A Method for Solving Non-Linear Initial Value Problems by Adomian Decomposition

To solve first and second-order nonlinear initial value problems involving variable system coefficients, we provide the Adomian decomposition position approach in this article. We look at different approaches for the Adomian decomposition series and the series of Adomian polynomials to find the answers to first- and second-order nonlinear initial value problems. To consistently compute the Taylor expansion series of the solution using easily integrable terms, we have introduced a novel modified recursion method, which slows down the Adomian decomposition series significantly. We demonstrate the appropriate nonlinear recurrence relations for the coefficients of the solutions. We then go on to study the errors and acceleration of convergence as they correspond to the sequence of solution approximations. We conclude by looking at many illustrative cases to show how quickly the two sets of data converge.