On the Usage of Orthogonal Polynomials with Picard Iteration Method to Find Numerical Solutions of Neutrosophic Non-Linear Ordinary and Partial Differential Equations

aspg-24-2807-1791482487 20261008180127 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Journal of Neutrosophic and Fuzzy Systems JNFS 2771-6430 2771-6449 2024 8 2 On the Usage of Orthogonal Polynomials with Picard Iteration Method to Find Numerical Solutions of Neutrosophic Non-Linear Ordinary and Partial Differential Equations Arwa Hajjari Cairo University, Cairo, Egypt This research aims to modify the Picard iteration method by hybridizing it with some orthogonal polynomials and then applying the hybrid method in solving neutrosophic nonlinear elementary value problems. This method is based on modifying the Picard iteration method by approximating the right-hand side of the neutrosophic differential equation of the studied problem either by Legendre polynomials or by Chebyshev polynomials of the first kind to obtain two different hybrids of the Picard iteration method. Also, we apply this modification to neutrosophic elementary value problems represented by neutrosophic nonlinear and right-handed nonlinear differential equations to demonstrate the reliability and efficiency of the proposed modified method. For this goal, we prove how effective this method is, we calculate the neutrosophic absolute error of approximate solutions resulting from the application of the proposed modification of the Picard iteration method and with the exact solution. 2024 31 37 2807 https://creativecommons.org/licenses/by/4.0/ 10.54216/JNFS.080204 https://www.americaspg.com/journal/24/article/2807 https://www.americaspg.com/storage/21715968035.pdf