Semi-analytical methods for solving non-linear differential equations: A review

Abstract

This article develops a new semi-analytical technique based on the homotopy analysis approach for solving linear or non-linear differential equations and the results are compared to the well-known approaches such as the Adomian decomposition method (ADM), homotopy perturbation method (HPM), homotopy analysis method (HAM), and optimized decomposition method (ODM). We discuss the decomposition of the non-linear operator to expedite the HAM solution's convergence to its precise values by using the convergence control parameter. The theoretical convergence analysis and the error estimates are studied. Numerical illustrations show that our proposed scheme improves the accuracy of the non-linear problems discussed in the recently published articles [30] and [31] to an excellent extent and also indicate rapid convergence.