ABSTRACT
In this chapter, we consider the numerical solution of two kinds of singular integral equations: first, the singular integral equations of first kind with Cauchy kernel and, second, the hypersingular integral equations of first kind.
Singular integral equations (CSIEs) with Cauchy kernel play a vital role in studying many problems of aerodynamics, fracture mechanics, neutron transport, wave propagation etc. Analogous to Cauchy singular integral equations, the hypersingular integral equations are equally important. Several problems occurring in the field of aerodynamics, aeronautics, interference, or interaction problems such as wing-tail surfaces problem are reducible into hypersingular integral equations or their system.
The analytic solutions of such equations are known when these equations are dominant equations. But these analytic solutions are of limited use as it is a nontrivial task to use it practically due to the presence of singularity in the known solutions itself. Further, there are many real-world problems such as crack problems occurring in the field of fracture mechanics which may not be always reducible as dominant equations. This is one of the reasons why there is a need to develop numerical methods. Although, various methods are available to find the approximate solution of Cauchy and hypersingular integral equations, search for numerical methods which are better than the available methods in some sense is always there. Hence, we propose numerical methods to find the approximate solution of Cauchy singular integral equations and hypersingular integral equations. The proposed methods convert the singular integral equations into a system of linear algebraic equations which can be solved easily. The convergence of sequence of approximate solutions is proved for both kinds of singular integral equations considered in this chapter. The derived convergence helps to obtain theoretical error bound for the error between the exact and approximate solutions. Hadamard conditions of well posedness are also established for each of the system of linear algebraic equations which is obtained as a result of approximation of corresponding singular integral equations. Finally, all the derived theoretical results are validated with the help of numerical examples.
