In this work, we study a telegraph integro-differential equation with a weighted integral condition. By means of the Galerkin method, we establish the existence and uniqueness of a generalized solution.
MSC: 35L05, 35L20, 35L99.
Keywords:integro-differential equation; integral conditions; approximate solution; Galerkin method
In this work, we consider the following hyperbolic integro-differential equation with integral conditions:
and the weighted integral conditions
where f, φ, ψ, h, a, c, α and K are given functions.
Various problems arising in heat conduction [1-5], chemical engineering , thermoelasticity , and plasma physics  can be modeled by the nonlocal problems. Boundary value problems with integral conditions constitute a very interesting and important class of problems. These nonlocal conditions arise mostly when the data on the boundary cannot be measured directly. Recall that the presence of an integral term in boundary conditions can complicate the application of classical methods of functional analysis in the theoretical study of nonlocal problems, therefore, several methods have been proposed for overcoming the difficulties arising from nonlocal conditions; see Beilin , Cannon et al.[2,8], and Dehghan et al.[3,4,9].
Numerical solutions are introduced to obtain approximations for the solution of partial differential equations when the analytical solutions are difficult or impossible to obtain due to complicated geometry or boundary conditions. In the area of numerical analysis, the Galerkin method is a class of methods for converting a continuous operator problem to a discrete problem. In principle, it is the equivalent of applying the method of a variation of parameters to a function space, by converting the equation to a weak formulation, hence in this approach we choose a system of linearly independent functions such that they satisfy the given homogeneous boundary condition, and they are dense in a function space containing the exact solution of the above boundary value problem.
The advantage of this approach is not only to establish the existence and uniqueness of the solution, but it is also a very effective method in the study of the approximate solution and its convergence.
In this paper, we study the hyperbolic integro-differential equation (1.1) with a Volterra operator of the form in the second member, which appears in the modelling of the quasi-static flexure of a thermo-elastic rod and has been studied in [9,10] under different boundary conditions, by means of the Rothe method. Let us mention that different methods are used to solve similar integro-differential equations, for example, in [11,12] the authors have established the existence and uniqueness of the solution using Rothe’s method of an integro-differential equation. In [10,13], the authors have used Rothe’s method and the techniques of  to prove the existence, uniqueness and continuous dependence of a strong solution to a quasi-linear integro-differential equation. In , the local existence and uniqueness of a classical solution of an abstract second-order integro-differential equation in a Banach space have been investigated by using the theory of an analytic semi-groups and contraction mapping theorem. In [14,15] the authors investigated a telegraph equation with non-local integral conditions by means of the Galerkin method.
This paper is organized as follows: In the next section, we define the generalized solution and the functional spaces. In Section 3, we prove that the generalized solution if it exists is unique. The existence of the generalized solution by using the Galerkin method is established in the fourth section, and for this, we construct an approximation solution of the problem (1.1)-(1.4). We prove that we can extract a subsequence, which converges to the desired generalized solution. An application is included to illustrate that corresponding assumptions are satisfied.
2 Notation and definition
Let be the usual space of Lebesgue square integrable real functions on whose inner product and norm will be denoted respectively by and . Denote by the Sobolev space consisting of all functions having weak derivatives in , with the norm
Let us define the generalized solution of the problem (1.1)-(1.4). Suppose that u is a solution of this problem, multiply both sides of equation (1.1) by , where , integrate by parts the resultant equation over the domain Q, use the conditions (1.2), (1.3), (1.4) and the fact that , we obtain
3 Uniqueness of generalized solution
For solving the problem, we make the following hypotheses:
(H1) The functions a and c are nonnegative and satisfy on Q
Now we shall show that the generalized solution of problem (1.1)-(1.4) if it exists is unique.
Proof Suppose that there exists two different generalized solutions and of the problem (1.1)-(1.4), then is a generalized solution of the problem (1.1)-(1.4) with and second member . We shall prove that in Q. Let and denote for .
Substituting v into identity (2.1), it follows
Integrating by parts it yields
Applying Cauchy inequality, ϵ-inequality and the hypotheses on the operator K to the last term in the right-hand side of (3.2), we get
then (3.3) becomes
Gronwall inequality implies
4 Existence of generalized solution
In order to prove the existence of the generalized solution we apply Galerkin method.
Substituting (4.1) in (4.3), we get
then (4.5) becomes
Consequently, we obtain a Cauchy system of second-order integro-differential equations with smooth coefficients, so it has one and only one solution that for every n there exists a unique sequence that satisfies (4.3). □
Integrating (4.6) over t from 0 to τ we obtain
Thanks to Cauchy inequality, ϵ-inequality, the hypotheses on the operator K to the last term in the right-hand side of (4.7), we get
Using similar inequalities for the second, the third and the fourth terms in the right-hand side of (4.7), then regrouping the same terms yields
then (4.8) becomes
Now, we apply Gronwall lemma to get
Remark 5 We have proved that the sequence is bounded, so we can extract a subsequence, which we denote by that is weakly convergent. Now we prove that its limit is the desired solution of the problem (1.1)-(1.4).
Proof We shall prove that the limit of the subsequence satisfies the identity (2.1). Let , such that , let us prove that identity (2.1) holds for any functions . Since the set is such that is dense in , it suffices to prove (2.1) for . Multiplying (4.3) by , summing according to k from 1 to n, then integrating over t from 0 to T, we obtain
Finally, by passing to the limit in (4.12), we get that the limit u satisfies (2.1). □
subject to the initial conditions
and the weighted integral condition
where . It is easy to prove that assumptions (H1)-(H3) are satisfied, then from Theorems 2 and 3, and we deduce that the problem (4.13)-(4.16) has a unique generalized solution in the sense of Definition 1. Moreover, the function is the solution of this problem.
The authors declare that they have no competing interests.
All authors typed, read and approved the final manuscript.
The authors would like to thank the referees for their valuable suggestions.
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