Improved Solutions of OHAM Approximate Procedure for Classes of Nonlinear ODEs

The primary purpose of this study is to apply the Optimal Homotopy Asymptotic Method (OHAM) to various nonlinear initial value problems of different orders to evaluate its accuracy, convergence, and computational efficiency. OHAM is considered a highly effective technique for solving nonlinear differential equations and is commonly used in scientific and engineering disciplines. It combines the strengths of homotopy and asymptotic methods. OHAM offers a straightforward approach to controlling and adjusting the convergence of the series solution. This is achieved through the utilization of an auxiliary function that incorporates multiple convergent control parameters with one order of approximation, which are optimally determined. The OHAM approach heavily relies on the auxiliary function H(p) which allows for the flexible and efficient solving of nonlinear differential equations. By carefully constructing and optimizing the parameters of H( p), the convergence of the solution series can be effectively controlled. As a result, OHAM proves to be a versatile and effective approach for solving various mathematical and engineering problems. Several examples have been solved. Numerical comparisons, displayed in tables and shown graphically in figures, prove and confirm the capability, efficiency, and better accuracy with less computational work.