Approximate Solution of Painlevé Equation I by Natural Decomposition Method and Laplace Decomposition Method
Muhammad Amir
Orcid profile, Jamil Abbas Haider
Orcid profile, Shahbaz Ahmad
Orcid profile, Sana Gul
Orcid profile and Asifa Ashraf
Orcid profile 
Published Online: Jul 15, 2023
Page range: 417 - 422
Received: Dec 08, 2022
Accepted: Mar 19, 2023
DOI: https://doi.org/10.2478/ama-2023-0048
Keywords
© 2023 Muhammad Amir et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
The Painlevé equations and their solutions occur in some areas of theoretical physics, pure and applied mathematics. This paper applies natural decomposition method (NDM) and Laplace decomposition method (LDM) to solve the second-order Painlevé equation. These methods are based on the Adomain polynomial to find the non-linear term in the differential equation. The approximate solution of Painlevé equations is determined in the series form, and recursive relation is used to calculate the remaining components. The results are compared with the existing numerical solutions in the literature to demonstrate the efficiency and validity of the proposed methods. Using these methods, we can properly handle a class of non-linear partial differential equations (NLPDEs) simply.
Novelty
One of the key novelties of the Painlevé equations is their remarkable property of having only movable singularities, which means that their solutions do not have any singularities that are fixed in position. This property makes the Painlevé equations particularly useful in the study of non-linear systems, as it allows for the construction of exact solutions in certain cases. Another important feature of the Painlevé equations is their appearance in diverse fields such as statistical mechanics, random matrix theory and soliton theory. This has led to a wide range of applications, including the study of random processes, the dynamics of fluids and the behaviour of non-linear waves.