New Numerical Iteration Schemes Based on Perturbation Iteration Algorithms
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2024
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Abstract
Perturbation-Iteration algorithms (PIA) have been developed recently to solve differential equations analytically. A continuous solution in terms of closed form functions as an approximation of the original equation can be found using the method. The method has been implemented to algebraic equations, ordinary and partial differential equations successfully. Based on the formalism developed previously, in this work, purely numerical versions of the perturbation-iteration algorithms are proposed for the first time for first-order nonlinear ordinary differential equations. The new algorithms are called as the Numerical Perturbation-Iteration Algorithms (NPIA) to distinguish them from the continuous analytical ones (PIA) and the method in general will be called the Numerical Perturbation-Iteration Method (NPIM). Iteration schemes based on one correction term in the perturbation solution and first-order derivatives in the Taylor series expansions (NPIA(1,1)), and one correction term in the perturbations and second-order derivatives in the series expansions (NPIA(1,2)) are derived. The numerical results are compared with the exact solutions and a very good match is observed. NPIA(1,2) produced slightly better results compared to NPIA(1,1) with total errors being at least O(h2), h being the step size for both methods. An improvement suggestion for NPIA(1,1) is proposed in the last section to eliminate unrealistic blow-up solutions which are encountered for some specific equations. © 2024 World Scientific Publishing Company.
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Algebra , Initial value problems , Iterative methods , Nonlinear equations , Numerical methods , Perturbation techniques , Taylor series , Closed form , Correction terms , Form function , Initial-value problem , Iteration algorithms , Iteration techniques , Numerical iteration , Numerical iteration scheme , Numerical iteration technique , Perturbation–iteration algorithm , Ordinary differential equations