-adjoint positive definite operator A. The stability estimates for the solutions of these difference schemes are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered. The stability estimates for the solutions of these difference schemes for the nonlocal boundary value hyperbolic problem are established. Finally, a numerical method proposed and numerical experiments, analysis of the errors, and related execution times are presented in order to verify theoretical statements.


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## Body

## 1. Introduction

Hyperbolic partial differential equations play an important role in many branches of science and engineering and can be used to describe a wide variety of phenomena such as acoustics, electromagnetics, hydrodynamics, elasticity, fluid mechanics, and other areas of physics (see [1–5] and the references given therein).While applying mathematical modelling to several phenomena of physics, biology, and ecology, there often arise problems with nonclassical boundary conditions, which the values of unknown function on the boundary are connected with inside of the given domain. Such type of boundary conditions are called nonlocal boundary conditions. Over the last decades, boundary value problems with nonlocal boundary conditions have become a rapidly growing area of research (see, e.g., [6–16] and the references given therein).In the present work, we consider the nonlocal boundary value problem(1.1)d2u(t)dt2+Au(t)=f(t)(0≤t≤1),u(0)=∑j=1n‍αju(λj)+φ,ut(0)=∑j=1n‍βjut(λj)+ψ,0<λ1<λ2<⋯<λn≤1,
where A is a self-adjoint positive definite operator in a Hilbert space H.A functionu(t) is called a solution of the problem (1.1), if the following conditions are satisfied:(i)
u(t) is twice continuously differentiable on the segment [0,1]. The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.(ii)
The elementu(t) belongs to D(A), independent of t, and dense in H for all t∈[0,1] and the function Au(t) is continuous on the segment [0,1].(iii)
u(t) satisfies the equation and nonlocal boundary conditions (1.1).In the paper of [8], the following theorem on the stability estimates for the solution of the nonlocal boundary value problem (1.1) was proved.Theorem 1.1.
Suppose thatφ∈D(A), ψ∈D(A1/2), and f(t) is a continuously differentiable function on [0,1] and the assumption
(1.2)∑k=1n‍|αk+βk|+∑m=1n‍|αm|∑k=1k≠mn‍|βk|<|1+∑k=1n‍αkβk|
holds. Then, there is a unique solution of problem (1.1) and the stability inequalities
(1.3)max⁡0≤t≤1‖u(t)‖H≤M[‖φ‖H+‖A-1/2ψ‖H+max⁡0≤t≤1‖A-1/2f(t)‖H],max⁡0≤t≤1‖A1/2u(t)‖H≤M[‖A1/2φ‖H+‖ψ‖H+max⁡0≤t≤1‖f(t)‖H],max⁡0≤t≤1‖d2u(t)dt2‖H+max⁡0≤t≤1‖Au(t)‖H≤M[‖Aφ‖H+‖A1/2ψ‖H+‖f(0)‖H+∫01‖f′(t)‖Hdt]
hold, where M does not depend on φ,  ψ, and f(t),  t∈[0,1].Moreover, the first order of accuracy difference scheme(1.4)τ-2(uk+1-2uk+uk-1)+Auk+1=fk,fk=f(tk),tk=kτ,1≤k≤N-1,Nτ=1,u0=∑r=1n‍αru[λr/τ]+φ,τ-1(u1-u0)=∑r=1n‍βr(u[λr/τ]+1-u[λr/τ])1τ+ψ,
for the approximate solution of problem (1.1) was presented. The stability estimates for the solution of this difference scheme, under the assumption
(1.5)∑k=1n‍|αk|+∑k=1n‍|βk|+∑k=1n‍|αk|∑k=1n‍|βk|<1,
were established.In the development of numerical techniques for solving PDEs, the stability has been an important research topic (see [6–31]). A large cycle of works on difference schemes for hyperbolic partial differential equations, in which stability was established under the assumption that the magnitude of the grid steps τ and h with respect to the time and space variables, are connected. In abstract terms, this particularly means that τ∥Ah∥→0 when τ→0.We are interested in studying the high order of accuracy difference schemes for hyperbolic PDEs, in which stability is established without any assumption with respect to the grid stepsτ and h. Particularly, a convenient model for analyzing the stability is provided by a proper unconditionally absolutely stable difference scheme with an unbounded operator.In the present paper, the second order of accuracy unconditionally stable difference schemes for approximately solving boundary value problem (1.1) is presented. The stability estimates for the solutions of these difference schemes and their first and second order difference derivatives are established. This operator approach permits one to obtain the stability estimates for the solutions of difference schemes of nonlocal boundary value problems, for one-dimensional hyperbolic equation with nonlocal boundary conditions in space variable and multidimensional hyperbolic equation with Dirichlet condition in space variables.Some results of this paper without proof were presented in [7].Note that nonlocal boundary value problems for parabolic equations, elliptic equations, and equations of mixed types have been studied extensively by many scientists (see, e.g., [11–16, 20–24, 32–38] and the references therein).


## 2. The Second Order of Accuracy Difference Scheme Generated byA2

Throughout this paper for simplicityλ1>2τ and λn<1 will be considered. Let us associate boundary value problem (1.1) with the second order of accuracy difference scheme
(2.1)τ-2(uk+1-2uk+uk-1)+Auk+τ24A2uk+1=fk,fk=f(tk),tk=kτ,1≤k≤N-1,Nτ=1,u0=∑m=1n‍αm{u[λm/τ]+τ-1(u[λm/τ]-u[λm/τ]-1)×(λm-[λmτ]τ)}+φ,(I+τ2A2)τ-1(u1-u0)-τ2(f0-Au0)=∑k=1n‍βk{τ-1(u[λk/τ]-u[λk/τ]-1)+(τ2+(λk-[λkτ]τ))×(f[λk/τ]-Au[λk/τ])}+ψ,f0=f(0).
A study of discretization, over time only, of the nonlocal boundary value problem also permits one to include general difference schemes in applications, if the differential operator in space variables A is replaced by the difference operator Ah that act in the Hilbert space and are uniformly self-adjoint positive definite in h for 0<h≤h0.In general, we have not been able to obtain the stability estimates for the solution of difference scheme (2.1) under assumption (1.5). Note that the stability of solution of difference scheme (2.1) will be obtained under the strong assumption
(2.2)∑k=1n‍|αk|(1+|λk-[λkτ]τ|)+∑k=1n‍|βk|(1+|λk-[λkτ]τ|)+∑k=1n‍|αk|∑k=1n‍|βk|(1+|λk-[λkτ]τ|)+∑k=1n‍|αk||λk-[λkτ]τ|∑k=1n‍|βk||λk-[λkτ]τ|<1.
Now, let us give some lemmas that will be needed in the sequel.Lemma 2.1.
The following estimates hold:(2.3)‖R‖H↦H≤1,‖R~‖H↦H≤1,‖R~-1R‖H↦H≤1,‖R-1R~‖H↦H≤1,‖τA1/2R‖H↦H≤1,‖τA1/2R~‖H↦H≤1.
Here, H is the Hilbert space, R=(I+iτA1/2-(τ2/2)A)-1, and R~=(I-iτA1/2-(τ2/2)A)-1.Lemma 2.2.
Suppose that assumption (2.2) holds. Denote
(2.4)Bτ=∑m=1nαm(I+τ2A2)-1‍[R[λm/τ]-1(I-iτA1/2)+R~[λm/τ]-1(I+iτA1/2)2]+∑k=1n‍βk(I+τ2A2)-1[R~[λk/τ]-1R-1+R[λk/τ]-1R~-12]-12∑m=1n‍∑k=1n‍αmβk(I+τ2A2)-1[R[λm/τ]-1R~[λk/τ]-1+R~[λm/τ]-1R[λk/τ]-1]+∑m=1n‍αm(λm-[λmτ]τ)(iA1/2)(I+τ2A2)-1×[R~[λm/τ]-1(I-(iτ/2)A1/2)(I+iτA1/2)-R[λm/τ]-1(I+(iτ/2)A1/2)(I-iτA1/2)2]+∑k=1n‍βk(λk-[λkτ]τ)(iA1/2)(I+τ2A2)-1[R~[λk/τ]-1R-1-R[λk/τ]-1R~-12]+14∑m=1n‍∑k=1n‍αmβk(λm-[λmτ]τ)(I+τ2A2)-2×iA1/2[R[λm/τ]-1R~[λk/τ]-1(I+iτ2A1/2)-R~[λm/τ]-1R[λk/τ]-1(I-iτ2A1/2)]-12∑m=1n‍∑k=1n‍αmβk(λk-[λkτ]τ)(iA1/2)(I+τ2A2)-1×[R[λm/τ]-1R~[λk/τ]-1-R~[λm/τ]-1R[λk/τ]-1]-12∑m=1n‍∑k=1n‍αmβk(λm-[λmτ]τ)(λk-[λkτ]τ)A(I+τ2A2)-1×[R~[λm/τ]-1R[λk/τ]-1(I-iτ2A1/2)+R[λm/τ]-1R~[λk/τ]-1(I+iτ2A1/2)].
Then, the operator I-Bτ has an inverse
(2.5)Tτ=(I-Bτ)-1,
and the following estimate holds:
(2.6)‖Tτ‖H↦H≤M.Proof.
The proof of estimate (2.6) is based on the estimate
(2.7)‖I-Bτ‖H↦H≥1-∑k=1n‍|αk|(1+|λk-[λkτ]τ|)-∑k=1n‍|βk|(1+|λk-[λkτ]τ|)-∑k=1n‍|αk|∑k=1n‍|βk|(1+|λk-[λkτ]τ|)-∑k=1n‍|αk||λk-[λkτ]τ|∑k=1n‍|βk||λk-[λkτ]τ|.
Estimate (2.7) follows from the triangle inequality and estimate (2.3). Lemma 2.2 is proved.
Now, we will obtain the formula for the solution of problem (2.1). It is easy to show that (see, e.g., [18]) there is unique solution of the problem
(2.8)τ-2(uk+1-2uk+uk-1)+Auk+τ24A2uk+1=fk,fk=f(tk),tk=kτ,1≤k≤N-1,Nτ=1,(I+τ2A2)τ-1(u1-u0)-τ2(f0-Au0)=ω,f0=f(0),u0=μ,
and the following formula holds:
(2.9)u0=μ,u1=(I+τ2A2)-1(μ+τω+τ22f0],uk=(I+τ2A2)-1(Rk-1(I-iτA1/2)+R~k-1(I+iτA1/2)2)μ+(I+τ2A2)-1(iA1/2)-1(R~k-1R-1-Rk-1R~-12)(ω+τ2f0)-∑s=1k-1‍τ2iA-1/2(Rk-s-R~k-s)fs,2≤k≤N.
Applying formula (2.9) and the nonlocal boundary conditions in problem (2.1), we get
(2.10)μ=Tτ{[∑k=1n‍βk(τ2+(λk-[λkτ]τ))∑m=1n‍αmτ2f0×((I+τ2A2)-1(iA1/2)-1×R~[λm/τ]-1R-1-R[λm/τ]-1R~-12+∑m=1n‍αm(λm-[λmτ]τ)(I+τ2A2)-1×(R~[λm/τ]-1R-1(I-(iτ/2)A1/2)+R[λm/τ]-1R~-1(I+(iτ/2)A1/2)2)(I+τ2A2)-1)-(∑m=1n‍αm∑s=1[λm/τ]-1‍τ2iA-1/2(R[λm/τ]-s-R~[λm/τ]-s)fs-∑m=1n‍αm(λm-[λmτ]τ)×∑s=1[λm/τ]-2‍τ2iA1/2(iA-1/2)(R[λm/τ]-s(I-iτ2A1/2)+R~[λm/τ]-s(I-iτ2A1/2))fs+∑m=1n‍αmτ(λm-[λmτ]τ)RR~f[λm/τ]-1-φ)]×[I+∑k=1n‍βk(τ2+(λk-[λkτ]τ))A(I+τ2A2)-1(iA1/2)-1×(R~[λk/τ]-1R-1-R[λk/τ]-1R~-12)-∑m=1n‍αm(λm-[λmτ]τ)(I+τ2A2)-1×(R~[λm/τ]-1R-1(I-(iτ/2)A1/2)+R[λm/τ]-1R~-1(I+(iτ/2)A1/2)2)∑k=1n‍βk(τ2+(λk-[λkτ]τ))]+[∑m=1n‍αm(I+τ2A2)-1(iA1/2)-1R~[λm/τ]-1R-1-R[λm/τ]-1R~-12+∑m=1n‍αm(λm-[λmτ]τ)(I+τ2A2)-1×(R~[λm/τ]-1R-1(I-(iτ/2)A1/2)+R[λm/τ]-1R~-1(I+(iτ/2)A1/2)2)∑k=1n‍βk(τ2+(λk-[λkτ]τ))]×[τ2f0(∑k=1n‍βk(I+τ2A2)-1×(R~[λk/τ]-1R-1(I-(iτ/2)A1/2)2+R[λk/τ]-1R~-1(I+(iτ/2)A1/2)2)-∑k=1n‍βk(τ2+(λk-[λkτ]τ))A×(I+τ2A2)-1(iA1/2)-1×R~[λk/τ]-1R-1-R[λk/τ]-1R~-12∑k=1n‍βk(I+τ2A2)-1)+∑k=1n‍βk∑s=1[λk/τ]-2τ2i‍×A-1/2(iA1/2)(R[λk/τ]-s(I+iτ2A1/2)+R~[λk/τ]-s(I-iτ2A1/2))fs+∑k=1n‍βkτRR~f[λk/τ]-1+∑k=1n‍βk(τ2+(λk-[λkτ]τ))A∑s=1[λk/τ]-1τ2i‍A-1/2×(R[λk/τ]-s-R~[λk/τ]-s)fs+∑k=1n‍βk(τ2+(λk-[λkτ]τ))f[λk/τ]-1+ψ]},ω=Tτ{[I-∑m=1n‍αm(I+τ2A2)-1×(R[λm/τ]-1(I-iτA1/2)2+R~[λm/τ]-1(I+iτA1/2)2)-∑m=1n‍αm(λm-[λmτ]τ)(I+τ2A2)-1(iA1/2)×R~[λm/τ]-1(I-(iτ/2)A1/2)(I+iτA1/2)-R[λm/τ]-1(I+(iτ/2)A1/2)(I-iτA1/2)2]×[τ2f0(∑k=1n‍βk(I+τ2A2)-1×(R~[λk/τ]-1R-1(I-(iτ/2)A1/2)2+R[λk/τ]-1R~-1(I+(iτ/2)A1/2)2)-∑k=1n‍βk(τ2+(λk-[λkτ]τ))A×(I+τ2A2)-1(iA1/2)-1×R~[λk/τ]-1R-1-R[λk/τ]-1R~-12∑k=1n‍βk(I+τ2A2)-1)+∑k=1n‍βk∑s=1[λk/τ]-2‍τ2i×A-1/2(iA1/2)(R[λk/τ]-s(I+iτ2A1/2)+R~[λk/τ]-s(I-iτ2A1/2))fs+∑k=1n‍βkRR~f[λk/τ]-1+∑k=1n‍βk(τ2+(λk-[λkτ]τ))A∑s=1[λk/τ]-1‍τ2i×A-1/2(R[λk/τ]-s-R~[λk/τ]-s)fs+(τ2+(λk-[λkτ]τ))f[λk/τ]-1+ψ(I+τ2A2)-1]-[∑k=1n‍βkτ2A(I+τ2A2)-1R[λk/τ]-1(I-iτA1/2)+R~[λk/τ]-1(I+iτA1/2)2-12∑k=1n‍βk(I+τ2A2)-1(R~[λk/τ]-1(I-iτ2A1/2)(I+iτA1/2)-R[λk/τ]-1(I+iτ2A1/2)(I-iτA1/2))(I+τ2A2)-1]×[τ2f0(∑m=1n‍αm(I+τ2A2)-1(iA1/2)-1R~[λm/τ]-1R-1-R[λm/τ]-1R~-12+∑m=1n‍αm(λm-[λmτ]τ)(I+τ2A2)-1×(R~[λm/τ]-1R-1(I-(iτ/2)A1/2)+R[λm/τ]-1R~-1(I+(iτ/2)A1/2)2)∑m=1n‍αm(I+τ2A2)-1)-(∑m=1n‍αmτ(λm-[λmτ]τ)∑m=1n‍αm∑s=1[λm/τ]-1‍τ2iA-1/2(R[λm/τ]-s-R~[λm/τ]-s)fs+∑m=1n‍αm(λm-[λmτ]τ)×∑s=1[λm/τ]-2‍τ2iA-1/2(iA1/2)[R[λm/τ]-s(I+iτ2A1/2)+R~[λm/τ]-s(I-iτ2A1/2)]fs+∑m=1n‍αmτ(λm-[λmτ]τ)RR~f[λm/τ]-1-φ)]}.
Thus, formulas (2.9) and (2.10) give a solution of problem (2.1).Theorem 2.3.
Suppose that assumption (2.2) holds and φ∈D(A),  ψ∈D(A1/2). Then, for the solution of difference scheme (2.1) the stability inequalities
(2.11)max⁡0≤k≤N‖uk‖H≤M{∑k=0N-1‍‖A-1/2fk‖Hτ+‖A-1/2ψ‖H+‖φ‖H},(2.12)max⁡0≤k≤N‖A1/2uk‖H≤M{∑k=0N-1‍‖fk‖Hτ+‖A1/2φ‖H+‖ψ‖H},(2.13)max⁡1≤k≤N-1‖τ-2(uk+1-2uk+uk-1)‖H+max⁡0≤k≤N-1‖Auk+τ24A2uk+1‖H≤M{∑k=1N-1‍‖fk-fk-1‖H+‖f0‖H+‖A1/2ψ‖H+‖Aφ‖H}
hold, where M does not depend on τ,  φ,  ψ, and fk,  0≤k≤N-1.Proof.
By [18], the following estimates
(2.14)max⁡0≤k≤N‖uk‖H≤M{∑k=0N-1‍‖A-1/2fk‖Hτ+‖A-1/2ω‖H+‖μ‖H},(2.15)max⁡0≤k≤N‖A1/2uk‖H≤M{∑k=0N-1‍‖fk‖Hτ+‖A1/2μ‖H+‖ω‖H},(2.16)max⁡1≤k≤N-1‖τ-2(uk+1-2uk+uk-1)‖H+max⁡0≤k≤N-1‖Auk+τ24A2uk+1‖H≤M{∑k=1N-1‍‖fk-fk-1‖H+‖f0‖H+‖A1/2ω‖H+‖Aμ‖H}
hold for the solution of (2.8). Using formulas of μ, ω, and (2.3) and (2.6) the following estimates obtained
(2.17)‖μ‖H≤M{∑s=0N-1‍‖A-1/2fs‖Hτ+‖A-1/2ψ‖H+‖φ‖H},(2.18)‖A-1/2ω‖H≤M{∑s=0N-1‍‖A-1/2fs‖Hτ+‖A-1/2ψ‖H+‖φ‖H}.
Estimate (2.11) follows from (2.14), (2.17), and (2.18). In a similar manner, we obtain
(2.19)max⁡0≤k≤N‖A1/2uk‖H≤M{∑k=0N-1‍‖fk‖Hτ+‖A1/2φ‖H+‖ψ‖H}.
Now, we obtain the estimates for ∥Aμ∥H and ∥A1/2ω∥H. First, applying A to the formula of μ and using Abel’s formula, we can write
(2.20)Aμ=Tτ{∑s=2[λk/τ]-1‍[τ2f0(∑m=1n‍αm(I+τ2A2)-1A1/2R~[λm/τ]-1R-1-R[λm/τ]-1R~-12i+∑m=1n‍αm(λm-[λmτ]τ)A(I+τ2A2)-1×R~[λm/τ]-1R-1(I-(iτ/2)A1/2)+R[λm/τ]-1R~-1(I+(iτ/2)A1/2)2∑m=1n‍αm(I+τ2A2)-1)-(12∑m=1n‍αm∑s=2[λm/τ]-1(R[λm/τ]-s(I+iτ2A1/2)-1+R~[λm/τ]-s(I-iτ2A1/2)-1)‍×(fs-fs-1)+∑m=1n‍αm12(R[λm/τ]-1(I+iτ2A1/2)-1+R~[λm/τ]-1(I-iτ2A1/2)-1)f1-∑m=1n‍αm(I+τ2A4)-1f[λm/τ]-1+∑m=1n‍αm(λm-[λmτ]τ)×[∑s=2[λm/τ]-212iA1/2‍×(R[λm/τ]-s+R~[λm/τ]-s)(fs-fs-1)+12iA1/2(R[λm/τ]-1+R~[λm/τ]-1)f1+iA1/2f[λm/τ]-2∑s=2[λm/τ]-212iA1/2‍]+∑m=1n‍αmτ(λm-[λmτ]τ)ARR~f[λm/τ]-1-Aφ)∑m=1n‍αm(I+τ2A2)-1(I+τ2A4)-1]×[I+∑k=1n‍βk(τ2+(λk-[λkτ]τ))A(I+τ2A2)-1(iA1/2)-1×(R~[λk/τ]-1R-1-R[λk/τ]-1R~-12)-∑m=1n‍αm(λm-[λmτ]τ)(I+τ2A2)-1×R~[λm/τ]-1R-1(I-(iτ/2)A1/2)+R[λm/τ]-1R~-1(I+(iτ/2)A1/2)2(I+τ2A4)-1]+[∑m=1n‍αm(I+τ2A2)-1R~[λm/τ]-1R-1-R[λm/τ]-1R~-12i+∑m=1n‍αm(λm-[λmτ]τ)×A1/2(I+τ2A2)-1×R~[λm/τ]-1R-1(I-(iτ/2)A1/2)+R[λm/τ]-1R~-1(I+(iτ/2)A1/2)2(I+τ2A4)-1]×[τ2f0(∑s=2[λm/τ]-212iA1/2‍∑k=1n‍βkA1/2(I+τ2A2)-1×(R~[λk/τ]-1R-1(I-(iτ/2)A1/2)2+R[λk/τ]-1R~-1(I+(iτ/2)A1/2)2)+∑k=1n‍βki(τ2+(λk-[λkτ]τ))A×(I+τ2A2)-1×R~[λk/τ]-1R-1-R[λk/τ]-1R~-12∑m=1n‍αm(I+τ2A2)-1)-∑k=1n‍βk∑s=2[λk/τ]-212i‍(R[λk/τ]-s+R~[λk/τ]-s)(fs-fs-1)-∑k=1n‍βk12i(R[λk/τ]-1+R~[λk/τ]-1)f1-∑k=1n‍βkif[λm/τ]-2+∑k=1n‍βkA1/2τRR~f[λk/τ]-1+∑k=1n‍βk(τ2+(λk-[λkτ]τ))×[12A-1/2∑s=2[λk/τ]-1‍(R[λk/τ]-s(I+iτ2A1/2)-1+R~[λk/τ]-s(I-iτ2A1/2)-1)×(fs-fs-1)+12A-1/2(R[λk/τ]-1(I+iτ2A1/2)-1+R~[λk/τ]-1(I-iτ2A1/2)-1)f1-A-1/2(I+τ2A4)-1f[λk/τ]-1∑m=1n‍αm(I+τ2A2)-1]+∑k=1n‍βk(τ2+(λk-[λkτ]τ))A1/2f[λk/τ]-1+A1/2ψ∑m=1n‍αm(I+τ2A2)-1∑s=2[λk/τ]-1‍]}.
Second, applying A1/2 to the formula of ω and using Abel’s formula, we can write
(2.21)A1/2ω=Tτ{[I-∑m=1n‍αm(I+τ2A2)-1×(R[λm/τ]-1(I-iτA1/2)2+R~[λm/τ]-1(I+iτA1/2)2)-∑m=1n‍αm(λm-[λmτ]τ)(I+τ2A2)-1(iA1/2)×R~[λm/τ]-1(I-(iτ/2)A1/2)(I+iτA1/2)-R[λm/τ]-1(I+(iτ/2)A1/2)(I-iτA1/2)2∑m=1n‍αm(I+τ2A2)-1]×[(I+τ2A2)-112f0(∑k=1n‍βkA1/2τ(I+τ2A2)-1×(R~[λk/τ]-1R-1(I-(iτ/2)A1/2)2+R[λk/τ]-1R~-1(I+(iτ/2)A1/2)2)+∑k=1n‍βk(τ2+(λk-[λkτ]τ))(I+τ2A2)-1×A1/2iR~[λk/τ]-1R-1-R[λk/τ]-1R~-12∑m=1n‍αm(I+τ2A2)-1)-∑k=1n‍βk∑s=2[λk/τ]-212i‍(R[λk/τ]-s+R~[λk/τ]-s)×(fs-fs-1)-∑k=1n‍βk12i(R[λk/τ]-1+R~[λk/τ]-1)f1-∑k=1n‍βkif[λk/τ]-2+∑k=1n‍βkA1/2τRR~f[λk/τ]-1+∑k=1n‍βk(τ2+(λk-[λkτ]τ))A-1/2×[(I+τ2A2)-112‍∑s=2[λk/τ]-1(R[λk/τ]-s(I+iτ2A1/2)-1+R~[λk/τ]-s(I-iτ2A1/2)-1)×(fs-fs-1)+12(R[λk/τ]-1(I+iτ2A1/2)-1+R~[λk/τ]-1(I-iτ2A1/2)-1)f1-(I+τ2A4)-1f[λk/τ]-1]+A1/2ψ]-[∑k=1n‍βkA1/2τ2(I+τ2A2)-1×(R[λk/τ]-1(I-iτA1/2)+R~[λk/τ]-1(I+iτA1/2)2)-∑k=1n‍βkA-1/2(I+τ2A2)-1×(R~[λk/τ]-1(I-(iτ/2)A1/2)(I+iτA1/2)2-R[λk/τ]-1(I+(iτ/2)A1/2)(I-iτA1/2)2)]×[12f0(∑m=1n‍αm(I+τ2A2)-1×τA1/2R~[λm/τ]-1R-1-R[λm/τ]-1R~-12i+∑m=1n‍αm(λm-[λmτ]τ)A(I+τ2A2)-1×(R~[λm/τ]-1R-1(I-(iτ/2)A1/2)+R[λm/τ]-1R~-1(I+(iτ/2)A1/2)2))-(12∑m=1n‍αmA-1/2×∑s=2[λm/τ]-1‍(R[λm/τ]-s(I+iτ2A1/2)-1+R~[λm/τ]-s(I-iτ2A1/2)-1)×(fs-fs-1)+12∑m=1n‍αmA-1/2×(R[λm/τ]-1(I+iτ2A1/2)-1+R~[λm/τ]-1(I-iτ2A1/2)-1)f1-∑m=1n‍αmA-1/2(I+τ2A4)-1f[λm/τ]-1-∑m=1n‍αm(λm-[λmτ]τ)×[∑s=2[λm/τ]-212iA1/2‍×(R[λm/τ]-s+R~[λm/τ]-s)(fs-fs-1)+12iA1/2(R[λm/τ]-1+R~[λm/τ]-1)f1+iA1/2f[λm/τ]-2∑s=2[λm/τ]-212iA1/2‍]+∑m=1n‍αmτ(λm-[λmτ]τ)ARR~f[λm/τ]-1-Aφ)(I+τ2A2)-1]}.
The following estimates
(2.22)‖Aμ‖H≤M{∑s=1N-1‍‖fs-fs-1‖H+‖f0‖H+‖A1/2ψ‖H+‖Aφ‖H},‖A1/2ω‖H≤M{∑s=1N-1‍‖fs-fs-1‖H+‖f0‖H+‖A1/2ψ‖H+‖Aφ‖H}
are obtained by using formulas (2.20), (2.21), (2.3), and (2.6).
Estimate (2.13) follows from (2.16), and (2.22). Theorem 2.3 is proved.
Now, let us consider the applications of Theorem2.3. First, the nonlocal the mixed boundary value problem for hyperbolic equation
(2.23)utt-(a(x)ux)x+δu=f(t,x),0<t<1,0<x<1,u(0,x)=∑m=1n‍αmu(λm,x)+φ(x),0≤x≤1,ut(0,x)=∑k=1n‍βkut(λk,x)+ψ(x),0≤x≤1,u(t,0)=u(t,1),ux(t,0)=ux(t,1),0≤t≤1,
under assumption (2.2), is considered. Here ar(x),  (x∈(0,1)),  φ(x),ψ(x)(x∈[0,1]) and f(t,x)(t∈(0,1),x∈(0,1)) are given smooth functions and ar(x)≥a>0,  δ>0. The discretization of problem (2.23) is carried out in two steps.
In the first step, the grid space is defined as follows:(2.24)[0,1]h={x:xr=rh,0≤r≤K,Kh=1}.
We introduce the Hilbert space L2h=L2([0,1]h),  W2h1=W2h1([0,1]h) and W2h2=W2h2([0,1]h) of the grid functions φh(x)={φr}1K-1 defined on [0,1]h, equipped with the norms
(2.25)‖φh‖L2h=(∑r=1K-1‍|φh(x)|2h)1/2,‖φh‖W2h1=‖φh‖L2h+(∑r=1K-1‍|(φh)x,¯j|2h)1/2,‖φh‖W2h2=‖φh‖L2h+(∑r=1K-1‍|(φh)x,¯j|2h)1/2+(∑r=1K-1‍|(φh)xx,¯j|2h)1/2,
respectively. To the differential operator A generated by problem (2.23), we assign the difference operator Ahx by the formula
(2.26)Ahxφh(x)={-(a(x)φx-)x,r+δφr}1K-1,
acting in the space of grid functions φh(x)={φr}0K satisfying the conditions φ0=φK,  φ1-φ0=φK-φK-1. With the help of Ahx, we arrive at the nonlocal boundary value problem
(2.27)d2vh(t,x)dt2+Ahxvh(t,x)=fh(t,x),0≤t≤1,x∈[0,1]h,vh(0,x)=∑j=1n‍αjvh(λj,x)+φh(x),x∈[0,1]h,vth(0,x)=∑j=1n‍βjvth(λj,x)+ψh(x),x∈[0,1]h,
for an infinite system of ordinary differential equations.
In the second step, we replace problem (2.27) by difference scheme (2.28)
(2.28)uk+1h(x)-2ukh(x)+uk-1h(x)τ2+Ahxukh(x)+τ24(Ahx)2uk+1h(x)=fkh(x),x∈[0,1]h,fkh(x)=fh(tk,x),tk=kτ,1≤k≤N-1,Nτ=1,u0h(x)=∑j=1n‍αj{u[λj/τ]h(x)+τ-1(u[λj/τ]h(x)-u[λj/τ]-1h(x))(λj-[λjτ]τ)}+φh(x),x∈[0,1]h,(I+τ2Ahx2)u1h(x)-u0h(x)τ-τ2(f0h(x)-Ahxu0h(x))=∑j=1n‍βj{u[λj/τ]h(x)-u[λj/τ]-1h(x)τ+(τ2+(λj-[λjτ]τ))×(f[λj/τ]h(x)-Ahxu[λj/τ]h(x))}+ψh(x),f0h(x)=fh(0,x),x∈[0,1]h.Theorem 2.4.
Letτ and h be sufficiently small positive numbers. Suppose that assumption (2.2) holds. Then, the solution of difference scheme (2.28) satisfies the following stability estimates:
(2.29)max⁡0≤k≤N‖ukh‖L2h+max⁡0≤k≤N‖(ukh)x‖L2h≤M1[max⁡0≤k≤N-1‖fkh‖L2h+‖ψh‖L2h+‖φx-h‖L2h],max⁡1≤k≤N-1‖τ-2(uk+1h-2ukh+uk-1h)‖L2h+max⁡0≤k≤N‖(ukh)x-x‖L2h≤M1[‖f0h‖L2h+max⁡1≤k≤N-1‖τ-1(fkh-fk-1h)‖L2h+‖ψx-h‖L2h+‖(φx-h)x‖L2h].
Here, M1 does not depend on τ,  h,  φh(x),  ψh(x) and fkh,  0≤k<N.The proof of Theorem2.4 is based on abstract Theorem 2.3 and symmetry properties of the operator Ahx defined by (2.26).Second, for them-dimensional hyperbolic equation under assumption (2.2) is considered. Let Ω be the unit open cube in the m-dimensional Euclidean space ℝm{x=(x1,…,xm):0<xj<1,1≤j≤m} with boundary S,Ω¯=Ω∪S. In [0,1]×Ω, the mixed boundary value problem for the multidimensional hyperbolic equation
(2.30)∂2u(t,x)∂t2-∑r=1m‍(ar(x)uxr)xr=f(t,x),x=(x1,…,xm)∈Ω,0<t<1,u(0,x)=∑j=1n‍αju(λj,x)+φ(x),x∈Ω;¯ut(0,x)=∑k=1n‍βkut(λk,x)+ψ(x),x∈Ω;¯u(t,x)=0,x∈S
is considered.Here,ar(x),  (x∈Ω),  φ(x),  ψ(x)  (x∈Ω¯) and f(t,x)  (t∈(0,1),x∈Ω) are given smooth functions and ar(x)≥a>0. The discretization of problem (2.30) is carried out in two steps. In the first step, let us define the grid sets
(2.31)Ω~h={x=xr=(h1r1,…,hmrm),r=(r1,…,rm),0≤rj≤Nj,hjNj=1,j=1,…,m},Ωh=Ω~h∩Ω,Sh=Ω~h∩S.
We introduce the Banach space L2h=L2(Ω~h),  W2h1=W2h1(Ω~h) and W2h2=W2h2(Ω~h) of the grid functions φh(x)={φ(h1r1,…,hmrm)} defined on Ω~h, equipped with the norms
(2.32)‖φh‖L2(Ω~h)=(∑x∈Ωh¯‍|φh(x)|2h1⋯hm)1/2,‖φh‖W2h1=‖φh‖L2h+(∑x∈Ωh¯‍∑‍mr=1|(φh)xr¯,jr|2h1⋯hm)1/2,‖φh‖W2h2=‖φh‖L2h+(∑x∈Ωh¯‍∑‍mr=1|(φh)xr¯|2h1⋯hm)1/2+(∑x∈Ωh¯‍∑‍mr=1|(φh)xrxr¯,jr|2h1⋯hm)1/2,
respectively. To the differential operator A generated by problem (2.27), we assign the difference operator Ahx by the formula
(2.33)Ahxuh=-∑r=1m‍(ar(x)uxr-h)xr,jr,
acting in the space of grid functions uh(x), satisfying the conditions uh(x)=0 for all x∈Sh. Note that Ahx is a self-adjoint positive definite operator in L2(Ω~h). With the help of Ahx we arrive at the nonlocal boundary value problem
(2.34)d2vh(t,x)dt2+Ahxvh(t,x)=fh(t,x),0<t<1,x∈Ωh,vh(0,x)=∑l=1n‍αlvh(λl,x)+φh(x),x∈Ω~h,dvh(0,x)dt=∑l=1n‍βlvth(λl,x)+ψh(x),x∈Ω~,
for an infinite system of ordinary differential equations.In the second step, we replace problem (2.34) by difference scheme (2.35)
(2.35)uk+1h(x)-2ukh(x)+uk-1h(x)τ2+Ahxukh(x)+τ24(Ahx)2uk+1h(x)=fkh(x),x∈Ωh,fkh(x)=fh(tk,x),tk=kτ,1≤k≤N-1,Nτ=1,u0h(x)=∑l=1n‍αl{u[λl/τ]h(x)+τ-1(u[λl/τ]h(x)-u[λl/τ]-1h(x))(λl-[λl/τ]τ)}+φh(x),x∈Ω~h,(I+τ2Ahx2)u1h(x)-u0h(x)τ-τ2(f0h(x)-Ahxu0h(x))=∑l=1n‍βl{u[λl/τ]h(x)-u[λl/τ]-1h(x)τ+(τ2+(λl-[λlτ]τ))×(f[λl/τ]h(x)-Ahxu[λl/τ]h(x))}+ψh(x),f0h(x)=fh(0,x),x∈Ω~h.Theorem 2.5.
Letτ and h be sufficiently small positive numbers. Suppose that assumption (2.2) holds. Then, the solution of difference scheme (2.35) satisfies the following stability estimates:
(2.36)max⁡0≤k≤N‖ukh‖L2h+max⁡0≤k≤N∑r=1m‍‖(ukh)xr,jr‖L2h≤M1[max⁡0≤k≤N-1‖fkh‖L2h+‖ψh‖L2h+∑r=1m‍‖φxr-,jrh‖L2h],max⁡1≤k≤N-1‖τ-2(uk+1h-2ukh+uk-1h)‖L2h+max⁡0≤k≤N∑r=1m‍‖(ukh)xr-xr,jr‖L2h≤M1[‖f0h‖L2h+max⁡1≤k≤N-1‖τ-1(fkh-fk-1h)‖L2h+∑r=1m‍‖ψxr-,jrh‖L2h+∑r=1m‍‖φxr-xr,jrh‖L2h].
Here, M1 does not depend on τ,  h,  φh(x),  ψh(x) and fkh,  0≤k<N.The proof of Theorem2.5 is based on abstract Theorem 2.3, symmetry properties of the operator Ahx defined by formula (2.33), and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h.Theorem 2.6.
For the solutions of the elliptic difference problem(2.37)Ahxuh(x)=ωh(x),x∈Ωh,uh(x)=0,x∈Sh,
the following coercivity inequality holds [21]:
(2.38)∑r=1m‍‖uxrx-r,jrh‖L2h≤M‖ωh‖L2h.


## 3. The Second Order of Accuracy Difference Scheme Generated byA

Note that the difference scheme (2.1) generated by A2 is based on the second order approximation formula for the differential equation u′′(t)+Au(t)=f(t) and second order of approximation formulas for nonlocal boundary conditions. Using the differential equation above, one can construct the second order of approximation formula for this differential equation. Thus, we obtain the following difference scheme:
(3.1)τ-2(uk+1-2uk+uk-1)+12Auk+14A(uk+1+uk-1)=fk,fk=f(tk),tk=kτ,1≤k≤N-1,Nτ=1,u0=∑m=1n‍αm{u[λm/τ]+τ-1(u[λm/τ]-u[λm/τ]-1)×(λm-[λmτ]τ)}+φ,(I+τ2A4)[(I+τ2A4)τ-1(u1-u0)-τ2(f0-Au0)]=∑k=1n‍βk{τ-1(u[λk/τ]-u[λk/τ]-1)+(τ2+(λk-[λkτ]τ))×(f[λk/τ]-Au[λk/τ])}+ψ,f0=f(0).
In exactly the same manner using the method of Section 2 one provesTheorem 3.1.
Suppose that assumption (2.2) holds and φ∈D(A),  ψ∈D(A1/2). Then, for the solution of difference scheme (3.1) the following stability inequalities
(3.2)max⁡0≤k≤N‖uk‖H≤M{∑k=0N-1‍‖A-1/2fk‖Hτ+‖A-1/2ψ‖H+‖φ‖H},(3.3)max⁡0≤k≤N‖A1/2uk‖H≤M{∑k=0N-1‍‖fk‖Hτ+‖A1/2φ‖H+‖ψ‖H},(3.4)max⁡1≤k≤N-1‖τ-2(uk+1-2uk+uk-1)‖H+max⁡1≤k≤N-1‖12Auk+14A(uk+1+uk-1)‖H≤M{∑k=1N-1‍‖fk-fk-1‖H+‖f0‖H+‖A1/2ψ‖H+‖Aφ‖H}
hold, where M does not depend on τ,  φ,  ψ and fk,  0≤k≤N-1.Moreover, we can construct the difference schemes of a second order of accuracy with respect to one variable for approximate solution of nonlocal boundary value problems (2.30) and (2.34). Therefore, abstract Theorem 3.1 permits us to obtain the stability estimates for the solution of these difference schemes.


## 4. Numerical Analysis

In this section, several numerical examples are presented for the solution of problem (4.1), demonstrating the accuracy of the difference schemes. In general, we have not been able to determine a sharp estimate for the constants figuring in the stability inequalities. However, the numerical examples are presented for the following nonlocal boundary value problem
(4.1)∂2u(t,x)∂t2-∂2u(t,x)∂x2+u(t,x)=(6t+2t3)sin⁡x,0<t<1,0<x<π,u(0,x)=14u(1,x)-14u(12,x)-732sin⁡x,0≤x≤π,ut(0,x)=14ut(1,x)-14ut(12,x)-916sin⁡x,0≤x≤π,u(t,0)=u(t,π)=0,0≤t≤1.
The exact solution of this problem is u(t,x)=t3sinx.We consider the set[0,1]τ×[0,π]h of a family of grid points depending on the small parameters τ and h: [0,1]τ×[0,π]h={(tk,xn):tk=kτ,0≤k≤N,Nτ=1,xn=nh,0≤n≤M,Mh=π}. For the approximate solution of problem (4.1), we have applied the first order and the two different types of second orders of accuracy difference schemes, respectively. Solving these difference equations we have applied a procedure of modified Gauss elimination method with respect to n with matrix coefficients.First let us consider the first order of accuracy in time difference scheme (1.4) for the approximate solution of the Cauchy problem (4.1). Using difference scheme (1.4), we obtain
(4.2)unk+1-2unk+unk-1τ2-un+1k+1-2unk+1+un-1k+1h2+unk+1=f(tk+1,xn),tk=kτ,1≤k≤N-1,Nτ=1,xn=nh,1≤n≤M-1,Mh=π,un0=14unN-14un(N/2)+1-732sin⁡x,1≤n≤M-1,τ-1(un1-un0)=14τ-1(unN-unN-1)-14τ-1(un(N/2)+1-unN/2)-916sin⁡x,xn=nh,1≤n≤M-1,u0k=uMk=0,0≤k≤N,
for the approximate solution of Cauchy problem (4.1). We have (N+1)×(N+1) system of linear equation in (4.2) and we can write them in the matrix form as
(4.3)AUn+1+BUn+CUn-1=Dφn,1≤n≤M-1,U0=0~,UM=0~,
where
(4.4)A=[0000⋅00000a0⋅000000a⋅000⋅⋅⋅⋅⋅⋅⋅⋅0000⋅0a00000⋅000](N+1)×(N+1),B=[10000⋅140⋅000-14bcd00⋅00⋅00000bcd0⋅00⋅000000bcd⋅00⋅0000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00000⋅00⋅000000000⋅00⋅bcd000000⋅00⋅0bcd-11000⋅-1414⋅0014-14](N+1)×(N+1),C=A,D=[10⋅001⋅0⋅⋅⋅⋅00⋅1](N+1)×(N+1),φnk=[φn0φn1⋮φnN](N+1)×1,0≤k≤N,φn0=-732sin⁡(xn),φnN=-916sin⁡(xn),φnk={(6(tk+1)+2(tk+1)3)sin⁡(xn),f(tk+1,xn),1≤k≤N-1,Usk=[us0us1⋮usN](N+1)×1,0≤k≤N,s=n±1,n.
Here,
(4.5)a=-1h2,b=1τ2,c=-2τ2,d=1τ2+2h2+1.
For the solution of difference equation (4.2), we have applied the modified Gauss elimination method. Therefore, we seek a solution of the matrix equation by using the following iteration formula:
(4.6)un=αn+1un+1+βn+1,n=M-1,…,2,1,0,
where αj,  βj(j=1,…,M) are (N+1)×(N+1) square matrices andγj are (N+1)×1 column matrices. Now, we obtain formula of αn+1,  βn+1,
(4.7)αn+1=-(B+Cαn)-1A,βn+1=(B+Cαn)-1(Dφn-Cβn),n=1,2,3,…M-1.
Note that
(4.8)α1=[0⋅0⋅⋅⋅0⋅0](N+1)×(N+1),β1=[0⋅0⋅⋅⋅0⋅0](N+1)×(N+1),
and UM=0~.Thus, using the formulas and matrices above we obtain the difference scheme first order of accuracy int and second order of accuracy in x for approximate solution of nonlocal boundary value problem (4.1).Second, let us consider the second order of accuracy in time implicit difference scheme (3.1) for the approximate solution of problem (4.1). Using difference scheme (3.1), we obtain
(4.9)Unk+1-2Unk+Unk-1τ2-Un+1k-2Unk+Un-1k2h2-Un+1k+1-2Unk+1+Un-1k+14h2-Un+1k-1-2Unk-1+Un-1k-14h2=(6(tk)+2(tk)3)sin⁡(xn),tk=kτ,1≤k≤N-1,Nτ=1,xn=nh,1≤n≤M-1,Mh=π,un0=14uN-14u(N/2)+1-732sin⁡x,1≤n≤M-1,(I+τ2Ahx4)[(I+τ2Ahx4)(u1h(xn)-u0h(xn))+τ2(fh(0,xn)-Ahxu0h(xn))]=14(2τ)-1(-uN-2h(xn)+4uN-1h(xn)-3uNh(xn))-14(2τ)-1(-3u(N/2)+1h(xn)+4u(N/2)+1h(xn)-u(N/2)+3h(xn))-916sin⁡(xn),1≤n≤M-1,u0k=uMk=0,0≤k≤N,
for the approximate solution of problem (4.1). We have again the same linear system (4.3); therefore, we use exactly the same method that we have used for the solution of difference equation (4.2), but matrices are different as follows:
(4.10)A=[0000⋅000zwz0⋅0000zwz⋅000⋅⋅⋅⋅⋅⋅⋅⋅0000⋅zwz0000⋅000](N+1)×(N+1),B=[1000⋅1400⋅00-14bcd0⋅000⋅0000bcd⋅000⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅0000⋅000⋅0000000⋅000⋅cd00000⋅000⋅bcd-32τ42τ-12τ0⋅-38τ48τ-18τ⋅-18τ48τ-38τ](N+1)×(N+1),C=A,D=[10⋅001⋅0⋅⋅⋅⋅00⋅1](N+1)×(N+1).
Here,
(4.11)x=1τ2+12h2+14,y=-2τ2+1h2+12,z=-14h2,w=-12h2.
Thus, using the above matrices we obtain the difference scheme second order of accuracy in t and x for approximate solution of nonlocal boundary value problem (4.1).Next, let us consider the second order of accuracy in time implicit difference scheme (3.1) for the approximate solution of problem (4.1). Using difference scheme (3.1), we obtain
(4.12)unk+1-2unk+unk-1τ2-un+1k-2unk+un-1kh2+unk+τ24(un+2k+1-4un+1k+1+6unk+1-4un-1k+1+un-2k+1h4-2un+1k+1-2unk+1+un-1k+1h2)=(6(tk)+2(tk)3)sin⁡⁡(xn),tk=kτ,1≤k≤N-1,Nτ=1,xn=nh,1≤n≤M-1,Mh=π,un0=14uN-14u(N/2)+1-732sin⁡x,1≤n≤M-1,(I+τ2Ahx4)[(I+τ2Ahx4)(u1h(xn)-u0h(xn))+τ2(fh(0,xn)-Ahxu0h(xn))]=14(2τ)-1(-uN-2h(xn)+4uN-1h(xn)-3uNh(xn))-14(2τ)-1(-3u(N/2)+1h(xn)+4u(N/2)+1h(xn)-u(N/2)+3h(xn))-916sin⁡(xn),1≤n≤M-1,u0k=uMk=0,0≤k≤N.u1k=45u2k-15u3k,uM-1k=45uM-2k-15uM-3k,0≤k≤N,
for the approximate solution of problem (4.1). We have (N+1)×(N+1) system of linear equation in (4.12) and we can write in the matrix form as
(4.13)AUn+2+BUn+1+CUn+DUn-1+EUn-2=Rφn,2≤n≤M-2,U0=UM=0~,U3k=4U2k-5U1k,UM-3k=4UM-2k-5UM-1k,0≤k≤N,
where
(4.14)A=[0000⋅00000x0⋅000000x⋅000⋅⋅⋅⋅⋅⋅⋅⋅0000⋅0x00000⋅000](N+1)×(N+1),B=[0000⋅0000wy0⋅00000wy⋅000⋅⋅⋅⋅⋅⋅⋅⋅0000⋅wy00000⋅000](N+1)×(N+1),C=[1000⋅1400⋅00-14vtz0⋅000⋅0000vtz⋅000⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅0000⋅000⋅0000000⋅000⋅tz00000⋅000⋅vtz-32τ42τ-12τ0⋅-38τ48τ-18τ⋅-18τ48τ-38τ](N+1)×(N+1),D=B,E=A,R=[10⋅001⋅0⋅⋅⋅⋅00⋅1](N+1)×(N+1),Usk=[us0us1⋮usN](N+1)×1,0≤k≤N,s=n±2,n±1,n.
Here,
(4.15)x=τ24h4,y=-τ2h4,z=1τ2+3τ22h4,t=-2τ2,d=1τ2+2h2+1,v=1τ2,w=-1h2.
For the solution of the linear system (4.13), we use the modified variant Gauss elimination method and seek a solution of the matrix equation by the following form:
(4.16)un=αn+1un+1+βn+1un+2+γn+1,n=M-2,…,2,1,0,
where αj,  βj  (j=1:M-1) are (N+1)×(N+1) square matrices and γj-s are (N+1)×1 column matrices. We obtain the following formulas of αn+1,  βn+1,  γn+1 from linear system (4.13) by using formula (4.16):
(4.17)αn+1=[C+Dαn+Eαn-1αn+Eβn-1]-1×[-B-Dβn-Eαn-1βn],βn+1=-[C+Dαn+Eαn-1αn+Eβn-1]-1A,γn+1=[C+Dαn+Eαn-1αn+Eβn-1]-1×[Rφn-Dγn+Eαn-1γn+Eγn-1].
We have α1,  β1,  γ1,  α2,  β2,  γ2:(4.18)α1=[0⋅0⋅⋅⋅0⋅0](N+1)×(N+1),β1=[0⋅0⋅⋅⋅0⋅0](N+1)×(N+1),γ1=[00⋮0](N+1)×(1),γ2=[00⋮0](N+1)×(1),α2=[450⋅0045⋅0⋅⋅⋅⋅00⋅45](N+1)×(N+1),β2=[-150⋅00-15⋅0⋅⋅⋅⋅00⋅-15](N+1)×(N+1),UM=0~,UM-1=[(βM-2+5I)-(4I-αM-2)αM-1]-1[(4I-αM-2)γM-1-γM-2].
Thus, using the above matrices we obtain the difference scheme second order of accuracy in t and x for approximate solution of nonlocal boundary value problem (4.1).Matlab is a programming language for numeric scientific computation. One of its characteristic features is the use of matrices as the only data type. Therefore, the implementations of numerical examples are carried out by Matlab.Now, we will give some numerical results for the solutions of (4.2), (4.9), and (4.12) for different M=N values where N and M are the step numbers for the time and space variables, respectively. Note that the grid step numbers N and M in the given examples are chosen equal for clarity and this is not necessary for the stability and solutions of the difference schemes.The errors are computed by the following formula:(4.19)EMN=max⁡1≤k≤M-1(∑k=1N-1‍|u(tk,xn)-unk|2h)1/2.
Here, u(tk,xn) represents the exact solution and unk represents the approximate solution at (tk,xn). The errors and the related CPU times are presented in Tables 1 and 2, respectively, for N=M=20,  30,  40,  80,  300, and 500. The implementations are carried out by MATLAB 7.1 software package and obtained by a PC System 64 bit, Pentium (R) Core(TM) i5 CPU, 3.20 6 Hz, 3.19 Hz, 4000 Mb of RAM.Table 1
Comparison of errors for approximate solutions.
Difference Scheme
20
30
40
80
300
500
F
0.0494
0.0324
0.0241
0.0119
0.0032
0.0019
S
1
0.0020
0.0008
0.0004918
0.0001230
0.0000087
0.0000031
S
2
0.0031
0.0015
0.0007875
0.0001971
0.0000140Table 2
CPU times.
Difference Scheme
20
30
40
80
300
500
F
0.0080
0.0261
0.0542
0.4518
69.384
545.411
S
1
0.0086
0.0267
0.0572
0.4631
69.933
532.852
S
2
0.0112
0.0441
0.1042
0.9842
169.880Let us denote the first order of accuracy difference scheme (4.2) as F, the second order of accuracy difference scheme generated by three points (4.9) as S1, and the second order of accuracy difference scheme generated by five points (4.12) as S2.In order to get accurate results, CPU times are recorded by running each program100 times for small values N=M=20,  30,  40,  80 and taking the average of the elapsed time.The following conclusions can be noted for the comparison of the numerical results presented in the tables above.(1)
(i)
In the tables, it is noted that almost the same accuracy is achieved byS1 with data error=0.0020,N=20 and by F with data error=0.0019,N=500 in different CPU times; 545s and 0.0086s, respectively. This means the use of the difference scheme S1 accelerates the computation with a ratio of more than 545/0.0086≅63372 times, that is, S1 is considerably faster than F.(ii)
It is also noted that almost the same accuracy is achieved by the difference schemeS2 with data error =0.0031,N=20, and by the difference scheme F with data error=0.0032,N=300 in different CPU times; 0.0112s and 69s, respectively. Thus, the use of the difference scheme S2 accelerates the computation with a ratio of more than 69/0.0112≅6160 times, that is, S2 is considerably faster than F.(iii)
When we consider almost the same CPU times forF and S1 as 0.0080s ≅0.0086s, F computes the solution with an error=0.0494, for N=20, where the difference scheme S1 computes the solution with an error=0.0020, which is almost 25 times smaller error than the computation error of F. Namely, S1 yields 25 times more accurate results than F. For S2, this ratio reduces to 0.0112/0.0080≅1.4 times.(iv)
While both types of the second order difference schemes reach approximately the same accuracy, the CPU time for the difference schemeS2 is always greater than S1.(v)
The CPU times of difference schemesF,  S1, and S2 recorder for N≥100 exceed ones. Matlab gives “out of memory” error for S2 with values N=500 which means the memory of the computer is not enough. It is not necessary to have results for S2 when N=500, since it is obvious that CPU time of S2 is more than that of F and S1.(vi)
It is observed from the tables that for largerN values the numerical results become approximately the same for each difference scheme in the reliable range of the CPU times. This indicates that the approximation made for the solution of problem (4.1) is valid.In conclusion, the second order difference schemes are much more accurate than the first order difference scheme, and the second order difference scheme generated by three points is more convenient than the second order difference scheme generated by five points when considering the CPU times and the error levels. Comparing with many other numerical methods, our method is not based on the relationship between the grid step sizes of time and space variables (see [25–30] and the references therein).



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*Source: 846582-2012-09-06.xml*