A third order of accuracy absolutely stable difference schemes is presented for nonlocal boundary value hyperbolic problem of the differential equations in a Hilbert space H with self-adjoint positive definite operator A. Stability estimates for solution of the difference scheme are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions is considered.

1. Introduction

In modeling several phenomena of physics, biology, and ecology mathematically, there often arise problems with nonlocal boundary conditions (see [1–5] and the references given therein). Nonlocal boundary value problems have been a major research area in the case when it is impossible to determine the boundary conditions of the unknown function. Over the last few decades, the study of nonlocal boundary value problems is of substantial contemporary interest (see, e.g., [6–14] and the references given therein).

We consider the nonlocal boundary value problem

()

for hyperbolic equations in a Hilbert space H with self-adjoint positive definite linear operator A with domain D(A).

A function u(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 element u(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 nonlocal boundary conditions (1).

Here, φ(x), ψ(x) (x ∈ [0,1]) and f(t, x) (t, x ∈ [0,1]) are smooth functions.

In the study of numerical methods for solving PDEs, stability is an important research area (see [6–27]). Many scientists work on difference schemes for hyperbolic partial differential equations, in which stability was established under the assumption that the magnitudes of the grid steps τ and h with respect to the time and space variables are connected. This particularly means that τ∥A_h∥ → 0 when τ → 0.

We are interested in studying high order of accuracy unconditionally stable difference schemes for hyperbolic PDEs.

In the present paper, third order of accuracy difference scheme generated by integer power of A for approximately solving nonlocal boundary value problem (1) is presented. The stability estimates for solution of the difference scheme are established.

In [8], some results of this paper, without proof, were presented.

The well posedness of nonlocal boundary value problems for parabolic equations, elliptic equations, and equations of mixed types have been studied extensively by many scientists (see, e.g., [11–14, 19–32] and the references therein).

2. Third Order of Accuracy Difference Scheme Subject to Nonlocal Conditions

In this section, we obtain stability estimates for the solution of third order of accuracy difference scheme

()

for numerical solution of nonlocal boundary value problem (1). Here,

()

We study the stability of solutions of difference scheme (2) under the following assumption:

()

We give a lemma that will be needed in the sequel which was presented in [18]. First, let us present the following operators:

()

and its conjugate

()

and its conjugate

()

and its conjugate

, and

()

and its conjugate

We consider the following operators:

()

and its conjugate

()

and its conjugate

Lemma 1. The following estimates hold:

()

Now let us give, without proof, the second lemma.

Lemma 2. The following estimates hold:

()

Throughout the section, for simplicity, we denote

()

Lemma 3. Suppose that assumption (4) holds. Then, the operator I − B_τ has an inverse . From symmetry and positivity properties of operator A, the following estimate is satisfied:

()

Proof. Using the definitions of , estimates (11), and the following simple estimates,

()

and the triangle inequality, we get

()

where

()

Since q < 1, the operator I − B_τ has a bounded inverse and

()

Lemma 3 is proved.

Now, let us obtain formula for the solution of problem (2). Using the results of [18], one can obtain the following formula:

()

for the solution of difference scheme

()

Applying formula (19) and nonlocal boundary conditions

()

one can write

()

Using formulas in (22), we obtain

()

So, formulas (19) and (23) give a solution of problem (2).

Unfortunately, the estimates for

, and

cannot be obtained under the conditions

()

Nevertheless, we have the following theorem.

Theorem 4. Suppose that assumption (4) holds and φ ∈ D(A^3/2), ψ ∈ D(A^1/2). Then, for solution of difference scheme (2), the following stability estimates hold:

()

where M does not depend on τ, φ, ψ, f_1,1(x), and f_s(x), 1 ≤ s ≤ N − 1.

Proof. Using formulas in (23) and estimates (11), (12), and (14), we obtain

()

Applying A^1/2 to formulas in (23), we get

()

Now, applying Abel’s formula to (23), we obtain the following formulas:

()

Next, let us obtain the estimates for and . First, applying A to formula (28) and using estimates (11), (12), and (14) and the triangle inequality, one can obtain

()

Second, applying A^1/2 to formula (29) and using estimates (11), (12), and (14) and the triangle inequality, we get

()

Now, we will prove estimates (25). Using formula (19), estimates (11), (12), (26), and (27), and the triangle inequality, we obtain

()

for any k ≥ 2. Applying A^1/2 to (19), we get

()

for k ≥ 2. Now, applying Abel’s formula to (19), we have

()

Applying A to formula (34) and using estimates (11) and (12) and the triangle inequality, we obtain

()

for k ≥ 2. Theorem 4 is proved.

Note that the stability estimates obtained previously permit us to get the convergence estimate of difference scheme (2) under the smoothness property of solution (1). Actually, under the condition u(t) ∈ C([0,1], H), we can obtain the third order of accuracy for the error of difference scheme (2). Since , this condition is satisfied under the given data φ ∈ D(A³), ψ ∈ D(A^5/2), , and f(0) ∈ D(A³).

Now, let us give application of this abstract result for nonlocal boundary value problem

()

for hyperbolic equation. Problem (36) has a unique smooth solution u(t, x), δ > 0 and the smooth functions a(x) ≥ a > 0 (a(0) = a(1), 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 (36) to nonlocal boundary value problem (1) in a Hilbert space H = L₂[0,1] with a self-adjoint positive definite operator A^x defined by (36).

The discretization of problem (36) is carried out in two steps. In the first step, let us define the grid space

()

We introduce Hilbert space L_2h = L₂([0,1] _h),

, and

of the grid functions

defined on [0,1] _h, and we assign the difference operator

by the formula

()

acting in the space of grid functions

satisfying the conditions φ₀ = φ_K, φ₁ − φ₀ = φ_K − φ_K−1.

With the help of

, we arrive at the nonlocal boundary value problem

()

for a system of ordinary differential equations.

In the second step, we replace problem (2) with difference scheme (40)

()

Theorem 5. Let τ and h be sufficiently small numbers. Then, the solution of difference scheme (40) satisfies the following stability estimates:

()

Here, M₁ does not depend on

, and

The proof of Theorem 5 is based on the proof of abstract Theorem 4 and the symmetry property of operator defined by (38).

Acknowledgments

The authors would like to thank Professor Pavel E. Sobolevskii and reviewers for their helpful comments.

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On Stability of a Third Order of Accuracy Difference Scheme for Hyperbolic Nonlocal BVP with Self-Adjoint Operator

Abstract

1. Introduction

2. Third Order of Accuracy Difference Scheme Subject to Nonlocal Conditions

Acknowledgments

References

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On Stability of a Third Order of Accuracy Difference Scheme for Hyperbolic Nonlocal BVP with Self-Adjoint Operator

Abstract

1. Introduction

2. Third Order of Accuracy Difference Scheme Subject to Nonlocal Conditions

Acknowledgments

References

Citing Literature

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