Abstract

We consider a telegraph equation with nonlocal boundary conditions, and using the application of Galerkin's method we established the existence and uniqueness of a generalized solution.

1. Introduction

The initial work devoted to the second-order partial differential equations with nonlocal integral conditions goes back to Cannon [1]. Subsequent investigations include [1–10] and many other references therein.

Nonlocal problems for some classes of partial differential equations have attracted much interest in the last few years, and several papers have been devoted to this subject. The problem of existence and uniqueness of solutions is of growing interest, as well as methods of obtaining explicit or approximate solutions for this kind of problems. The main reason of such a wide interest in this type of problems is that the non-local conditions occur in various equations of mathematical physics and in mathematical biology.

Nonlocal conditions come up when values of the function on the boundary are connected to values inside the domain. There are various types of non-local integral conditions for hyperbolic, parabolic, or elliptic equations; we point out an important non-local problem (see [11, 12]), studied for a parabolic equation with the following non local conditions. connected with quasistationary thermoelasticity theory. The non-local boundary conditions of type arise in modeling the technology of integral circuits, studied in [13].

Problems with nonlocal integral conditions for multidimensional hyperbolic equations have been intensively studied in recent years. However, the nonlocal integral conditions for equations were in the form of second kind as or where is a bounded domain in with a smooth boundary, is the inward normal vector to ; see [14, 15]. Non-local integral conditions of first kind as are studied in [15]. Under some assumptions on the kernel , existence and uniqueness of the generalized solution are established.

The present work deals with the application of the Galerkin method to determine a function , is a bounded domain with a smooth boundary , , which satisfies, in weak sense, the telegraph equation subject to the initial conditions and the integral condition where is the inward normal vector to . This equation arises in the analysis of propagation of electrical signals in a cable transmission line. Both the current and voltage satisfy an equation of this form. This equation arises also in the propagation of pressure waves in the study of pulsate blood flow in arteries.

Cannon et al. [5] have developed and analyzed finite-difference and finite-element Galerkin’s methods for the solution of one-dimensional heat equation with the Neumann and integral conditions. Cannon and van der Hoek [4] presented numerical schemes based on the finite difference method. In [16], the authors have applied the Galerkin method to a parabolic problem with an integral boundary condition and established the existence, uniqueness, and continuous dependence upon the data of a weak solution. In [17–21], the authors have used the method of semidiscretization in time and have established the existence and uniqueness of a weak solution. In [22], the authors have discussed some existence uniqueness results for a neutral functional differential equation with a nonlocal initial condition via the Galerkin approximation.

The paper is organized as follows. In Section 2, we specify notations, state some inequalities and make the sense of the desired solution precise. In Section 3, we establish the uniqueness of the solution. Finally, Section 4, is devoted to the construction of the approximate solution and its existence via the Galerkin method.

2. Notation and Definition

Let be the usual space of Lebesgue square integrable real functions on whose inner product and norm will be denoted by and , respectively.

is the Sobolev space consisting of functions such that all derivatives lower than one belong to equipped with the norm

We define the space .

2.1. Some Useful Inequalities: Gronwall Inequalities [23]

Let and be two integrable nonnegative functions on the interval with nondecreasing. If where is a positive constant, then .

Cauchy Schwarz inequality:

Cauchy inequality:

Trace inequality:
where is a positive constant that depends only on and on the domain .

2.2. Definition of Generalized Solution

We define the generalized solution of the problem (1.6)–(1.8) as a weak solution that satisfies for all Integrating by parts, we get Regrouping the identities (2.7), then (2.6) becomes

Definition 2.1. By a generalized solution of problem (1.6)–(1.8) one means a function such that the identity (2.8) holds for all function .

3. Uniqueness of the Generalized Solution

Now we will show that the generalized solution of problem (1.6)–(1.8), if it exists, is unique.

Theorem 3.1. Assume that , ,, , and then the generalized solution of problem (1.6)–(1.8), if it exists, is unique.

Proof. Suppose that there exist two different generalized solutions and for the problem (1.6)–(1.8); then obviously their difference is a generalized solution of the problem (1.6)–(1.8) with homogeneous equation and homogeneous initial and non local conditions, that is, . We will prove that in . Let , and denote . Consider the function The identity (2.8) becomes Substituting into (3.3), integrating by parts, and then using the fact that , it follows that Using the assumption on the function , we get where . Now, applying Cauchy Schwarz inequality to the first term in the right-hand side of (3.5), then Cauchy inequality with , yields Using the trace inequality, we obtain Considering the function it is easy to see that , , and . Consequently, substituting in (3.7), we get Seting it follows that Since is arbitrary chosen, let ; then (3.11) becomes Applying Gronwall Lemma, we get Consequently, we obtain , for all and .
If , then in . In the case where , we see that , where , is the entire part of ; then repeating the preceding reasoning for , we get , for all , and then in . Thus, the uniqueness is proved.

Remark 3.2. It should be noted that we can prove the uniqueness theorem by using a priori estimates, so we establish that the generalized solution, if it exists, satisfies the inequality

4. Existence of Generalized Solution

Theorem 4.1. Assume that the assumptions of Theorem 4.1 hold; then the non-local problem (1.6)–(1.8) has a unique solution .

Proof. In order to prove the existence of the generalized solution we apply Galerkin’s method. Let be a fundamental system in , such that . Now we will try to find an approximate solution of the problem (1.6)–(1.8) in the form The approximates of the functions and are denoted, respectively, by Substituting the approximation solution in (1.6), multiplying both sides by , and then integrating according to on , we get Substituting (4.1) in (4.3), we obtain This implies that Denote Then (4.5) becomes We obtain a system of differential equations of second order according to the variable with smooth coefficients and the initial conditions Consequently, we get a Cauchy problem of linear differential equations with smooth coefficients that is uniquely solvable. So it has a unique solution satisfying (4.3).

Lemma 4.2. The sequence is bounded.

Proof. Multiplying (4.4) by and summing over from 1 to , we get Integrating by parts the left-hand side of (4.9) over on yields Substituting the four last identities into (4.9), we get Applying Cauchy Schwarz inequality, Cauchy inequality, the hypothesis on the operator , and the fact that to the second term in the right hand side of (4.11), we get We may obtain similar inequalities for the third and the fourth terms in the right-hand side of (4.11). Now if we apply Cauchy Schwarz inequality to the last term in (4.11) we get Using the result inequalities in (4.11) and then regrouping the same terms yield Choose such that and and let then (4.14) becomes Now, we apply Gronwall Lemma to get Since and are the orthogonal projections of and , respectively, from the Pythagorean relation we have that and . Integrating (4.17) according to on yields Consequently, the sequence is bounded.