On a new reliable modification of homotopy analysis method

In this paper, a new modification of the homotopy analysis method (HAM) is presented and applied to homogeneous or non-homogeneous differential equations with constant or variable coefficients. A comparative study between the new modified homotopy analysis method (MHAM) and the classical HAM is conducted. The main advantage of MHAM is that one can avoid the uncontrollability problems of the non-zero endpoint conditions encountered in the traditional HAM. Several illustrative examples are given to demonstrate the effectiveness and reliability of MHAM.