Methods of solution of singular integral equations

Purpose: This paper investigates a different method to evaluate different real improper integrals and also to obtain the solutions of various types of Cauchy-type singular integral equations of the first kind. Methods: Methods using the analysis of functions of real variables only are reviewed and utilized for the above purpose. These methods clearly demonstrate that details of complex function theory which are normally employed in handling such integral equations for their solutions can be avoided altogether. Also, some approximate methods of solution of such integral equations are developed. Results: The solutions of real singular integral equations over different intervals such as (-1, 1); (a, b); (0, a) boolean OR (b, c); (-1, k) boolean OR (k, 1); (-infinity, b); (a,+infinity); (-infinity,+infinity); infinite intervals with a gap are obtained by using the proposed methods. Conclusion: The proposed methods are new and each has its own structure.