Abstract
In this work, we study a telegraph integrodifferential 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:
integrodifferential equation; integral conditions; approximate solution; Galerkin method1 Introduction
In this work, we consider the following hyperbolic integrodifferential equation with integral conditions:
for all
and the weighted integral conditions
where f, φ, ψ, h, a, c, α and K are given functions.
Various problems arising in heat conduction [15], chemical engineering [6], thermoelasticity [7], and plasma physics [8] 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 [1], 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 integrodifferential equation (1.1) with a
Volterra operator of the form
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
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
and
Calculating
Definition 1 By a generalized solution of problem (1.1)(1.4), we mean a function
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
The function α is continuous and denote
(H2) The function
(H3) The operator
Now we shall show that the generalized solution of problem (1.1)(1.4) if it exists is unique.
Theorem 2Assume that
Proof Suppose that there exists two different generalized solutions
Substituting v into identity (2.1), it follows
Integrating by parts it yields
Applying Cauchy inequality, ϵinequality and the hypotheses on the operator K to the last term in the righthand side of (3.2), we get
Applying similar inequalities with
denote
then (3.3) becomes
Gronwall inequality implies
hence
4 Existence of generalized solution
In order to prove the existence of the generalized solution we apply Galerkin method.
Theorem 3Assume that the assumptions of Theorem 2 hold, then the problem (1.1)(1.4) has a unique solution
Proof Let
We have to find for each
Denote
the approximate of the functions
Substituting (4.1) in (4.3), we get
Integrating by parts in
Denote
then (4.5) becomes
Consequently, we obtain a Cauchy system of secondorder integrodifferential equations
with smooth coefficients, so it has one and only one solution that for every n there exists a unique sequence
Lemma 4The sequence
Proof Multiplying (4.3) by
Integrating (4.6) over t from 0 to τ we obtain
Thanks to Cauchy inequality, ϵinequality, the hypotheses on the operator K to the last term in the righthand side of (4.7), we get
Using similar inequalities for the second, the third and the fourth terms in the righthand side of (4.7), then regrouping the same terms yields
Let
then (4.8) becomes
Now, we apply Gronwall lemma to get
Integrating (4.10) according to τ on
Thus inequality (4.11) implies the boundedness of the sequence
Remark 5 We have proved that the sequence
Lemma 6The limit of the subsequence
Proof We shall prove that the limit of the subsequence
Denote by u the weak limit of the subsequence
Finally, by passing to the limit in (4.12), we get that the limit u satisfies (2.1). □
Example 7 Consider the following boundary value problem for hyperbolic integrodifferential
equation for
subject to the initial conditions
and the weighted integral condition
where
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors typed, read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for their valuable suggestions.
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