Numerical Techniques for Solving Integral Equations with Carleman Kernel

Integral equations with Carleman type kernel arise frequently in physics and engineering,theory of elasticity, mathematical problems of radiativeheat transformationsand radiative equilibrium. In this work we focus our attention mainly on the numerical handling of the Fredholm and Volterra integral equations with Carleman kernel. The numerical treatment of such equations can be achieved by using the following numerical techniques, namely; Toeplitz matrix method, Product Nystrom method, sinc-collocation method and Laplace Adomian decomposition method. To test the efficiency of these methods, we consider some numerical test cases. Numerical results have shown that product Nystrom method is one of the most powerful numerical techniques for solving Fredholm integral equation with a Carleman kernel in comparison with the other numerical techniques. On the other hand, we see clearly that the Laplace Adomian decomposition method is a very reliable and efficient method for solving Volterra integral equations with Carleman kernel. Introduction In recent years singular integral equations have attracted the attention of many scientists and researchers due to their wide range of applications in science and technology. Many problems related to mathematical physics, engineering, theory of elasticity and the mixed problems of continuous media lead to integral equations of the first or second kind [7,21]. The solution of these problems can be obtained analytically using the theory developed by Muskhelishvili . On the other hand, numerical methods play a very important rule in solving singular integral equations. Abdalkhani obtained a numerical solution of the nonlinear Volterra integral equation with Carleman kernel. Carleman was the first scientist who realized the importance of the Fredholm integral equation and its applications . The importance of Carleman kernel came from the work of Arytiunian[8] who has shown that the contact problem of nonlinear theory of plasticity in the first approximation reduces to a Fredholm integral equation of the first kind with Carleman kernel. Krein’s technique [5] is used to find the relationship between integral which has Carleman kernel and integral with logarithmic kernel . Abdou, in his work [1,4] used a series in the Legendre polynomials form to obtain the solution of Fredholm-Volterra integral equation of the second kind under certain conditions.Guoqiang obtained a numerical solution of two dimensional Volterra integral equations by collocation and iterated collocation method. Brunner has used the sinc-collocation method to solve Fredholm integral equation of the second kind. Hendi and Al-Hazmi have obtained the solution of Volterra integral equation using Laplace Adomian decomposion method. Gragam applied the Galerkin method to obtain the solution of singular integral equation. Mohamed and Ismail implemented Toeplitz matrix method to obtain numerical solution. Furthermore, Orsi has used product Nystrom as numerical method. This thesis is organized as follows: chapter one presents some of the fundamental principles of integral equations and their classifications such as linearity, homogeneity and singularity. Carleman kernel is also included in this chapter. Some of numerical techniques used to solve integral equations with Carleman kernel are introduced in chapter two. These techniques are Laplace Adomian decomposition method, Product Nystrom method, Toeplitz matrix method and Sinc-collocation method. In chapter three, the simulation results for solving the Fredholm integral equation with a Carleman kernel using the previous techniques are presented. Finally, chapter four involves solving the Volterra integral equations of the second kind with a Carleman kernel using Laplace Adomian decomposition method, Toeplitz matrix method, product Nystrom method and sinc-collocation method.