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 equations. 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.