Approximate solution of painlevé equation i by natural decomposition method and laplace decomposition method

• Autorzy: Amir Muhammad , Haider Jamil Abbase , Ahmad Shahbaz , Ashraf Asifa , Gul Sana

Identyfikatory

DOI

10.2478/ama-2023-0048

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

natural decomposition method; Laplace decomposition method; series solution; Adomain polynomial; Painlevéequation

• Wydawca: Oficyna Wydawnicza Politechniki Białostockiej
• Czasopismo: Acta Mechanica et Automatica
• Rocznik: 2023
• Tom: Vol. 17, no. 3
• Strony: 417--422
• Opis fizyczny: Bibliogr. 32 poz., tab.

Twórcy

autor

Amir Muhammad

• Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan

• https://orcid.org/0009-0002-4871-4312

autor

Haider Jamil Abbase

• Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan

• https://orcid.org/0000-0002-7008-8576

autor

Ahmad Shahbaz

• Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan

• https://orcid.org/0000-0002-9901-0924

autor

Ashraf Asifa

• Department of Mathematics, University of Management and Technology Lahore, Pakistan

• https://orcid.org/0009-0005-4786-7757

autor

Gul Sana

• Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan

• https://orcid.org/0000-0002-5075-2599

Bibliografia

• 1. Painleve P. Sur les équationsdifférentielles du second ordre et d'ordresupérieurdontl'intégralegénéraleestuniforme. Acta mathema-tica. 1902 Dec;25:1-85.
• 2. Borisov AV, Kudryashov NA. Paul Painlevé and his contribution to science. Regular and Chaotic Dynamics. 2014 Feb;19:1-9.
• 3. Segur H, Ablowitz MJ. Asymptotic solutions of nonlinear evolution equations and a Painlevé transcedent. Physica D: Nonlinear Phe-nomena. 1981 Jul 1;3(1-2):165-84.
• 4. Kanna T, Sakkaravarthi K, Kumar CS, Lakshmanan M, Wadati M. Painlevé singularity structure analysis of three component Gross–Pitaevskii type equations. Journal of mathematical physics. 2009 Nov 25;50(11):113520.
• 5. Cao X, Xu C. ABäcklund transformation for the Burgers hierarchy. InAbstract and Applied Analysis 2010 Jan 1 (Vol. 2010). Hindawi.
• 6. Lee SY, Teodorescu R, Wiegmann P. Viscous shocks in Hele–Shaw flow and Stokes phenomena of the Painlevé I transcendent. Physica D: Nonlinear Phenomena. 2011 Jun 15;240(13):1080-91.
• 7. Dai D, Zhang L. On tronquée solutions of the first Painlevé hierarchy. Journal of Mathematical Analysis and Applications. 2010 Aug 15;368(2):393-9.
• 8. Florjańczyk M, Gagnon L. Exact solutions for a higher-order nonlin-ear Schrödinger equation. Physical Review A. 1990 Apr 1;41(8):4478.
• 9. Ablowitz MJ, Segur H. Solitons and the inverse scattering transform. Society for Industrial and Applied Mathematics; 1981 Jan 1.
• 10. Tajiri M, Kawamoto S. Reduction of KdV and cylindrical KdV equa-tions to Painlevé equation. Journal of the Physical Society of Japan. 1982 May 15;51(5):1678-81.
• 11. Dehghan M, Shakeri F. The numerical solution of the second Painlevé equation. Numerical Methods for Partial Differential Equa-tions: An International Journal. 2009 Sep;25(5):1238-59.
• 12. Clarkson PA. Special polynomials associated with rational solutions of the fifth Painlevé equation. Journal of computational and applied mathematics. 2005 Jun 1;178(1-2):111-29.
• 13. El-Gamel M, Behiry SH, Hashish H. Numerical method for the solu-tion of special nonlinear fourth-order boundary value problems. Ap-plied Mathematics and Computation. 2003 Dec 25;145(2-3):717-34.
• 14. Ellahi R, Abbasbandy S, Hayat T, Zeeshan A. On comparison of series and numerical solutions for second Painlevé equation. Numer-ical Methods for Partial Differential Equations. 2010 Sep;26(5): 1070-8.
• 15. Gromak VI, Laine I, Shimomura S. Painlevé differential equations in the complex plane. InPainlevé Differential Equations in the Complex Plane 2008 Aug 22. de Gruyter.
• 16. Bobenko AI, Eitner U, editors. Painlevé equations in the differential geometry of surfaces. Berlin, Heidelberg: Springer Berlin Heidelberg; 2000 Dec 12.
• 17. Dehghan M, Shakeri F. The numerical solution of the second Painlevé equation. Numerical Methods for Partial Differential Equa-tions: An International Journal. 2009 Sep;25(5):1238-59.
• 18. Saadatmandi A. Numerical study of second Painlevé equation. Comm. Numer. Anal. 2012;2012.
• 19. Sierra-Porta D, Núnez LA. On the polynomial solution of the first Painlevé equation. Int. J. of Applied Mathematical Research. 2017;6(1):34-8.
• 20. Ahmad H, Khan TA, Yao S. Numerical solution of second order Painlevé differential equation. Journal of Mathematics and Computer Science. 2020;21(2):150-7.
• 21. Izadi M. An approximation technique for first Painlevé equation.
• 22. Khan ZH, Khan WA. N-transform properties and applications. NUST journal of engineering sciences. 2008 Dec 31;1(1):127-33.
• 23. Belgacem FB, Silambarasan R. Theory of natural transform. Math. Engg. Sci. Aeros. 2012 Feb 25;3:99-124.
• 24. Spiegel MR. Laplace transforms. New York: McGraw-Hill; 1965.
• 25. Belgacem FB, Karaballi AA. Sumudu transform fundamental proper-ties investigations and applications. International Journal of Stochas-tic Analysis. 2006;2006.
• 26. Maitama S, Hamza YF. An analytical method for solving nonlinear sine-Gordon equation. Sohag Journal of Mathematics. 2020;7(1):5-10.
• 27. Elbadri M, Ahmed SA, Abdalla YT, Hdidi W. A new solution of time-fractional coupled KdV equation by using natural decomposition method. InAbstract and Applied Analysis 2020 Sep 1 (Vol. 2020, pp. 1-9). Hindawi Limited.
• 28. Maitama S, Kurawa SM. An efficient technique for solving gas dy-namics equation using the natural decomposition method. InInterna-tional Mathematical Forum 2014 (Vol. 9, No. 24, pp. 1177-1190). Hikari, Ltd..
• 29. Amir M, Awais M, Ashraf A, Ali R, Ali Shah SA. Analytical Method for Solving Inviscid Burger Equation. Punjab University Journal of Math-ematics. 2023 Dec 3;55(1).
• 30. Behzadi SS. Convergence of iterative methods for solving Painlevé equation. Applied Mathematical Sciences. 2010;4(30):1489-507.
• 31. Hesameddini E, Peyrovi A. The use of variational iteration method and homotopy perturbation method for Painlevé equation I. Applied Mathematical Sciences. 2009;3(37-40):1861-71.
• 32. Behzadi SS. Convergence of iterative methods for solving Painlevé equation. Applied Mathematical Sciences. 2010;4(30):1489-507.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).