A generalize Picard’s successive iteration method for the nth order initial valued problem

Abstract

This research extends Picard's successive iteration method from solving first-order initial value problems to a generalized approach for nth-order problems. A recursive formula is introduced, eliminating the need for linearization and discretization by leveraging previous point values for subsequent iterations. The method's convergence is rigorously established and validated through numerous tests and examples. Notably, as the order of the problem approaches infinity, the iterations consistently converge to the exact solution, demonstrating the method's scalability and efficacy. This advancement offers a robust and accurate approach for solving higher-order initial value problems.