Although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. Some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.Here, after criticizing previous works, a sucient condition for convergence ofhomotopy methods is presented. Finally, examples are given to show that evenif the homotopy method leads to a convergent series, it may not converge tothe exact solution of the equation under consideration. Manuscript profile

Although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. Some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.Here, after criticizing previous works, a sucient condition for convergence ofhomotopy methods is presented. Finally, examples are given to show that evenif the homotopy method leads to a convergent series, it may not converge tothe exact solution of the equation under consideration. Manuscript profile

In this paper we apply an approximate method based on Galerkin approach with Legendre wavelets basis, on a class of fourth order boundary value problems. The approach reduces the main equation to a system of linear algebraic equations that could be solved numerically. The operational matrix of the method is obtained, and the convergence of the method is proved. we approximate the solution and its higher order derivatives, for some special examples and compare the results with some other numerical methods. The results show the effectiveness of the proposed method. Manuscript profile