The theory and application of integral equations is an important subject within applied mathematics. Integral equations are used as mathematical models for many physical situations. I am interested in a numerical solution of the Fredholm integral equation of the second kind. The integral operator T is either a linear integral operator or a nonlinear Urysohn integral operator. The eigenvalue problem associated with a linear integral operator T is of equal interest.

As the exact solution can be obtained rarely, in order to find an approximate solution, the integral operator T is replaced by finite-rank operator Tn. The integral equation associated with Tn can then be reduced to a finite system of linear /non-linear equations and the associated eigenvalue problem is reduced to matrix eigenvalue problem, which can be solved using a computer. There are two principal approaches to construct a sequence of finite rank operators Tn approximating the integral operator. The first approach is to replace the integral by a convergent quadrature formula. This give rise to the Nystrom operator. The other approach is base on a sequence of projection operators converging to the Identity  operator pointwise. The classical Galerkin method and its variants fall under this category. In order to prove the convergence and the rate of convergence of the approximate solution to the exact solution, results from Fractional Analysis are used.

My research work involves the theoretical work such as obtaining the orders of convergence for various methods and validating the theoretical results by implementation in specific cases using a computer. I have proposed a method called Modified Projection Method which improves the order of convergence significantly as compared to the Galerkin method. This improvement as achieved while retaining the computational complexity essentially the same as in the case of Galerkin method.