On the stability of a weighted ﬁnite difference scheme for wave equation with nonlocal boundary conditions ˚

. We consider the stability of a weighted ﬁnite difference scheme for a linear hyperbolic equation with nonlocal integral boundary condition. By studying the spectrum of the transition matrix of the three-layered difference scheme we obtain a sufﬁcient stability condition in a special matrix norm.


Introduction
Nonlocal problems is a major research area in many branches of modern physics, biotechnology, chemistry and engineering, which arises when it is impossible to determine the boundary values of unknown function and its derivatives.Increasingly often, there arise problems with nonlocal integral boundary conditions, especially in particle diffusion [1] and heat conduction [2,3].Partial differential equations of the hyperbolic type with integral conditions often occur in problems related to fluid mechanics [4] (dynamics and elasticity), linear thermoelasticity [5], vibrations [6] etc.
Hyperbolic problems with nonlocal conditions have not been studied so broadly as, say, parabolic or elliptic problems.The paper [7] dealt with the new technique (Adomian Decomposition Method) for solving wave equation with integral boundary conditions.
The author applied the Adomian decomposition method for the solution of the wave equation.This algorithm is simple and easy to implement.The obtained results confirmed a good accuracy of the method and the calculations are simpler and faster than traditional techniques.
The stability of difference schemes for the nonlocal hyperbolic problems was studied in [8].Multidimensional hyperbolic equation with Dirichlet condition is considered `ar pxqu xr ˘xr " f pt, xq, x " px 1 , . . ., x m q P Ω, 0 ă t ă 1, up0, xq " n ÿ j"1 α j upλ j , xq `φpxq, u t p0, xq " β k u t pλ k , xq `ψpxq, x P Ω, upt, xq " 0, x P S, here Ω :" tx " px 1 , . . ., x m q: 0 ă x j ă 1, 1 ď j ď mu is the open unit cube in the m-dimensional Euclidean space R m with boundary S, Ω " Ω Y S. Stability conditions in a special norm } ¨}L 2h were obtained and numerical analysis was made.The spectrum and characteristic functions for eigenvalues of Sturm-Liouville problem are widely investigated in [9][10][11][12].For example, the following problems ´u2 " λu, t P p0, 1q, with one classical boundary condition up0q " 0 and other nonlocal boundary condition where γ P R, 0 ă ξ ă 1, were investigated.There were found eigenvalues of two types: the first type eigenvalues do not depending on γ, and the second type eigenvalues which do not depend on γ and exist only for some rational ξ.The authors introduced a method of generalized characteristic functions [13].The complex eigenvalues exist for these problems.Complex eigenvalues of these Sturm-Liouvile problems (in the case integral boundary condition) were investigated in [14,15].
In the present paper, we give a sufficient condition for the stability of a weighted difference scheme for a hyperbolic equation with nonlocal integral boundary conditions.By using a method applied earlier to explicit difference scheme for hyperbolic equations with nonlocal boundary conditions [16], we rewrite a three-layer difference scheme in the form of an equivalent two-layer scheme.By analyzing the spectrum of the transition matrix of the two-layer scheme, we obtain sufficient conditions for the stability of the three-layer scheme depending on the parameters occurring in the integral boundary conditions and not depending on the weight parameter used in scheme.We have generalized and clarified the results presented in [16].We note that for the investigated problem all eigenvalues are real.
To obtain the stability estimates of a difference nonlocal hyperbolic problem, we use a weighted three-layer difference scheme and approximate the nonlocal integral conditions by the trapezoid quadrature formula.By representing this scheme in the form of a second-order operator-difference equation and by using some transformations, one can obtain a two-layer scheme equivalent to this three-layer scheme [17, p. 364].To study the spectrum of the transition matrix of the two-layer scheme, we define special norms of matrices and vectors.The analysis of the structure of the spectrum of the transition matrix [18] and the use of a generalized nonlinear eigenvalue problem permit one to state the main result of the present paper, a sufficient condition for the stability of a weighted difference scheme for hyperbolic equations with integral boundary conditions.

A weighted finite difference scheme for nonlocal hyperbolic problem 2.1 Differential problem with integral conditions
Consider the hyperbolic equation where Ω " p0, Lq, with the classical initial conditions u| t"0 " φpxq, x P Ω :" r0, Ls, where f px, tq, φpxq, ψpxq, v l ptq, and v r ptq are given functions, and γ 0 and γ 1 are given real parameters.We are interested in sufficiently smooth solutions of the nonlocal problem (1)-( 5).Later we use notation γ :" γ 0 `γ1 .

Notations
We introduce grids where N `1 and M `1 are the numbers of grid points for x and t directions, accordingly, and N, M ě 2.
We use the notation U j i :" U px i , t j q for the function defined on the grid (or parts of the grid) ω h ˆωτ .Instead of writing indices, we denote q U j :" U j´1 and p U j :" U j`1 on grids r ω τ and ω τ Y tt 0 u, respectively.Later in this paper, we use the following notations: U pσq " σ q U `p1 ´2σqU `σ p U , σ P R. We define a space grid operator and the time grid operators Let H and H be a spaces of grid functions on ω h and ω h , respectively.We define the inner products rU, V s :" We can investigate problem (1)-( 5) in the interval r0, 1s instead of r0, Ls using transformation x " Lx 1 .Then new c 1 " c{L.Further we consider c 1 " 1 for simplicity.
We use inner products for functions e ˘ızxi , z P C, x P ω h : As a result, we obtain formulas and, using the fact that trapezoid formula is exact for linear polynomials, we also have Later we also use some inner products with discrete functions U i " p´1q i and U i " We use the following vector notation: U " pU 1 , U 2 , . . ., U N ´1q .Let P be a nonsingular matrix (det P ‰ 0); we define the norm of any m ˆm matrix M as follows: }M} ˚" }P ´1MP} 2 , where }M} 2 " pmax 1ďiďm λ i pM ˚Mqq 1{2 is the classical matrix norm and M ˚is the adjoint matrix.We define the associated vector norm by the formula where the r V i are the coordinates of the vector P ´1V.If all eigenvectors V 1 , V 2 , . . ., V m of any nonsymmetric mˆm matrix S are linearly independent, then we form the nonsingular matrix T " pV 1 , . . ., V m q.We have the relation where J " diagpµ 1 , . . ., µ m q, µ i , i " 1, m, are the eigenvalues of S and pSq is its spectral radius.
The vector norm associated with the matrix norm ( 10) is defined by identity (9) with i " 1, m.We use the theorems proved in [19, p. 168] and define the norms of matrices and vectors to be used in the stability analysis of the difference scheme.

Three-layer finite difference scheme
Now we state a difference analogue of the differential problem (1)- (5).We define a weighted finite difference scheme (FDS) approximating the original differential equation (1): where σ is a weight parameter.The initial conditions are approximated as follows: We rewrite the boundary conditions using the defined inner product: In problem ( 11)-( 15), we approximate functions f , φ, ψ, v l , and v r by grid functions F , Φ, Ψ , V l , and V r .
Matrix A ´1 exists for such σ.
We represent the three-layer scheme (18) as an equivalent two-layer scheme (e.g., see [17]) using notations According to [20,21], one can study the stability conditions for the two-layer difference scheme ( 21) by analyzing the spectrum of the matrix S. Note that the matrices S and Λ are nonsymmetric (matrix Λ is nonsymmetric except the classical case γ 1 " 0 and γ 2 " 0).
3 The structure of the spectrum of the matrix Λ Eigenvalue problem ΛU " λU for pN ´1q ˆpN ´1q matrix Λ is in general equivalent to the eigenvalue problem for the difference operator with nonlocal boundary conditions Lemma 1. (See [22].)For arbitrary values of the parameters γ 0 , γ 1 P R, all eigenvalues λ of the matrix Λ are real and simple, moreover, the following assertions hold: 1) if γ " γ 0 `γ1 ă 2, then all eigenvalues are positive; 2) if γ " 2, then there exists one zero eigenvalue, other eigenvalues are positive; 3) if 2 ă γ ă 2{h, then there exists one negative eigenvalue, all other are positive.
Using expression (6), we get an equation for q: In this formula, functions sin pqh{2q and cospqh{2q are never equal to zero in C q zt0, π{hu (since a sine function has only real zero points in a complex plane and function sinpqh{2q has no zero points in the interval p0, π{hq).We rewrite Eq. ( 27) in the form The roots of Eq. ( 28) can be found from two equations: The roots of the first type satisfy Eq. (29a).They are called constant points (see [13]) because they don't depend on γ: If sin pq{2q " 0, then cos pq{2q " ˘1.So, the roots of Eq. (29b) depend on γ.Such type of roots is called second type roots.Now we divide this equation by sin pq{2q and get expression for γ: A function γ " γpqq is called complex-real characteristic function [13].The roots of the second type q 2k`1 , k " 0, N 1 , N 1 :" tN {2u can be found as γ-values of the characteristic function (31).
Case (ii): λ " q " 0. In this case, the general solution of ( 23) is By substituting it into (24), we have So, we have zero eigenvalue when γ " 2.
Case (iii): q " π{h (λ " 4{h 2 ).Now the general solution of ( 23) is By substituting it into Eq.( 24), for the case when N is odd (when N is even, there are no nonzero solutions), we obtain: Solving this system using formulas (8), we obtain: The solution of Eq. ( 33) is defined only if N is odd: γ " 2{h 2 .
In general (except the case of γ " 2{h), the eigenvectors are real and form the complete eigenvector system tV 1 , . . ., V N ´1u (we have N ´2 eigenvectors tV 2 , . . ., V N ´1u when γ " 2{h).We call two eigenvectors equal if they are linearly dependent.These eigenvectors can be expressed by general formula: Note that q k " q k pγq.So, V ki also depends on γ.Eq. ( 34) can be rewritten as V ki " sin `qk p1 ´xi q ˘´γ 1 " 1, sin `qk px i ´xq ˘‰, k P 1, N ´1.
Lemma 2. The matrices A, B, and C have a common system of eigenvectors.More precisely, the eigenvectors of the matrix Λ are the eigenvectors of the matrices A, B, and C.
Proof.The eigenvectors of the matrix Λ are also the eigenvectors of the unit matrix I. So, since A, B, and C are the linear combination of matrices I and Λ, the formulated lemma is valid.
Let µ be the eigenvalue of the 2pN ´1q ˆ2pN ´1q matrix S (see Eq. ( 22)).We consider the eigenvalue problem We simplify determinant in Eq. ( 40) and get a characteristic equation for the eigenvalues of the generalized nonlinear eigenvalue problem Problem ( 41) is rather well studied for the case of symmetric matrices A, B, and C (e.g., see [23, p. 23]).We note that the eigenvalues µ of the matrix S coincide with the eigenvalues of the generalized nonlinear eigenvalue problem (41).The number of eigenvalues of problem (41) is 2pN ´1q.Let us clarify the relationship between the eigenvalues µ of the matrix S and the eigenvalues λ of the matrix Λ.
By substituting an eigenvector V k of matrix Λ (see Eq. ( 34)) into Eq.( 41), we obtain So, eigenvalues of the matrix S satisfy the quadratic equation and µ 2 k of the matrix S: Proof.Using relations (19), we calculate λ k pAq " λ k pCq " 1 `τ 2 σλ k , λ k pBq " ´2 `τ 2 p1 ´2σqλ k .By substituting these values into (43) and solving the resulting equation, we obtain relations (44) for eigenvalues of matrix S.
are linearly independent eigenvectors of the matrix S.
Remark 9.The obtained inequality (53) is an analogue of the stability inequality for three-layered difference schemes with classical Dirichlet boundary conditions (see [17]).
‚ The (stability) condition for the weight σ is the same as in the classical case γ 0 " γ 1 " 0.
‚ The spectrum of the matrix Λ is investigated.Eigenvalues are real, and eigenvectors form a complete system (except the case of γ " 2{h).
‚ The spectrum of Λ is qualitatively different (in some sence) for the cases of odd and even number of grid points N .
‚ If γ ą 2{h 2 and the number of grid points N is odd, then the spectrum of matrix Λ is in the interval p0, 4{h 2 q (as well as in the case of γ ă 2).

Remark 6 .Lemma 4 .
Equation (44)  determines the relation between eigenvalues µ m k and λ k .The value of µ m k can be complex as well as real, depending on the parameters σ, τ and eigenvalues λ k .Let λ k and V k be an eigenvalue and an eigenvector of the matrix Λ, respectively.Let µ 1 k and µ 2 k be the eigenvalues of matrix S corresponding to λ sin `αk px i ´xq ˘‰;