Application of Laplace Transform Method for Solving Weakly-Singular Integro-Differential Equations

In this paper, the aim is to study the weakly singular integro-differential equations, typically its significance appears in various applications of engineering and science. The contributions of this paper have been presented through the study of existence, uniqueness, and different stability of solution of the weakly singular integro-differential equations, and solving this type of equations analytically. We propose an analytic method based on Laplace transform to solve the weakly singular integro-differential equations, whose advantages lie in simplicity application and obtaining exact solutions. Some suitable examples have been provided to better understand this work and the results of experiments exhibited the proposed method as a simple approach and superiority in accuracy and efficiency of solution.


INTRODUCTION
Weakly singular integro-differential equations (WSIDE) have been growing importance enormously as it plays a key role to represent many typically in phenomena of physics and engineering as mathematical modeling.For instance, diffusion of separated particles in turbulent fluids [1], elasticity and fracture in mechanics [2], biosciences [3], potential problems, thermal conductivity problems, materials, radiative equilibrium, the Dirichlet problems [4][5][6] and so on.
So a variety works focused its interest to tackle the equations of this form which are usually numerical methods, including an operational method [7], Bernstein series [8], Block boundary value method [9], Partition of the interval and introduction of additional parameters [10], Smoothing transformation and spline collocation [11], The asymptotic estimations of the solution [12], Product integration [13], collocations methods as Spline, Piecewise Polynomial, and Spectral respectively [14][15][16], but it is well known that the results of numerical methods have an error rate.
To address the WSIDE with different orders of derivatives without error rate in results, in this paper we propose analytic efficient and simple technique to yield exact solution which is Laplace transform method (LTM).Procedure of Laplace transform (LT) with differential equations and integral equations is changing them to polynomial equations and easily can be solved, and thus the solution of the considered equation is obtained by taking the inverse Laplace transform (LT) for it [17,18].Our motivation of this work is to present exact solution for many real-world problems are described as WSIDE by using efficient and simple method.
The remainder of this paper is organized as follows.In Section 2, the definition of concept WSIDE is expressed.In Section 3, we discuss the existence and uniqueness of solution for WSIDE.Section 4 discusses the various stability of solution for WSIDE.In Section 5, we introduce the concept of Laplace transform and it is followed up implementation it to solve WSIDE.In Section 6, the exact solutions of the WSIDE are obtained using the Laplace transform by solving several examples.Finally, conclusions are drawn in Section 7.

THE WEAKLY-SINGULAR INTEGRO-DIFFERENTIAL EQUATIONS (WSIDE)
Consider the following standard form of th-order linear integro-differential equation of the second kind: where  () () =      ,  < ∞, (), (, ) are given functions and () is the unknown function, also (, ) is denoted the kernel of the integro equation.We usually propose that the functions () and () are continuous or square integrable on [, ].Furthermore, Eq. ( 1) is also denoted a singular integral equation if the kernel K(x, t) becomes infinite at one or more points in the domain of integration.
Motivation of present work is the desire to obtain exact analytic solution by using LTM for a linear weakly singular Volterra integro-differential equation (WSVIDE) of th order and the second kind, where the integrand is denote as a weakly singular in the sense that its integral is continuous at the singular point, that is, its kernel (, ) =  → , where, 0 <  < 1 is positive real constant.

EXISTENCE AND UNIQUENESS SOLUTION OF WSIDE
This section discusses the existence and uniqueness of solution for the WSIDE and toward meeting that, it will be beneficial considering following procedure: Let  () () = (), where () is a continuous real valued function defined on [, ] .Integrating both sides  -times from  to  and using the initial conditions, yields to: Eq. ( 1) can be written as: Suppose, So that Eq. ( 3) is transformed into, Now, having the existence theory of solution at our disposal, we discuss it.

STABILITY SOLUTION OF WSIDE
In this section we investigate the stability of the solution for the WSIDE, for this using the idea of the variation of parameters formula for linear differential systems to obtain an integral equation for the solutions of Eq. (1).For this purpose, let () be a fundamental matrix solution of: so that any solution of Eq. ( 1) with the initial functions  on [ 0 , ] is given by: and prove stability results for the system (1).We begin with giving a suitable definition of stability, then lemma and the theorems.
Consequently |( 1 )| <  which is a contradiction.Thus, the solution of Eq. ( 1) is uniformly stable, completing the proof.Then the solution of Eq. ( 1) is asymptotically stable.
Proof: We first show that stability of the solution.From Eq. ( 11), there exists a positive constant  such that: Thus, |( 1 )| <  , which is a contradiction.Thus, the solution of Eq. ( 1) is asymptotically stable.Thus from (a) and (b), the solution of Eq. ( 1) is exponentially stable.

Laplace transform method (LTM)
In general, the idea of a transformation is a very important in problem solving, where the difficult problem is changed in some way into an easier problem and then solve that easier problem to obtain solution and apply it to original problem.
The Laplace transform method (LTM) aims to seek solve the differential equations and integral equations easily by changing it to polynomial, and then taking the inverse Laplace transform, which lead for the solution of intended equation.
Here we present a basic concept of LTM, for a given function () with respect  ≥ 0, the Laplace transform method can be described as: where,  is real, and  denotes the Laplace transform operator.Furthermore, a vanishment () as  approaches infinity is an important and necessary condition for the existence of the Laplace transform ().This means that: As well as, there are key properties of the Laplace transforms which are used in the proposed framework are given briefly as follows: 1. Constant multiple: { ()} =  {()},  is constant.There are also elementary Laplace transforms, some of which can be summarized briefly in Table 1.