In this paper, modified homotopy perturbation method (MHPM) is applied to solve the general Fredholm-Volterra integro-differential equations (FV-IDEs) of order m with initial conditions. Selective functions and unknown parameters allowed us to obtain two step iterations. It is found that MHPM is a semi-analytical method for FV-IDEs and could avoid complex computations. Numerical examples are given to show the efficiency and reliability of the method. Proof of the convergence of the proposed method is also given.

Simple and efficient convex homotopy perturbation method (HPM) is presented to obtain an approximate solution of hyper-singular integral equations of the first kind. Convergence and error estimate of HPM are obtained. Three numerical examples were provided to verify the effectiveness of the HPM. Comparisons with reproducing kernel method (Chen et al., 2011) for the same number of iteration is also presented. Numerical examples reveal that the convergence of HPM can still be achieved for some problems even if the condition of convergence of HPM is not satisfied. (C) 2017 Ain Shams University.

Modified homotopy perturbation method (HPM) was used to solve the hypersingular integral equations (HSIEs) of the first kind on the interval [-1,1] with the assumption that the kernel of the hypersingular integral is constant on the diagonal of the domain. Existence of inverse of hypersingular integral operator leads to the convergence of HPM in certain cases. Modified HPM and its norm convergence are obtained in Hilbert space. Comparisons between modified HPM, standard HPM, Bernstein polynomials approach Mandal and Bhattacharya (Appl Math Comput 190: 1707-1716, 2007), Chebyshev expansion method Mahiub et al. (Int J Pure Appl Math 69(3): 265-274, 2011) and reproducing kernel Chen and Zhou (Appl Math Lett 24: 636-641, 2011) are made by solving five examples. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM and others. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points.