Homotopy Regularization Method to Solve the Singular Volterra Integral Equations of the First Kind

Abstract

In this paper, by combining the regularization method (RM) and the homotopy analysis method (HAM) a new approach is presented to solve the generalized Abel’s integral equations (AIEs) of the first kind which is called the HAM-regularization method (HRM). The RM is applied to transform the first kind integral equations (FKIEs) to the second kind (SK) which depends on the regularization parameter and the HAM is used to find the solution of AIEs of the SK. The validation of results are illustrated by presenting a convergence analysis theorem. Also, the efficiency and accuracy of the HRM are shown by solving three examples.

Key words and phrases. Homotopy analysis method, Regularization method, Abel’s singular integral equation of the first kind.

2000 Mathematics Subject Classification. 65R20, 45E10