Homotopy Analysis Method Based on Optimal Value of the Convergence Control Parameter for Solving Semi-Differential Equations

In this paper, homotopy analysis method is directly extended to investigate nth order semi-differential equations and to derive their numerical solutions which is introduced by replacing some integer-order space derivatives by fractional derivatives. the fractional derivatives are described in the caputo sense. so the homotopy analysis method for differential equations of integer-order is directly extended to derive explicit and numerical solutions of the fractional differential equa- tions. an optimal value of the convergence control parameter is given through the square residual error. comparison is made between homotopy perturbation method, collocation spline method, and the present method.