# On Third Order Stable Difference Scheme for Hyperbolic Multipoint Nonlocal Boundary Value Problem

Authors: Ozgur Yildirim; Meltem Uzun
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

This paper presents a third order of accuracy stable difference scheme for the approximate solution of multipoint nonlocal boundary value problem of the hyperbolic type in a Hilbert space with self-adjoint positive definite operator. Stability estimates for solution of the difference scheme are obtained. Some results of numerical experiments that support theoretical statements are presented.


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## Body

## 1. Introduction

Hyperbolic partial differential equations (PDEs) are of growing interest in several areas of engineering and natural sciences such as acoustic, electromagnetic, hydrodynamic, elasticity, fluid mechanics, and other areas of physics (see, e.g., [1–5] and the references given therein). In some cases, classical boundary conditions cannot describe process or phenomenon precisely. Therefore, mathematical models of various physical, chemical, biological, or environmental processes often involve nonclassical conditions. Such conditions are usually identified as nonlocal boundary conditions and project situations when the data on the domain boundary cannot be measured directly or when the data on the boundary depend on the data inside the domain. Many researchers study nonlocal boundary value problems (NBVPs) for hyperbolic and mixed types of partial differential equations. There are many results (see [6–10]) on integral inequalities with two dependent limits to theory of integral-differential equations of the hyperbolic type and of difference schemes for the approximate solution of these problems. Stability has been actively studied in the development of numerical methods for the approximate solutions of PDEs (see [5–31]). In particular, an appropriate model for the analysis of stability is provided by a suitable unconditionally stable difference scheme with an unbounded operator.It is known (see [29, 30]) that various multipoint NBVPs for hyperbolic equations can be reduced to the problem(1)d2utdt2+Aut=ft0≤t≤1,u0=∑r=1nαruλr+φ,ut0=∑r=1nβrutλr+ψ,0<λ1<λ2<⋯<λn≤1.Here A is a self-adjoint positive definite operator with domain DA in a Hilbert space H.A functionu(t) is called a solution of problem (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) for all t∈[0,1] and the function Au(t) is continuous on the segment [0,1].(iii)
u(t) satisfies the equations and the initial conditions (1).In the last decades, many scientists worked and published scientific papers in the field of finite difference method for the numerical solutions of hyperbolic PDEs. In this field, first order and two types of second order stable difference schemes for the solution of abstract hyperbolic problem (1) were presented in [17] and high order of accuracy difference schemes for the solution of the same problem were presented in [18]. However, the difference schemes presented in these references are generated by square roots of A. This action is not good for realization. Therefore, in spite of theoretical results the role of their application to a numerical solution of (1) is not great.In the present paper, a third order of accuracy unconditionally stable difference scheme generated by the integer power ofA for the approximate solution of the multipoint NBVP (1) is presented. Stability estimates for the solution of this difference scheme are established.Some results of this paper, without proof, are published in [16].Note that many scientists have studied the solutions of boundary value problems such as parabolic equations, elliptic equations, and equations of mixed types extensively (see, e.g., [6–10, 21–31] and the references therein).


## 2. Third-Order Accurate Stable Difference Scheme

In the present section a third order of accuracy stable difference scheme for the approximate solution of multipoint NBVP (1) is presented. We associate problem (1) with the corresponding third order of accuracy difference scheme:(2)τ-2uk+1-2uk+uk-1+23Auk+16Auk+1+uk-1+112τ2A2uk+1=fk,fk=23ftk+16ftk+1+ftk-1-112·τ2-Aftk+1+f′′tk+1,tk=kτ,1≤k≤N-1,Nτ=1,I-iτA2u0=∑k=1nαkuλk/τ+τ-1uλk/τ-uλk/τ-1λk-λkττ+32fλk/τ-Auλk/τλk-λkττ2+76fλk/τ′-τ-1Auλk/τ-uλk/τ-1·λk-λkττ3+φ,I+iτA2τ-1u1-u0=∑k=1nβk·τ-1uλk/τ-uλk/τ-1+fλk/τ-Auλk/τ·λk-λkττ+12!fλk/τ′-τ-1Auλk/τ-uλk/τ-1·λk-λkττ2+13!fλk/τ′′-Afλk/τ+A2uλk/τ·λk-λkττ3+ψ+f1,1τ.Here(3)f1,1=f0+-f0+τf′012-2f′0τ6.Note that throughout this paper for clarity λ1>2τ and λn<1 will be considered.We are interested in studying the stability of solution of difference scheme (2) under assumption(4)∑k=1nαk1+λkτ-λkτ+32λkτ-λkτ2+76λkτ-λkτ3+∑k=1nβk1+λkτ-λkτ+12λkτ-λkτ2+16λkτ-λkτ3+12∑k=1nαk·∑k=1nβk1+λkτ-λkτ2+712λkτ-λkτ4+736λkτ-λkτ6<1.Throughout this section for simplicity we denote(5)Bnτ=I+τ2A4-11-12∑k=1nαkR~1Rλk/τ-1-R1R~λk/τ-1-1τλk-λkττ+76τAλk-λkττ3+R~1Rλk/τ-R1R~λk/τ1+1τλk-λkττ-32Aλk-λkττ2-76τAλk-λkττ3-12·R2∑k=1nβkR~λk/τ-1-Rλk/τ-11τ-Aλk-λkττ-12τAλk-λkττ2+16A2λk-λkττ3+R~λk/τ-Rλk/τ1τ-Aλk-λkττ-12τAλk-λkττ2+16A2λk-λkττ3+14·R2∑k=1n∑k=1nαkβk-1τ+Aτλk-λkττ2+712A2τ3λk-λkττ4-736A3τ5λk-λkττ6R~1-R1·R~λk/τRλk/τ-1R-I-Rλk/τR~λk/τ-1R~-I.First, let us present some lemmas. We will consider the following operators:(6)R=I-13τ2A+iτA1/2I+172τ4A2·I+16τ2A+112τ4A2-1and their conjugate R~(7)R1=-5τ4144A2+7τ6216A3-iτA1/2I+τ212A+τ4144A2I+172τ4A2·-iτA1/2I+172τ4A2·I+τ212A+τ4144A2-1,and their conjugate R~1,(8)R2=I-τ212AI+τ26A+τ412A2·-iA1/2I+τ212A+τ4144A2I+172τ4A2-1,R3=I+τ26A+τ412A2I+τ212A+τ4144A2·-iτA1/2I+172τ4A2-1,R4=I+τ23A+τ49A2+τ672A3·-iA1/2I+172τ4A2I+τ26A+τ412A2·I+τ26A-1,R5=-τ22A-τ412A2+iτA1/2I+172τ4A2I+τ26·A+τ412A2-1,and their conjugate R~5,(9)R6=I-13τ2A+iτA1/2I+172τ4A2·τ22A+τ412A2-iτA1/2I+172τ4A2-1,and their conjugate R~6, (10)R7=-2iτA1/2I+172τ4A2I+τ26A+τ412A2-1.Now we consider the following lemma which was presented before in [14].Lemma 1.
The following estimates hold:(11)RH↦H≤1,R~H↦H≤1,R1H↦H≤1,R~1H↦H≤1,A1/2R2H↦H≤1,τA1/2R3H↦H≤1,A1/2R4H↦H≤1,A-1/2R5H↦H≤τ,A-1/2R~5H↦H≤τ,τA1/2R6H↦H≤1,τA1/2R~6H↦H≤1,A-1/2R~7H↦H≤τ.Proof.
Using the spectral property of self-adjoint positive definite operators, we write(12)I-13τ2A±iτA1/2I+172τ4A2·I+16τ2A+112τ4A2-1H↦H≤Supδ≤λ<∞⁡1-1/3τ2λ+iτλ1/21+1/72τ4λ21+1/6τ2λ+1/12τ4λ2.Since(13)1-1/3τ2λ2+τλ1/21+1/72τ4λ221+1/6τ2λ+1/12τ4λ22=1+1/3τ2λ+1/9τ4λ2+1/72τ6λ31+1/3τ2λ+7/36τ4λ2+1/36τ6λ3+1/144τ8λ4≤1,we have that(14)RH→H≤1,R~H→H≤1.In exactly the same manner, one can easily obtain all the other estimates of Lemma 1.Lemma 2.
Suppose that assumption (4) holds. Then, the operator I-Bnτ has an inverse Tτ=I-Bnτ-1. From the symmetry and positivity properties of the operator A the following estimate holds:(15)TτH↦H≤M.Proof.
Using the definitions ofBnτ, R, R~, estimates in (11), and the triangle inequality, we get(16)Bnτ≤12∑k=1nαkλkτ2R~5I+τ2A4-1H→HR~1Rλk/τ-1H→H·R5I+τ2A4-1H→H+R1R~λk/τ-1H→HR~5I+τ2A4-1H→H·λkτ-λkτ+76R~5τ2AI+τ2A4-1H→HR~1Rλk/τ-1H→H·R5τ2AI+τ2A4-1H→H+R1R~λk/τ-1H→HR~5τ2AI+τ2A4-1H→H·λkτ-λkτ3+τ2AI+τ2A4-1H→H·R~1Rλk/τH→H+R1R~λk/τH→H+32τ2AI+τ2A4-1H→H·R~1Rλk/τH→H+R1R~λk/τH→Hλkτ-λkτ2-12∑k=1nβkA1/2R2H→Hλkτ-λkτ3I+τ2A4-1H→H·R5τA1/2I+τ2A4-1H→HR~λk/τ-1H→HA-1/2τ-1R~5H→H+Rλk/τ-1H→HA-1/2τ-1R5H→H+12R~λk/τ-1H→HR~5τA1/2I+τ2A4-1H→H+Rλk/τ-1H→HR5τA1/2I+τ2A4-1H→H·λkτ-λkτ2+A1/2τI+τ2A4-1H→H·R~λk/τH→H+Rλk/τH→Hλkτ-λkτ+16τ3A3/2I+τ2A4-1H→H·R~λk/τH→H+Rλk/τH→Hλkτ-λkτ3-14∑k=1n∑k=1nαkβkA1/2R2H→H·λkτ6R~1-R1R~5τ-1A-1/2I+τ2A4-1H→HR~λk/τRλk/τ-1H→H·R~1-R1R5τ-1A-1/2I+τ2A4-1H→H+Rλk/τR~λk/τ-1H→H·R~1-R1R~5τ-1A-1/2I+τ2A4-1H→H·R~1-R1R~5τA1/2I+τ2A4-1H→HR~λk/τRλk/τ-1H→H·R~1-R1R5τ-1A1/2I+τ2A4-1H→H+Rλk/τR~λk/τ-1H→H·R~1-R1R~5τA1/2I+τ2A4-1H→H·λkτ-λkτ2+712·R~1-R1R~5τ3A3/2I+τ2A4-1H→HR~λk/τRλk/τ-1H→H·R~1-R1R5A3/2τ3I+τ2A4-1H→H+Rλk/τR~λk/τ-1H→H·R~1-R1R~5τ3A3/2I+τ2A4-1H→H·λkτ-λkτ4+736·R~1-R1R~5τ5A5/2I+τ2A4-1H→HR~λk/τRλk/τ-1H→H·R~1-R1R5τ5A5/2I+τ2A4-1H→H+Rλk/τR~λk/τ-1H→H·R~1-R1R~5τ5A5/2I+τ2A4-1H→H·λkτ-λkτ6≤q,where(17)q=∑k=1nαk1+λkτ-λkτ+32λkτ-λkτ2+76λkτ-λkτ3+∑k=1nβk1+λkτ-λkτ+12λkτ-λkτ2+16λkτ-λkτ3+12∑k=1nαk·∑k=1nβk1+λkτ-λkτ2+712λkτ-λkτ4+736λkτ-λkτ6.Since q<1, the operator I-Bnτ has a bounded inverse and(18)I-Bnτ-1H→H≤11-q.Lemma 2 is proved.Note that the stability estimates in the sequel are obtained by the method presented in the proofs of Lemmas1 and 2 to obtain stability estimates. For clarity and avoiding long equations the proofs of estimates are not put.Now, applying the following formula (see [14])(19)u0=μ,u1=I+τ212A+τ4144A2-1·I-512τ2A+τ4144A2μ+τI-τ212Aω+τ2f1,1,uk=12R~1Rk-R1R~kμ+12R~k-RkR2ω+12R~k-RkR3τ2f1,1+12R4∑s=1k-1R~k-s-Rk-sfsτ2,and the multipoint nonlocal boundary conditions in (2), we obtain(20)μ=TτI+iτ2A4-1λkτ-λkτ3I-12R2·∑k=1nβkλkτ-λkτ2τ-1R~λk/τ-1R~5+Rλk/τ-1R5+12τAR~λk/τ-1R~5+Rλk/τ-1R5·λkτ-λkτ2-AτR~λk/τ+Rλk/τλkτ-λkτ+16τ3A2R~λk/τ+Rλk/τλkτ-λkτ3·∑k=1n12αkR3τ2f1,1λkτ3R~λk/τ-1R~5+Rλk/τ-1R5·λkτ-λkτ-76Aτ2Rλk/τ-1R5+R~λk/τ-1R~5·λkτ-λkτ3+R~λk/τ+Rλk/τ-32τ2AR~λk/τ+Rλk/τλkτ-λkτ2+12R4∑k=1nαk∑s=1λk/τ-2fsτ2λkτ3R~λk/τ-s-1R~5+Rλk/τ-s-1R5·λkτ-λkτ-76Aτ2R~λk/τ-s-1R~5+Rλk/τ-s-1R5λkτ-λkτ3+12R4∑k=1nαk∑s=1λk/τ-1fsτ2R~λk/τ-s+Rλk/τ-s·1-32Aτ2λkτ-λkτ212R4·∑k=1nαkR7λkτ-λkτ-76Aτ2λkτ-λkτ3·fλk/τ-1τ2+∑k=1nαk32λkτ-λkτ2fλk/ττ2+76λkτ-λkτ3fλk/τ′τ3+φ∑k=1n12αkR3τ2f1,1λkτ3R~λk/τ-1R~5+Rλk/τ-1R5--12R2∑k=1nαkλkτ2R~1Rλk/τ-1R5+R1R~λk/τ-1R~5·λkτ-λkτ-76τ2AR~1Rλk/τ-1R5+R1R~λk/τ-1R~5·λkτ-λkτ3+R~1Rλk/τ+R1R~λk/τ-32Aτ2R~1Rλk/τ+R1R~λk/τλkτ-λkτ2∑k=1nαk·R3∑k=1n12βkf1,1τ2λkτ3τ-1R~λk/τ-1R~5+Rλk/τ-1R5-12τAR~λk/τ-1R~5+Rλk/τ-1R5λkτ-λkτ2-AτR~λk/τ+Rλk/τλkτ-λkτ+16A2τ3R~λk/τ+Rλk/τλkτ-λkτ3+12∑k=1n∑s=1λk/τ-2R4βkR~λk/τ-s-1R~5+Rλk/τ-s-1R5·1τ-12Aτλkτ-λkτ2fsτ2+12∑k=1n∑s=1λk/τ-1R4βkR~λk/τ-s+Rλk/τ-s·-Aτλkτ-λkτ+16Aτ2λkτ-λkτ3fsτ2+12∑k=1nR4βkR71τ-12Aτλkτ-λkτ2fλk/τ-1τ2+∑k=1nβk-16Aτ3λkτ-λkτ3fλk/τ+λkτ-λkτfλk/ττ+12λkτ-λkτ2fλk/τ′τ2+16τ3λkτ-λkτ3fλk/τ′′+ψ∑k=1n12βkf1,1τ2λkτ3τ-1R~λk/τ-1R~5+Rλk/τ-1R5,ω=TτI+iτ2A4-1·1-12∑k=1nak1-32Aτ2λkτ-λkτ2R~1Rλk/τ-1R5+R1R~λk/τ-1R~5·λkτ-λkτ-76Aτ2R~1Rλk/τ-1R5+R1R~λk/τ-1R~5·λkτ-λkτ3+R~1Rλk/τ+R1R~λk/τ·1-32Aτ2λkτ-λkτ2∑k=1nak·R3∑k=1n12βkf1,1τ2λkτ-λkτ3τ-1R~λk/τ-1R~5+Rλk/τ-1R5-12τAR~λk/τ-1R~5+Rλk/τ-1R5·λkτ-λkτ2-AτR~λk/τ+Rλk/τλkτ-λkτ+16A2τ3R~λk/τ+Rλk/τλkτ-λkτ3+12∑k=1n∑s=1λk/τ-2R4βkfsτ2R~λk/τ-s-1R~5+Rλk/τ-s-1R5·1τ-12Aτλkτ-λkτ2+12∑k=1n∑s=1λk/τ-1R4βkR~λk/τ-s+Rλk/τ-s·-Aτλkτ-λkτ+16Aτ2λkτ-λkτ3fsτ2+12∑k=1nR4βkR71τ-12Aτλkτ-λkτ2fλk/τ-1τ2+∑k=1nβk-16Aτ3λkτ-λkτ3fλk/τ+λkτ-λkτfλk/ττ+12λkτ-λkτ2fλk/τ′τ2+16τ3λkτ-λkτ3fλk/τ′′+ψ∑k=1n12βkf1,1τ2λkτ-λkτ3τ-1R~λk/τ-1R~5+Rλk/τ-1R5--12∑k=1nβkλkτ-λkτ31τR~1Rλk/τ-1R5+R1R~λk/τ-1R~5+12AτR~1Rλk/τ-1R5+R1R~λk/τ-1R~5λkτ-λkτ2-AτR~1Rλk/τ+R1R~λk/τλkτ-λkτ+16A2τ3R~1Rλk/τ+R1R~λk/τλkτ-λkτ312∑k=1nβkλkτ-λkτ31τR~1Rλk/τ-1R5+R1R~λk/τ-1R~5·∑k=1n12αkR3τ2f1,1λkτ2R~λk/τ-1R~5+Rλk/τ-1R5·λkτ-λkτ+76Aτ2Rλk/τ-1R5+R~λk/τ-1R~5·λkτ-λkτ3+R~λk/τ+Rλk/τ-32τ2AR~λk/τ+Rλk/τλkτ-λkτ2+12R4∑k=1nαk∑s=1λk/τ-2fsτ2λkτ3R~λk/τ-s-1R~5+Rλk/τ-s-1R5·λkτ-λkτ+76Aτ2R~λk/τ-s-1R~5+Rλk/τ-s-1R5·λkτ-λkτ3+12R4·∑k=1nαk∑s=1λk/τ-1fsτ2R~λk/τ-s+Rλk/τ-s·1-32Aτ2λkτ-λkτ212R4·∑k=1nαkR7λkτ-λkτ-76Aτ2λkτ-λkτ3·fλk/τ-1τ2+∑k=1nαk32λkτ-λkτ2fλk/ττ2+76λkτ-λkτ3fλk/τ′τ3+φ∑k=1n12αkR3τ2f1,1λkτ-λkτ2R~λk/τ-1R~5+Rλk/τ-1R5.Formulas (19) and (20) give a solution of problem (2).Theorem 3.
Suppose that assumptions (4) and (11) hold and φ∈D(A3/2) and ψ∈D(A1/2). Then, for the solution of difference scheme (2), the following stability estimates hold:(21)max1≤k≤N⁡ukH≤M∑s=1k-1A-1/2fsHτ+φH+A-1/2ψH+τA-1/2f1,1H,max1≤k≤N⁡A1/2ukH≤M∑s=1k-1fsHτ+A1/2φH+ψH+τf1,1H,max1≤k≤N⁡AukH≤M∑s=2N-1fs-fs-1H+f1H+AφH+A1/2ψH+τA1/2f1,1H,where M does not depend on τ, φ, ψ, f1,1 and fs, 1≤s≤N-1.Proof.
Using formulas (20), estimates (11) and (15), and the method of Lemmas 1 and 2, we get(22)μH≤M∑s=1k-1A-1/2fsHτ+φH+A-1/2ψH+τA-1/2f1,1H,A-1/2ωH≤M∑s=1k-1A-1/2fsHτ+φH+A-1/2ψH+τA-1/2f1,1H.Applying A1/2 to formulas (20), we obtain(23)A1/2μH≤M∑s=1k-1fsHτ+A1/2φH+ψH+τf1,1H,ωH≤M∑s=1k-1fsHτ+A1/2φH+ψH+τf1,1H.Now, applying Abel’s formula to (20), we get(24)μ=TτI+iτ2A4-1·I-12R2∑k=1nβkλkτ-λkτ3τ-1R~λk/τ-1R~5+Rλk/τ-1R5+12τAR~λk/τ-1R~5+Rλk/τ-1R5λkτ-λkτ2-AτR~λk/τ+Rλk/τλkτ-λkτ+16τ3A2R~λk/τ+Rλk/τλkτ-λkτ3∑k=1n0·∑k=1n12αkR3τ2f1,1·λkτ-λkτ2R~λk/τ-1R~5+Rλk/τ-1R5λkτ-λkτ-76Aτ2Rλk/τ-1R5+R~λk/τ-1R~5λkτ-λkτ3+R~λk/τ+Rλk/τ-32τ2AR~λk/τ+Rλk/τλkτ-λkτ2+12R4τ2∑k=1nαk·∑s=2λk/τ-2λkτ3R5R6Rλk/τ-s-1-R~5R~6R~λk/τ-s-1·λkτ-λkτ-76Aτ2R5R6Rλk/τ-s-1-R~5R~6R~λk/τ-s-1·λkτ-λkτ3fs-fs-1+λkτ-λkτ3R~5R~6-R5R6λkτ-λkτ-76Aτ2R~5R~6-R5R6λkτ-λkτ3fλk/τ-2-λkτ-λkτ3R~5R~6R~λk/τ-1-R5R6Rλk/τ-1λkτ-λkτ+76Aτ2R~5R~6R~λk/τ-1-R5R6Rλk/τ-1·λkτ-λkτ3f1∑s=2λk/τ-2λkτ-λkτ3+12R4τ2∑k=1nαk·∑s=2λk/τ-1λkτ-λkτλkτ-λkτ2R6Rλk/τ-s-R~6R~λk/τ-s-32Aτ2R6Rλk/τ-s-R~6R~λk/τ-s·λkτ-λkτ2fs-fs-1+R~6-R6-32Aτ2R~6-R6λkτ-λkτ2fλk/τ-1-λkτ-λkτ2R~6R~λk/τ-1-R6Rλk/τ-1+32Aτ2R~6R~λk/τ-1-R6Rλk/τ-1λkτ-λkτ2f1∑s=2λk/τ-1R6Rλk/τ-s-R~6R~λk/τ-s·12R4∑k=1nαkR7λkτ-λkτ-76Aτ2λkτ-λkτ3·fλk/τ-1τ2+∑k=1nαk32λkτ-λkτ2fλk/ττ2+76λkτ-λkτ3fλk/τ′τ3+φ∑k=1n12αkR3τ2f1,1--12R2∑k=1nαk·λkτ-λkτ2R~1Rλk/τ-1R5+R1R~λk/τ-1R~5λkτ-λkτ-76τ2AR~1Rλk/τ-1R5+R1R~λk/τ-1R~5λkτ-λkτ3+R~1Rλk/τ+R1R~λk/τ-32Aτ2R~1Rλk/τ+R1R~λk/τλkτ-λkτ2∑k=1nαk·R3∑k=1n12βkf1,1τ2·λkτ-λkτ3τ-1R~λk/τ-1R~5+Rλk/τ-1R5-12τAR~λk/τ-1R~5+Rλk/τ-1R5λkτ-λkτ2-AτR~λk/τ+Rλk/τλkτ-λkτ+16A2τ3R~λk/τ+Rλk/τλkτ-λkτ3+12R4τ2∑k=1nβk·∑s=2λk/τ-2λkτ-λkτ21τR5R6Rλk/τ-s-1-R~5R~6R~λk/τ-s-1-12AτR5R6Rλk/τ-s-1-R~5R~6R~λk/τ-s-1·λkτ-λkτ2fs-fs-1+λkτ-λkτ21τR~5R~6-R5R6-12Aτλkτ-λkτ2·R~5R~6-R5R6λkτ-λkτ2fλk/τ-2-λkτ-λkτ21τR~5R~6R~λk/τ-1-R5R6Rλk/τ-1-12AτR~5R~6R~λk/τ-1-R5R6Rλk/τ-1·λkτ-λkτ2f1+12R4τ2∑k=1nβk·∑s=2λk/τ-1λkτ-λkτ3-AτR6Rλk/τ-s-R~6R~λk/τ-s·λkτ-λkτ+16Aτ2R6Rλk/τ-s-R~6R~λk/τ-s·λkτ-λkτ3fs-fs-1+λkτ-λkτ3-AτR~6-R6λkτ-λkτ+16Aτ2R~6-R6λkτ-λkτ3fλk/τ-1-λkτ3-AτR~6R~λk/τ-1-R6Rλk/τ-1λkτ-λkτ+16Aτ2R~6R~λk/τ-1-R6Rλk/τ-1λkτ-λkτ3f1∑s=2λk/τ-1λkτ-λkτ3-AτR6Rλk/τ-s-R~6R~λk/τ-s+12∑k=1nR4βkR71τ-12Aτλkτ-λkτ2fλk/τ-1τ2+∑k=1nβk·λkτ-λkτ3-16Aτ3λkτ-λkτ3fλk/τ+λkτ-λkτfλk/ττ+12λkτ-λkτ2fλk/τ′τ2+16τ3λkτ-λkτ3fλk/τ′′+ψ∑k=1n12βkf1,1τ2∑k=1nβkλkτ-λkτ3τ-1R~λk/τ-1R~5+Rλk/τ-1R5,(25)ω=TτI+iτ2A4-1·1-12∑k=1nak32Aτ2λkτ2R~1Rλk/τ-1R5+R1R~λk/τ-1R~5·λkτ-λkτ-76Aτ2R~1Rλk/τ-1R5+R1R~λk/τ-1R~5·λkτ-λkτ3+R~1Rλk/τ+R1R~λk/τ·1-32Aτ2λkτ-λkτ212∑k=1nak·R3∑k=1n12βkf1,1τ2·λkτ-λkτ3τ-1R~λk/τ-1R~5+Rλk/τ-1R5-12τAR~λk/τ-1R~5+Rλk/τ-1R5·λkτ-λkτ2-AτR~λk/τ+Rλk/τλkτ-λkτ+16A2τ3R~λk/τ+Rλk/τλkτ-λkτ3+12R4τ2∑k=1nβk·∑s=2λk/τ-2λkτ-λkτ2λkτ-λkτ31τR~λk/τ-s-1R5R6Rλk/τ-s-1-R~5R~6R~λk/τ-s-1-12AτR5R6Rλk/τ-s-1-R~5R~6R~λk/τ-s-1·λkτ-λkτ2fs-fs-1+λkτ-λkτ2λkτ-λkτ31τR~5R~6-R5R6-12Aτλkτ-λkτ2·R~5R~6-R5R6λkτ-λkτ2fλk/τ-2-λkτ-λkτ31τR~5R~6R~λk/τ-1-R5R6Rλk/τ-1-12AτR~5R~6R~λk/τ-1-R5R6Rλk/τ-1·λkτ-λkτ2f1∑s=2λk/τ-2λkτ-λkτ31τR5R6Rλk/τ-s-1+12R4τ2∑k=1nβk·∑s=2λk/τ-1λkτ3-AτR6Rλk/τ-s-R~6R~λk/τ-s·λkτ-λkτ+16Aτ2R6Rλk/τ-s-R~6R~λk/τ-sλkτ-λkτ3·fs-fs-1+λkτ-λkτ3-AτR~6-R6λkτ-λkτ+16Aτ2R~6-R6λkτ-λkτ3fλk/τ-1-λkτ-λkτ3-AτR~6R~λk/τ-1-R6Rλk/τ-1λkτ-λkτ+16Aτ2R~6R~λk/τ-1-R6Rλk/τ-1·λkτ-λkτ3f1∑s=2λk/τ-2λkτ-λkτ31τR5R6Rλk/τ-s-1+12∑k=1nR4βkR71τ-12Aτλkτ-λkτ2fλk/τ-1τ2+∑k=1nβk-16Aτ3λkτ-λkτ3fλk/τ+λkτ-λkτfλk/ττ+12λkτ-λkτ2fλk/τ′τ2+16τ3λkτ-λkτ3fλk/τ′′+ψ∑k=1n12βkf1,1τ2--12∑k=1nβk·λkτ-λkτ31τR~1Rλk/τ-1R5+R1R~λk/τ-1R~5+12AτR~1Rλk/τ-1R5+R1R~λk/τ-1R~5·λkτ-λkτ2-AτR~1Rλk/τ+R1R~λk/τ·λkτ-λkτ+16A2τ3R~1Rλk/τ+R1R~λk/τλkτ-λkτ3∑k=1nβk·λkτ-λkτ3∑k=1n12αkR3τ2f1,1·λkτ3R~λk/τ-1R~5+Rλk/τ-1R5λkτ-λkτ+76Aτ2Rλk/τ-1R5+R~λk/τ-1R~5λkτ-λkτ3+R~λk/τ+Rλk/τ-32τ2AR~λk/τ+Rλk/τ·λkτ-λkτ2+12R4τ2∑k=1nαk·∑s=2λk/τ-2λkτ3R5R6Rλk/τ-s-1-R~5R~6R~λk/τ-s-1·λkτ-λkτ-76Aτ2R5R6Rλk/τ-s-1-R~5R~6R~λk/τ-s-1λkτ-λkτ3·fs-fs-1+λkτ-λkτ3R~5R~6-R5R6·λkτ-λkτ-76Aτ2R~5R~6-R5R6·λkτ-λkτ3fλk/τ-2-λkτ-λkτR~5R~6R~λk/τ-1-R5R6Rλk/τ-1·λkτ-λkτ-76Aτ2·R~5R~6R~λk/τ-1-R5R6Rλk/τ-1·λkτ-λkτ3f1∑s=2λk/τ-2λkτ3R5R6Rλk/τ-s-1-R~5R~6R~λk/τ-s-1+12R4τ2∑k=1nαk·∑s=2λk/τ-1λkτ-λkτ3λkτ-λkτ2R6Rλk/τ-s-R~6R~λk/τ-s-32Aτ2R6Rλk/τ-s-R~6R~λk/τ-s·λkτ-λkτ2fs-fs-1+λkτ-λkτ2R~6-R6-32Aτ2R~6-R6·λkτ-λkτ2fλk/τ-1-λkτ-λkτ3R~6R~λk/τ-1-R6Rλk/τ-1-32Aτ2·R~6R~λk/τ-1-R6Rλk/τ-1·λkτ-λkτ2f1∑s=2λk/τ-1λkτ-λkτ3R6Rλk/τ-s-R~6R~λk/τ-s·12R4∑k=1nαkR7·λkτ-λkτ3λkτ-λkτ-76Aτ2λkτ-λkτ3fλk/τ-1τ2+∑k=1nαk32λkτ-λkτ2fλk/ττ2+76λkτ-λkτ3fλk/τ′τ3+φ.Let us obtain estimates for AμH and A1/2ωH using the method of lemmas. First, applying A to formula (24), using estimates (11) and (15) and the triangle inequality, we obtain(26)AμH≤M∑s=2N-1fs-fs-1H+f1H+AφH+A1/2ψH+τA1/2f1,1H.Second, applying A1/2 to formula (25), using estimates (11) and (15) and the triangle inequality, we obtain(27)A1/2ωH≤M∑s=2N-1fs-fs-1H+f1H+AφH+A1/2ψH+τA1/2f1,1H.The estimates (21) are proved for k=0,1. Now, we give the proofs of estimates (21) which were also presented in [14] for k≥2. Using formula (19), estimates (11), (22), and (23), the triangle inequality, and method of Lemmas 1 and 2, we obtain(28)ukH≤M∑s=1N-1A-1/2fsHτ+φH+A-1/2ψH+τA-1/2f1,1H.Applying A1/2 to (19), we get (29)A1/2ukH≤M∑s=1N-1fsHτ+A1/2I+iτA1/2φH+ψH+τf1,1H.Applying Abel’s formula to (19), we get(30)uk=12R~1Rk-R1R~kμ+12R~k-RkR2ω+12R~k-RkR3τ2f1,1+τ2R4·12∑s=2k-1R6Rk-s-R~6R~k-sfs-fs-1+R~6-R6fk-1-R~6R~k-1-R6Rk-1f1,2≤k≤N.Applying A to formula (30), using estimates (11) and the triangle inequality, we obtain(31)AukH≤M∑s=2N-1fs-fs-1H+f1H+AI+iτA1/2φH+A1/2ψH+τA1/2f1,1H.Theorem 3 is proved.We denote the mesh function of approximation byaτ=akk=0N. Then (I+τ2A4)-1a1H~a1H=o(τ) if we assume that τAa1H tends to 0 as τ→0 not slower than a1H. It takes place in applications by supplementary restriction of the smooth property of the data of space variables. Let us give applications of the abstract results presented in Theorem 3.First, the mixed multipoint NBVP for hyperbolic equation(32)utt-axuxx+δu=ft,x,0<t<1,0<x<1,u0,x=∑j=1nαjuλj,x+φx,0≤x≤1,ut0,x=∑j=1nβjutλj,x+ψx,0≤x≤1,ut,0=ut,1,uxt,0=uxt,1,0≤t≤1is considered. Problem (32) has a unique smooth solution u(t,x), δ>0, and the smooth functions a(x)≥a>0 (a0=a1, x∈(0,1)), φ(x), ψ(x) (x∈0,1) and f(t,x) (t,x∈0,1). This allows us to reduce mixed problem (32) to multipoint NBVP (1) in a Hilbert space H=L2[0,1] with a self-adjoint positive definite operator Ax defined by (32).The discretization of problem (32) is carried out in two steps. In the first step, we present the grid space(33)0,1h=x:xr=rh,0≤r≤K,Kh=1in a 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 (34)φhL2h=∑r=1K-1φhx2h1/2,φhW2h1=φhL2h+∑r=1K-1φhx,¯j2h1/2,φhW2h2=φhL2h+∑r=1K-1φhx,¯j2h1/2+∑r=1K-1φhxx,¯j2h1/2,respectively. To the differential operator A generated by the problem (32), we assign the difference operator Ahx by the formula(35)Ahxφhx=-axφx¯x,r+δφr1K-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 multipoint NBVP(36)d2vht,xdt2+Ahxvht,x=fht,x,0<t<1,x∈0,1h,vh0,x=∑j=1nαjvhλj,x+φhx,x∈0,1h,vth0,x=∑j=1nβjvthλj,x+ψhx,x∈0,1hfor an infinite system of ordinary differential equations.In the second step, we replace problem (36) by difference scheme (37):(37)τ-2uk+1hx-2ukhx+uk-1hx+23Ahxukhx+16·Ahxuk+1hx+uk-1hx+112τ2Ahx2uk+1hx=fkhx,fkhx=23fhtk,x+16fhtk+1,x+fhtk-1,x-112τ2-Afhtk+1,x+ftthtk+1,x,x∈0,1h,tk=kτ,Nτ=1,1≤k≤N-1,u0hx=∑j=1nαjuλj/τhx+τ-1uλj/τhx-uλj/τ-1hxλj-λjττ+32fλj/τ-Ahxuλj/τhxλj-λjττ2+76fλj/τ′-τ-1Ahxuλj/τhx-uλj/τ-1hx·λj-λjττ3+φhx,x∈0,1h,I+τ2Ahx4τ-1u1hx-u0hx=∑j=1nβj·τ-1uλj/τhx-uλj/τ-1hx+fλj/τ-Ahxuλj/τhxλj-λjττ+12!fλj/τ′-τ-1Ahxuλj/τhx-uλj/τ-1hx·λj-λjττ2+13!fλj/τ′′-Ahxfλj/τ+Ahx2uλj/τhx·λj-λjττ3+ψhx,x∈0,1h,f1,1hx=12fh0,x+τ6fth0,x.Theorem 4.
Letτ and h be sufficiently small numbers. Then, the solution of difference scheme (37) satisfies the following stability estimates:(38)max0≤k≤N⁡ukhL2h+max0≤k≤N⁡ukhW2h1≤M1max1≤k≤N-1⁡fkhL2h+ψhL2h+φhW2h1+τφhW2h2+τf1,1hL2h,max1≤k≤N-1⁡τ-2uk+1h-2ukh+uk-1hL2h+max0≤k≤N⁡ukhW2h2≤M1f1hL2h+max2≤k≤N-1⁡τ-1fkh-fk-1hL2h+ψhW2h1+φhW2h2+τφhW2h3+τf1,1hW2h1.Here M1 does not depend on τ, h, φh(x), ψh(x), f1,1h and fkh, 1≤k<N.The proof of Theorem4 is based on the proof of abstract Theorem 3 and the symmetry property of the operator Ahx defined by (35).Second, letΩ be the unit open cube in the m-dimensional Euclidean space Rmx=(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(39)∂2ut,x∂t2-∑r=1marxuxrxr=ft,x,x=x1,…,xm∈Ωh,0<t<1,uh0,x=∑j=1nαjuhλj,x+φhx,x∈Ω¯h,uth0,x=∑j=1nβjuthλj,x+ψhx,x∈Ω¯h,uht,x=0,x∈Sis considered. Here ar(x), (x∈Ω), φ(x), ψ(x) (x∈Ω¯h), and f(t,x) (t∈(0,1), x∈Ωh) are given smooth functions and ar(x)≥a>0.We introduce the Hilbert spaceL2(Ω¯) of all square integrable functions defined on Ω¯, equipped with the norm(40)fL2Ω¯=∫⋯∫x∈Ω¯fx2dx1⋯dxm1/2.Problem (39) has a unique smooth solution u(t,x), and φ(x), ψ(x), ar(x), and f(t,x) are smooth functions. This allows us to reduce the mixed problem (39) to multipoint NBVP (1) in a Hilbert space H=L2(Ω¯) with a self-adjoint positive definite operator Ax defined by (39).The discretization of problem (39) is carried out in two steps. In the first step, we define the grid sets(41)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=W2h1Ω~h, and W2h2=W2h2Ω~h of the grid functions φh(x)={φ(h1r1,…,hmrm)} defined on Ω~h, equipped with norms(42)φhL2Ω~h=∑x∈Ω¯hφhx2h1⋯hm1/2,φhW2h1=φhL2h+∑x∈Ω¯h∑r=1mφhx¯r,jr2h1⋯hm1/2,φhW2h2=φhL2h+∑x∈Ω¯h∑r=1mφhx¯r2h1⋯hm1/2+∑x∈Ω¯h∑r=1mφhxrx¯r,jr2h1⋯hm1/2,respectively. To the differential operator A generated by problem (39), we assign the difference operator Ahx by the formula (43)Ahxuxh=-∑r=1marxux¯rhxr,jracting in the space of grid functions uh(x), satisfying the conditions uh(x)=0 for all x∈Sh. It is known that Ahx is a self-adjoint positive definite operator in L2(Ω~h). With the help of Ahx we arrive at the multipoint NBVP:(44)d2vht,xdt2+Ahxvht,x=fht,x,0<t<1,x∈Ωh,vh0,x=∑j=1nαjvhλj,x+φhx,x∈Ω~h,dvh0,xdt=∑j=1nβjvthλj,x+ψhx,x∈Ω~hfor an infinite system of ordinary differential equations.In the second step, we replace problem (44) by difference scheme (45):(45)τ-2uk+1hx-2ukhx+uk-1hx+23Ahxukhx+16·Ahxuk+1hx+uk-1hx+112τ2Ahx2uk+1hx=fkhx,fkhx=23fhtk,x+16fhtk+1,x+fhtk-1,x-112τ2-Afhtk+1,x+ftthtk+1,x,tk=kτ,1≤k≤N-1,Nτ=1,x∈Ωh,u0hx=∑j=1nαjuλj/τhx+τ-1uλj/τhx-uλj/τ-1hxλj-λjττ+32fλj/τ-Ahxuλj/τhxλj-λjττ2+76fλj/τ′-τ-1Ahxuλj/τhx-uλj/τ-1hx·λj-λjττ3+φhx,x∈Ωh,I+τ2Ahx4τ-1u1hx-u0hx=∑j=1nβj·τ-1uλj/τhx-uλj/τ-1hx+fλj/τ-Ahxuλj/τhxλj-λjττ+12!fλj/τ′-τ-1Ahxuλj/τhx-uλj/τ-1hx·λj-λjττ2+13!fλj/τ′′-Ahxfλj/τ+Ahx2uλj/τhx·λj-λjττ3+ψhx,x∈Ωh,f1,1hx=12fh0,x+τ6fth0,x.Theorem 5.
Letτ and |h| be sufficiently small numbers. Then, the solution of difference scheme (45) satisfies the following stability estimates:(46)max0≤k≤N⁡ukhL2h+max0≤k≤N⁡ukhW2h1≤M1max1≤k≤N-1⁡fkhL2h+ψhL2h+φhW2h1+τφhW2h2+τf1,1hL2h,max1≤k≤N-1⁡τ-2uk+1h-2ukh+uk-1hL2h+max0≤k≤N⁡ukhW2h2≤M1f1hL2h+max2≤k≤N-1⁡τ-1fkh-fk-1hL2h+ψhW2h1+φhW2h2+τφhW2h3+τf1,1hW2h1.Here M1 does not depend on τ, h, φh(x), ψh(x), f1,1h, and fkh, 1≤k<N.Proof of Theorem5 is based on the proof of abstract Theorem 3, the symmetry property of the operator Ahx defined by formula (43), and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h.Theorem 6.
For the solutions of the elliptic difference problem(47)Ahxuhx=ωhx,x∈Ωh,uhx=0,x∈Shthe following coercivity inequality holds [25]:(48)∑r=1muxrx¯r,jrhL2h≤MωhL2h.Here M does not depend on h, ωh(x).


## 3. Numerical Results

In this section some numerical results are presented. The multipoint NBVP(49)∂2ut,x∂t2-∂2ut,x∂x2=t2-2t+3sin⁡x,0<t<1,0<x<π,u0,x=18u1,x+18u12,x+3132sin⁡x,0≤x≤π,ut0,x=18ut1,x+18ut12,x-158sin⁡x,0≤x≤π,ut,0=ut,π=0,0≤t≤1for one-dimensional hyperbolic equation is considered. The exact solution of (49) is(50)ut,x=1-t2sin⁡x.For the approximate solution of problem (49) the third order of accuracy difference scheme (2) is used, and the set 0,1τ×0,πh of a family of grid points depending on the small parameters τ and h is considered: [0,1]τ×[0,π]h={(tk,xn):tk=kτ,0≤k≤N,Nτ=1,xn=nh,0≤n≤M,Mh=π}. The implementations of the numerical experiments are carried out by Matlab.The errors are computed by the following formula:(51)EMN=max1≤k≤N-1,1≤k≤M-1⁡utk,xn-unk.Here,utk,xn represents the exact solution and unk represents the numerical solution at tk,xn. Errors are presented in Table 1.Table 1
Errors for the approximate solutions of problem (49).
Method
N
=
20, M=20
N
=
40, M=40
N
=
80, M=80
Difference scheme solution
0,0005962
0,0001554
0,00003946In the table, the results are presented for differentM and N values which are the step numbers for time and space variables, respectively. The results in Table 1 indicate that error decreases as M and N values get larger. Therefore it can be verified by the experimental results (Table 1) that the difference scheme is stable.


## 4. Conclusion

In this paper a third order of accuracy unconditionally stable difference scheme for the approximate solution of hyperbolic multipoint NBVP in a Hilbert space with self-adjoint positive definite operator is presented. The stability is established without any assumptions in respect of the grid stepsτ and h. Numerical results verifying the theoretical statements are presented. It is not easy to obtain arbitrary order stable difference schemes. Nevertheless one can obtain fourth order or higher order stable difference schemes using our results and method. It is possible to consider the same problem with fractional derivative or with small parameter ε.



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