A New Analytical Approximate Approach for Non-Linear Initial Value

The textbook "A New Analytical Approximate Approach for Non-Linear Initial Value" offers a comprehensive investigation of a novel analytical method for solving non-linear initial value problems. The work is based on the Taylor expansion technique and utilizes the derivatives of the Taylor series as a central element in the methodology. Through this innovative approach, analytical solutions for various non-linear problems have been developed, presented in the form of power series. The coefficients of these series represent the non-linear terms in the non-linear partial differential equations (NPDE). The effectiveness of the approach is demonstrated through numerous examples, including biological population models and the system of Burger's equations. The results show significant improvements in terms of computation time, convergence, and accuracy, supported by theoretical proofs of the convergence theorems. The tables and figures of the new analytical solutions illustrate the validity and utility of this approach as a powerful mathematical tool for solving non-linear partial differential equations.