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

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 aremany 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 numericalmethods 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 schemewith an unbounded operator. It is known (see [29, 30]) that various multipoint NBVPs for hyperbolic equations can be reduced to the problem


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][2][3][4][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][7][8][9][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 ).In particular, an appropriate model for the analysis of stability is provided by a suitable unconditionally stable difference scheme with an unbounded operator.
A function () is called a solution of problem (1) if the following conditions are satisfied: (i) () 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 element () belongs to () for all  ∈ [0, 1] and the function () is continuous on the segment [0, 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 .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 of  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].

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: Here Note that throughout this paper for clarity  1 > 2 and   < 1 will be considered.We are interested in studying the stability of solution of difference scheme (2) Throughout this section for simplicity we denote First, let us present some lemmas.We will consider the following operators: and their conjugate R and their conjugate R1 , and their conjugate R5 , and their conjugate R6 , Now we consider the following lemma which was presented before in [14].

Lemma 1.
The following estimates hold: Proof.Using the spectral property of self-adjoint positive definite operators, we write Since we have that In exactly the same manner, one can easily obtain all the other estimates of Lemma 1.
Lemma 2. Suppose that assumption (4) where Since  < 1, the operator  −    has a bounded inverse and Lemma 2 is proved.
Note that the stability estimates in the sequel are obtained by the method presented in the proofs of Lemmas 1 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]) and the multipoint nonlocal boundary conditions in (2), we obtain Formulas ( 19) and ( 20) give a solution of problem (2).
We denote the mesh function of approximation by The discretization of problem (32) is carried out in two steps.In the first step, we present the grid space respectively.To the differential operator  generated by the problem (32), we assign the difference operator   ℎ by the formula acting in the space of grid functions  ℎ () = {  }  0 satisfying the conditions  0 =   ,  1 −  0 =   −  −1 .With the help of   ℎ we arrive at the multipoint NBVP for an infinite system of ordinary differential equations.
The proof of Theorem 4 is based on the proof of abstract Theorem 3 and the symmetry property of the operator   ℎ defined by (35).
We introduce the Hilbert space  2 (Ω) of all square integrable functions defined on Ω, equipped with the norm Problem (39) has a unique smooth solution (, ), and (), (),   (), and (, ) are smooth functions.This allows us to reduce the mixed problem (39) to multipoint NBVP (1) in a Hilbert space  =  2 (Ω) with a self-adjoint positive definite operator   defined by (39).
The discretization of problem (39) is carried out in two steps.In the first step, we define the grid sets We introduce the Banach space  2ℎ =  2 ( Ωℎ ), respectively.To the differential operator  generated by problem (39), we assign the difference operator   ℎ by the formula acting in the space of grid functions  ℎ (), satisfying the conditions  ℎ () = 0 for all  ∈  ℎ .It is known that   ℎ is a self-adjoint positive definite operator in  2 ( Ωℎ ).With the help of   ℎ we arrive at the multipoint NBVP: for an infinite system of ordinary differential equations.

Proof of
Here, (  ,   ) represents the exact solution and    represents the numerical solution at (  ,   ).Errors are presented in Table 1.
In the table, the results are presented for different  and  values which are the step numbers for time and space variables, respectively.The results in Table 1 indicate that error decreases as  and  values get larger.Therefore it can be verified by the experimental results (Table 1) that the difference scheme is stable.

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 steps  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 .

Table 1 :
Errors for the approximate solutions of problem (49).