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Nonlocal boundary value hyperbolic problems involving integral conditions

Abstract

Stability estimates for the solution of the nonlocal boundary value problem with two integral conditions for hyperbolic equations in a Hilbert space H are established. In applications, stability estimates for the solution of the nonlocal boundary value problems for hyperbolic equations are obtained.

MSC: 35L10.

1 Introduction

It is well known that nonlocal boundary value problems with integral conditions are widely used for thermo-elasticity, chemical engineering, heat conduction, and plasma physics [1]–[4]. Some problems arising in dynamics of ground waters are defined as hyperbolic equations with nonlocal conditions [5] and [6]. The authors of [7] investigate nonclassical problems for multidimensional hyperbolic equation with integral boundary conditions and the uniqueness of classical solution. In [8] a linear second-order hyperbolic equation with forcing and integral constraints on the solution is converted to a nonlocal hyperbolic problem. Using the Riesz representation theorem and the Schauder fixed point theorem, existence and uniqueness of a generalized solution are proved. The solutions of hyperbolic equations with nonlocal integral conditions were investigated in [9]–[16]. The method of operators as a tool for investigation of the solution to hyperbolic equations in Hilbert and Banach spaces has been used extensively in [17]–[29].

In [30] the nonlocal boundary value problem

{ d 2 u ( t ) d t 2 + A u ( t ) = f ( t ) , 0 t 1 , u ( 0 ) = r = 1 n α r u ( λ r ) + φ , u t ( 0 ) = r = 1 n β r u t ( λ r ) + ψ , 0 < λ 1 λ 2 λ n 1

was investigated. Stability estimates for the solution of the problem were established. First order of accuracy difference schemes for the approximate solutions of the problem were presented. Stability estimates for the solution of these difference schemes were established. Theoretical statements were supported by numerical examples.

In the present paper, we consider the nonlocal boundary value problem with integral conditions

{ d 2 u ( t ) d t 2 + A u ( t ) = f ( t ) , 0 t 1 , u ( 0 ) = 0 1 α ( ρ ) u ( ρ ) d ρ + φ , u t ( 0 ) = 0 1 β ( ρ ) u t ( ρ ) d ρ + ψ
(1)

in a Hilbert space H with a self-adjoint positive definite operator A. We are interested in studying the stability of solutions of problem (1) under the assumption

|1+ 0 1 α(s)β(s)ds|> 0 1 ( | α ( s ) | + | β ( s ) | ) ds.
(2)

As in [30], a function u(t) is called a solution of problem (1) if the following conditions are satisfied:

  1. (i)

    u(t) is twice continuously differentiable on the interval (0,1) and continuously differentiable on the segment [0,1].

  2. (ii)

    The element u(t) belongs to D(A) for all t[0,1], and the function Au(t) is continuous on the segment [0,1].

  3. (iii)

    u(t) satisfies the equation and nonlocal boundary conditions (1).

2 The main theorem

Let H be a Hilbert space, A be a positive definite self-adjoint operator with AδI, where δ> δ 0 >0. Throughout this paper, {c(t),t0} is a strongly continuous cosine operator-function defined by

c(t)= e i t A 1 / 2 + e i t A 1 / 2 2 .

Then, from the definition of sine operator-function s(t),

s(t)u= 0 t c(ρ)udρ

it follows that

s(t)= A 1 / 2 e i t A 1 / 2 e i t A 1 / 2 2 i .

For the theory of cosine operator-function we refer to [23] and [24].

Lemma 2.1

The following estimates hold:

c ( t ) H H 1, A 1 / 2 s ( t ) H H 1.
(3)

Lemma 2.2

Suppose that assumption (2) holds. Then the operator T,

T= ( 1 + 0 1 0 1 α ( s ) β ( s ) d s d s ) I 0 1 ( β ( s ) + α ( s ) ) c(s)ds,

has the inverse

T 1 = [ ( 1 + 0 1 0 1 α ( s ) β ( s ) d s d s ) I 0 1 ( β ( s ) + α ( s ) ) c ( s ) d s ] 1

and the following estimate is satisfied:

T 1 H H 1 | 1 + 0 1 α ( s ) β ( s ) d s | 0 1 ( | α ( s ) | + | β ( s ) | ) d s .
(4)

Proof

Applying the triangle inequality and estimates (3), we obtain

( 1 + 0 1 0 1 α ( s ) β ( s ) d s d s ) I 0 1 ( β ( s ) + α ( s ) ) c ( s ) d s H H | 1 + 0 1 α ( s ) β ( s ) d s | 0 1 ( β ( s ) + α ( s ) ) c ( s ) d s H H | 1 + 0 1 α ( s ) β ( s ) d s | 0 1 ( | α ( s ) | + | β ( s ) | ) c ( s ) H H d s | 1 + 0 1 α ( s ) β ( s ) d s | 0 1 ( | α ( s ) | + | β ( s ) | ) d s > 0 .

Estimate (4) follows from this estimate. Lemma 2.2 is proved. □

Now, we will obtain the formula for the solution of problem (1) for φD(A) and ψD( A 1 / 2 ). It is clear that [23] the initial value problem

d 2 u d t 2 +Au(t)=f(t),0<t<1,u(0)= u 0 , u (0)= u 0
(5)

has a unique solution,

u(t)=c(t) u 0 +s(t) u 0 + 0 t s(tλ)f(λ)dλ,
(6)

where the function f(t) is not only continuous but also continuously differentiable on [0,1], u 0 D(A) and u 0 D( A 1 / 2 ).

Using (6) and the nonlocal boundary condition

u 0 = 0 1 α(y)u(y)dy+φ,

we get

u 0 = 0 1 α(y) ( c ( y ) u 0 + s ( y ) u 0 + 0 y s ( y λ ) f ( λ ) d λ ) dy+φ.

Then

( I 0 1 α ( y ) c ( y ) d y ) u 0 0 1 α(y)s(y)dy u 0 = 0 1 α(y) 0 y s(yλ)f(λ)dλdy+φ.
(7)

Differentiating both sides of (6), we obtain

u (t)=As(t) u 0 +c(t) u 0 + 0 t c(tλ)f(λ)dλ.

Using this formula and the integral condition

u 0 = 0 1 β(y) u (y)dy+ψ,

we get

u 0 = 0 1 β(y) ( A s ( y ) u 0 + c ( y ) u 0 + 0 y c ( y λ ) f ( λ ) d λ ) dy+ψ.

Thus,

0 1 β ( y ) A s ( y ) d y u 0 + ( I 0 1 β ( y ) c ( y ) d y ) u 0 = 0 1 β ( y ) 0 y c ( y λ ) f ( λ ) d λ d y + ψ .
(8)

Now, we have a system of equations (7) and (8) for the solution of u 0 and u 0 . Solving it, we get

u 0 = T 1 { [ I 0 1 β ( y ) c ( y ) d y ] [ 0 1 α ( y ) 0 y s ( y λ ) f ( λ ) d λ d y + φ ] + 0 1 α ( y ) s ( y ) d y [ 0 1 β ( y ) 0 y c ( y λ ) f ( λ ) d λ d y + ψ ] }
(9)

and

u 0 = T 1 { [ I 0 1 α ( y ) c ( y ) d y ] [ 0 1 β ( y ) 0 y c ( y λ ) f ( λ ) d λ d y + ψ ] 0 1 β ( y ) A s ( y ) d y [ 0 1 α ( y ) 0 y s ( y λ ) f ( λ ) d λ d y + φ ] } .
(10)

Hence, for the solution of the nonlocal boundary value problem (1) we have (6), (9), and (10).

Theorem 2.1

Suppose thatφD(A), ψD( A 1 / 2 ), andf(t)is a continuously differentiable on[0,1]; assumption (2) holds. Then there is a unique solution of problem (1) and the following stability inequalities:

max 0 t 1 u ( t ) H M [ φ H + A 1 / 2 ψ H + 0 1 A 1 / 2 f ( λ ) H d λ ] ,
(11)
max 0 t 1 d u ( t ) d t H + max 0 t 1 A 1 / 2 u ( t ) H M [ A 1 / 2 φ H + ψ H + 0 1 f ( λ ) H d λ ] ,
(12)
max 0 t 1 d 2 u ( t ) d t 2 H + max 0 t 1 A u ( t ) H M { A φ H + A 1 / 2 ψ H + f ( 0 ) H + 0 1 f ( λ ) H d λ }
(13)

are valid, where M does not depend onf(t), t[0,1], φ, and ψ.

Proof

We take the estimates

max 0 t 1 u ( t ) H [ u ( 0 ) H + A 1 / 2 u ( 0 ) H + 0 1 A 1 / 2 f ( t ) H d t ] ,
(14)
max 0 t 1 u ( t ) H + max 0 t 1 A 1 / 2 u ( t ) H M [ A 1 / 2 u ( 0 ) H + u ( 0 ) H + 0 1 f ( t ) H d t ] ,
(15)
max 0 t 1 d 2 u ( t ) d t 2 H + max 0 t 1 A u ( t ) H + max 0 t 1 A 1 / 2 u ( t ) H M [ A u ( 0 ) H + A 1 / 2 u ( 0 ) H + f ( 0 ) H + 0 1 f ( t ) H d t ]
(16)

from [26] for the solution of problem (5). The proof of Theorem 2.1 is based on estimates (14), (15), (16), and the estimates for the norms of u 0 , A 1 / 2 u 0 , A 1 / 2 u 0 , u 0 , A u 0 , A 1 / 2 u 0 .

First of all, let us find an estimate for u ( 0 ) H . By using (9) and estimates (3), (4), we obtain

u ( 0 ) H M 1 [ 0 1 A 1 / 2 f ( λ ) H d λ + φ H + A 1 / 2 ψ H ] .
(17)

Applying A 1 / 2 to (10), we get

A 1 / 2 u 0 = T 1 { [ I 0 1 α ( y ) c ( y ) d y ] [ 0 1 β ( y ) 0 y c ( y λ ) A 1 / 2 f ( λ ) d λ d y + A 1 / 2 ψ ] 0 1 β ( y ) A 1 / 2 s ( y ) d y × [ 0 1 α ( y ) 0 y A 1 / 2 s ( y λ ) A 1 / 2 f ( λ ) d λ d y + φ ] } .

Using estimates (3), (4), we obtain

A 1 / 2 u ( 0 ) H M 2 [ 0 1 A 1 / 2 f ( λ ) H d λ + A 1 / 2 ψ H + φ H ] .
(18)

Thus, estimates (14), (17), and (18) yield estimate (11).

Second, applying operator A 1 / 2 to (9), we get

A 1 / 2 u 0 = T 1 { [ I 0 1 β ( y ) c ( y ) d y ] [ 0 1 α ( y ) 0 y A 1 / 2 s ( y λ ) f ( λ ) d λ d y + A 1 / 2 φ ] + 0 1 α ( y ) A 1 / 2 s ( y ) d y [ 0 1 β ( y ) 0 y c ( y λ ) f ( λ ) d λ d y + ψ ] } .

Using estimates (3) and (4), we obtain

A 1 / 2 u ( 0 ) H M 1 [ 0 1 f ( λ ) H d λ + A 1 / 2 φ H + ψ H ] .
(19)

Using (10), and estimates (3), (4), we get

u ( 0 ) H M 2 [ 0 1 f ( λ ) H d λ + ψ H + A 1 / 2 φ H ] .
(20)

Then estimate (12) follows from estimates (15), (19), and (20).

Third, applying A to (9) and using Abel’s formula, we have

A u 0 = T 1 { [ I 0 1 β ( y ) c ( y ) d y ] × [ 0 1 α ( y ) [ f ( y ) c ( y ) f ( 0 ) 0 y c ( y λ ) f ( λ ) d λ ] d y + A φ ] + 0 1 α ( y ) A 1 / 2 s ( y ) d y × [ 0 1 β ( y ) [ A 1 / 2 s ( y ) f ( 0 ) + 0 y A 1 / 2 s ( y λ ) f ( λ ) d λ ] d y + A 1 / 2 ψ ] }

and using estimates (3), (4), we get

A u ( 0 ) H M 3 [ A φ H + A 1 / 2 ψ H + f ( 0 ) H + 0 1 f ( λ ) H d λ ] .
(21)

In the same manner, applying A 1 / 2 to (10) and using Abel’s formula, and estimates (3), (4), we obtain

A 1 / 2 u ( 0 ) H M 4 [ A φ H + A 1 / 2 ψ H + f ( 0 ) H + 0 1 f ( λ ) H d λ ] .
(22)

Thus, estimate (13) follows from estimates (16) and (21), and (22). □

3 Applications

Now, we consider the applications of Theorem 2.1. First, a nonlocal boundary value problem for a hyperbolic equation

{ u t t ( a ( x ) u x ) x + σ u = f ( t , x ) , 0 < t < 1 , 0 < x < 1 , u ( 0 , x ) = 0 1 α ( ρ ) u ( ρ , x ) d ρ + φ ( x ) , u t ( 0 , x ) = 0 1 β ( ρ ) u t ( ρ , x ) d ρ + ψ ( x ) , 0 x 1 , u ( t , 0 ) = u ( t , 1 ) , u x ( t , 0 ) = u x ( t , 1 ) , 0 t 1
(23)

under assumption (2) is considered. Problem (23) has a unique smooth solution u(t,x) for (2), smooth functions a(x)a>0 (x(0,1)), a(0)=a(1), φ(x), ψ(x) (x[0,1]) and f(t,x) (t,x[0,1]), σ a positive constant and under some conditions. This allows us to reduce problem (23) to nonlocal boundary value problem (1) in the Hilbert space H= L 2 [0,1] with a self-adjoint positive definite operator A x defined by (23).

Theorem 3.1

For the solution of problem (23), we have the following stability inequalities:

max 0 t 1 u x ( t , ) L 2 [ 0 , 1 ] M [ max 0 t 1 f ( t , ) L 2 [ 0 , 1 ] + φ x L 2 [ 0 , 1 ] + ψ L 2 [ 0 , 1 ] ] ,
(24)
max 0 t 1 u x x ( t , ) L 2 [ 0 , 1 ] + max 0 t 1 u t t ( t , ) L 2 [ 0 , 1 ] M [ max 0 t 1 f t ( t , ) L 2 [ 0 , 1 ] + f ( 0 , ) L 2 [ 0 , 1 ] + φ x x L 2 [ 0 , 1 ] + ψ x L 2 [ 0 , 1 ] ] ,
(25)

where M does not depend onφ(x), ψ(x), andf(t,x).

The proof of Theorem 3.1 is based on Theorem 2.1 and the symmetry properties of the space operator generated by problem (23).

Proof

Problem (23) can be written in the abstract form

{ d 2 u ( t ) d t 2 + A u ( t ) = f ( t ) ( 0 t 1 ) , u ( 0 ) = 0 1 α ( ρ ) u ( ρ ) d ρ + φ , u t ( 0 ) = 0 1 β ( ρ ) u t ( ρ ) d ρ + ψ
(26)

in the Hilbert space L 2 [0,1] of all square integrable functions defined on [0,1] with a self-adjoint positive definite operator A= A x defined by the formula

A x u(x)= ( a ( x ) u x ) x +σu(x)

with the domain

D ( A x ) = { u ( x ) : u , u x , ( a ( x ) u x ) x L 2 [ 0 , 1 ] , u ( 0 ) = u ( 1 ) , u ( 0 ) = u ( 1 ) } .

Here, f(t)=f(t,x) and u(t)=u(t,x) are known and unknown abstract functions defined on [0,1] with the values in H= L 2 [0,1]. Therefore, estimates (24) and (25) follow from estimates (11), (12), and (13). Thus, Theorem 3.1 is proved. □

Second, let Ω be the unit open cube in the m-dimensional Euclidean space R m :{x=( x 1 ,, x m ):0< x j <1,1jm} with boundary S, Ω ¯ =ΩS. In [0,1]×Ω, let us consider a boundary value problem for the multidimensional hyperbolic equation

{ 2 u ( t , x ) t 2 r = 1 m ( a r ( x ) u x r ) x r + σ u ( x ) = f ( t , x ) , x = ( x 1 , , x m ) Ω , 0 < t < 1 , u ( 0 , x ) = 0 1 α ( ρ ) u ( ρ , x ) d ρ + φ ( x ) , x Ω ¯ , u t ( 0 , x ) = 0 1 β ( ρ ) u t ( ρ , x ) d ρ + ψ ( x ) , x Ω ¯ , u ( t , x ) = 0 , x S ,
(27)

under assumption (2). Here, a r (x) (xΩ), φ(x), ψ(x) (x Ω ¯ ) and f(t,x), t(0,1), xΩ are given smooth functions and a r (x)a>0.

Let us introduce the Hilbert space L 2 ( Ω ¯ ) of all square integrable functions defined on  Ω ¯ , equipped with the norm

f L 2 ( Ω ¯ ) = { x Ω ¯ | f ( x ) | 2 d x 1 d x m } 1 2 .

Theorem 3.2

For the solution of problem (27), the following stability inequalities hold:

max 0 t 1 r = 1 m u x r ( t , ) L 2 ( Ω ¯ ) M [ max 0 t 1 f ( t , ) L 2 ( Ω ¯ ) + r = 1 m φ x r L 2 ( Ω ¯ ) + ψ L 2 ( Ω ¯ ) ] ,
(28)
max 0 t 1 r = 1 m u x r x r ( t , ) L 2 ( Ω ¯ ) + max 0 t 1 u t t ( t , ) L 2 ( Ω ¯ ) M [ max 0 t 1 f t ( t , ) L 2 ( Ω ¯ ) + f ( 0 , ) L 2 ( Ω ¯ ) + r = 1 m φ x r x r L 2 ( Ω ¯ ) + r = 1 m ψ x r L 2 ( Ω ¯ ) ] ,
(29)

where M does not depend onφ(x), ψ(x), andf(t,x) (t(0,1), xΩ).

Proof

Problem (27) can be written in the abstract form (26) in the Hilbert space L 2 ( Ω ¯ ) with a self-adjoint positive definite operator A= A x defined by the formula

A x u(x)= r = 1 m ( a r ( x ) u x r ) x r +σu(x)

with domain

D ( A x ) = { u ( x ) : u , u x r , ( a r ( x ) u x r ) x r L 2 ( Ω ¯ ) , 1 r m , u ( x ) = 0 , x S } .

Here, f(t)=f(t,x) and u(t)=u(t,x) are known and unknown abstract functions defined on Ω ¯ with the values in H= L 2 ( Ω ¯ ). So, estimates (28) and (29) follow from estimates (11), (12), (13), and the following theorem. □

Theorem 3.3

[29]

For the solution of the elliptic differential problem

A x u ( x ) = ω ( x ) , x Ω , u ( x ) = 0 , x S ,

the following coercivity inequality holds:

r = 1 m u x r x r L 2 ( Ω ¯ ) M ω L 2 ( Ω ¯ ) ,

where M is independent of ω.

4 Conclusion

This work is devoted to the study of the stability of the nonlocal boundary value problem with integral conditions for hyperbolic equations. For the solution of nonlocal boundary problem (1) in a Hilbert space H with a self-adjoint positive definite operator A, Theorem 2.1 is established. Two applications of Theorem 2.1 are given. Of course, stable two-step difference schemes for approximate solution of problem (1) can be presented. The methods given above permit us to establish the stability of these difference schemes. Applying [31], we can give a numerical support of the theoretical results.

References

  1. 1.

    Shi P: Weak solution to evolution problem with a nonlocal constraint. SIAM J. Math. Anal. 1993, 24: 46-58. 10.1137/0524004

    MathSciNet  Article  Google Scholar 

  2. 2.

    Choi YS, Chan KY: A parabolic equation with nonlocal boundary conditions arising from electrochemistry. Nonlinear Anal. 1992, 18: 317-331. 10.1016/0362-546X(92)90148-8

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cahlon B, Kulkarni DM, Shi P: Stepwise stability for the heat equation with a nonlocal constraint. SIAM J. Numer. Anal. 1995, 32: 571-593. 10.1137/0732025

    MathSciNet  Article  Google Scholar 

  4. 4.

    Samarski AA: Some problems in the modern theory of differential equations. Differ. Uravn. 1980, 16: 1221-1228.

    Google Scholar 

  5. 5.

    Beilin SA: Existence of solutions for one-dimensional wave equations with non-local conditions. Electron. J. Differ. Equ. 2001., 2001:

    Google Scholar 

  6. 6.

    Dehghan M: On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numer. Methods Partial Differ. Equ. 2005, 21: 24-40. 10.1002/num.20019

    Article  Google Scholar 

  7. 7.

    Avalishvili G, Avalishvili M, Gordeziani D: On integral nonlocal boundary value problems for some partial differential equations. Bull. Georgian Acad. Sci. 2011, 5(1):31-37.

    MathSciNet  Google Scholar 

  8. 8.

    Pulkina LS: A nonlocal problem with integral conditions for hyperbolic equations. Electron. J. Differ. Equ. 1999., 1999:

    Google Scholar 

  9. 9.

    Ramezani M, Dehghan M, Razzaghi M: Combined finite difference and spectral methods for the numerical solution of hyperbolic equation with an integral condition. Numer. Methods Partial Differ. Equ. 2008, 24: 1-8. 10.1002/num.20230

    MathSciNet  Article  Google Scholar 

  10. 10.

    Pulkina LS:On solvability in L 2 of nonlocal problem with integral conditions for hyperbolic equations. Differ. Uravn. 2000, 36(2):279-280.

    MathSciNet  Google Scholar 

  11. 11.

    Mesloub S, Bouziani A: On a class of singular hyperbolic equation with a weighted integral condition. Int. J. Math. Math. Sci. 1999, 22(3):511-519. 10.1155/S0161171299225112

    MathSciNet  Article  Google Scholar 

  12. 12.

    Shakeri F, Dehghan M: The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition. Comput. Math. Appl. 2008, 56: 2175-2188. 10.1016/j.camwa.2008.03.055

    MathSciNet  Article  Google Scholar 

  13. 13.

    Dehghan M, Saadatmandi A: Variational iteration method for solving the wave equation subject to an integral conservation condition. Chaos Solitons Fractals 2009, 41: 1448-1453. 10.1016/j.chaos.2008.06.009

    MathSciNet  Article  Google Scholar 

  14. 14.

    Dehghan M, Shokri A: A meshless method for numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions. Numer. Algorithms 2009, 52: 461-477. 10.1007/s11075-009-9293-0

    MathSciNet  Article  Google Scholar 

  15. 15.

    Dehghan M, Saadatmandi A: Numerical solution of the one-dimensional wave equation with an integral condition. Numer. Methods Partial Differ. Equ. 2007, 23: 282-292. 10.1002/num.20177

    MathSciNet  Article  Google Scholar 

  16. 16.

    Yousefi SA, Barikbin Z, Dehghan M: Bernstein Ritz-Galerkin method for solving an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numer. Methods Partial Differ. Equ. 2010, 26: 1236-1246.

    MathSciNet  Google Scholar 

  17. 17.

    Ashyralyev A, Aggez N: A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations. Numer. Funct. Anal. Optim. 2004, 25(5-6):1-24. 10.1081/NFA-200041711

    MathSciNet  Article  Google Scholar 

  18. 18.

    Sobolevskii PE, Semenov S: On some approach to investigation of singular hyperbolic equations. Dokl. Akad. Nauk SSSR 1983, 270(1):555-558.

    MathSciNet  Google Scholar 

  19. 19.

    Agarwal RP, Bohner M, Shakhmurov VB: Maximal regular boundary value problems in Banach-valued weighted space. Bound. Value Probl. 2004, 1: 9-42.

    MathSciNet  Google Scholar 

  20. 20.

    Ashyralyev A, Sobolevskii PE: New Difference Schemes for Partial Differential Equations. Birkhäuser, Basel; 2004.

    Book  Google Scholar 

  21. 21.

    Ashyralyev A, Sobolevskii PE: Well-Posedness of Parabolic Difference Equations. Operator Theory Advances and Applications. Birkhäuser, Basel; 1994.

    Book  Google Scholar 

  22. 22.

    Kostin VA: On analytic semigroups and strongly continuous cosine functions. Dokl. Akad. Nauk SSSR 1989, 307(4):796-799.

    MathSciNet  Google Scholar 

  23. 23.

    Fattorini HO: Second Order Linear Differential Equations in Banach Space. North-Holland, Amsterdam; 1985.

    Google Scholar 

  24. 24.

    Piskarev S, Shaw Y: On certain operator families related to cosine operator function. Taiwan. J. Math. 1997, 1(4):3585-3592.

    MathSciNet  Google Scholar 

  25. 25.

    Guidetti D, Karasözen B, Piskarev S: Approximation of abstract differential equations. J. Math. Sci. 2004, 122(2):3013-3054. 10.1023/B:JOTH.0000029696.94590.94

    MathSciNet  Article  Google Scholar 

  26. 26.

    Ashyralyev A, Sobolevskii PE: A note on the difference schemes for hyperbolic equations. Abstr. Appl. Anal. 2001, 6(2):63-70. 10.1155/S1085337501000501

    MathSciNet  Article  Google Scholar 

  27. 27.

    Soltanov H: A note on the Goursat problem for a multidimensional hyperbolic equation. Contemp. Anal. Appl. Math. 2013, 1(2):98-106.

    MathSciNet  Google Scholar 

  28. 28.

    Ashyralyev A, Aggez N: Finite difference method for hyperbolic equations with the nonlocal integral condition. Discrete Dyn. Nat. Soc. 2011., 2011: 10.1155/2011/562385

    Google Scholar 

  29. 29.

    Sobolevskii PE: Difference Methods for the Approximate Solution of Differential Equations. Izdat. Voronezh. Gosud. Univ., Voronezh; 1975.

    Google Scholar 

  30. 30.

    Ashyralyev A, Yildirim O: On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations. Taiwan. J. Math. 2010, 14(1):165-194.

    MathSciNet  Google Scholar 

  31. 31.

    Ashyralyev A, Aggez N: On the solution of NBVP for multidimensional hyperbolic equations. Sci. World J. 2014., 2014:

    Google Scholar 

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Acknowledgements

This work is supported by the Scientific Research Fund of Fatih University (Project No: P50041203-B).

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Ashyralyev, A., Aggez, N. Nonlocal boundary value hyperbolic problems involving integral conditions. Bound Value Probl 2014, 205 (2014). https://doi.org/10.1186/s13661-014-0205-4

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Keywords

  • hyperbolic equation
  • stability
  • nonlocal boundary value problems