Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 365904, 14 pages doi:10.1155/2012/365904 Research Article Numerical Solution of Weakly Singular Integrodifferential Equations on Closed Smooth Contour in Lebesgue Spaces Feras M. Al Faqih 1, 2 1 Department of Mathematics and Statistics, King Faisal University, Saudi Arabia 2 Department of Mathematics, Al-Hussein Bin Talal University, P. O. Box 20 Ma’an, Jordan Correspondence should be addressed to Feras M. Al Faqih, oxfer@yahoo.com Received 18 June 2012; Accepted 3 September 2012 Academic Editor: Yongkun Li Copyright q 2012 Feras M. Al Faqih. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The present paper deals with the justification of solvability conditions and properties of solutions for weakly singular integro-differential equations by collocation and mechanical quadrature methods. The equations are defined on an arbitrary smooth closed contour of the complex plane. Error estimates and convergence for the investigated methods are established in Lebesgue spaces. 1. Introduction Singular integral equations SIE and singular integro-differential equations with Cauchy kernels SIDE and systems of such equations model many problems in elasticity theory, aerodynamics, mechanics, thermoelasticity and queuing analysis see 1–6 and the literature cited therein. The general theory of SIE and SIDE has been widely investigated over the last decades 7–11. It is known that the exact solution for SIDE can be found only in some particular cases. That is why there is a necessity to elaborate approximation methods for solving SIDE. In the past, there was a lot of research in literature devoted to an approximate solution of SIE and SIDE by collocation and mechanical quadrature methods. The equations are defined on the unit circle centered at the origin or on the real axis, see for example 12–15. However, the case when the contour of integration is an arbitrary smooth closed curve has not been studied enough. It should be noted that conformal mapping from the arbitrary smooth closed contour to the unit circle does not solve the problem. Moreover, it makes it more difficult. In the