On the semi-analytic technique to deal with nonlinear fractional differential equations

• Autorzy: Khirsariya Sagar R. , Rao Snehal B.

Identyfikatory

DOI

10.17512/jamcm.2023.1.02

• Języki publikacji: EN

Abstrakty

EN

In this article, we present a novel hybrid approach, by combining the Sawi transform with the homotopy perturbation method, to achieve the approximate and analytic solutions of nonlinear fractional differential equations (ODE as well as PDE) using the time-fractional Caputo derivative. The proposed algorithm is faster and simple compared to other iterative methods. The Sawi transform is used along with the homotopy perturbation method to accelerate the convergence of the series solution. The results discussed using calculations, graphs and tables are compatible for comparison with other known methods like the residual power series method and the exact solution which are discussed in the literature.

Słowa kluczowe

EN

fractional differential equation; logistic equation; Fornberg-Whitham equation; homotopy perturbation method; Sawi transform

PL

ułamkowe równanie różniczkowe; równanie logistyczne; równanie Fornberga-Whithama; metoda perturbacji homotopii

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2023
• Tom: Vol. 22, nr 1
• Strony: 17--30
• Opis fizyczny: Bibliogr. 30 poz., rys., tab.

Twórcy

autor

Khirsariya Sagar R.

• Department of Mathematics, Marwadi University Rajkot-360003, Gujarat, India

autor

Rao Snehal B.

• Department of Applied Mathematics, Faculty of Technology and Engineering, The Maharaja Sayajirao University of Baroda, Vadodara-390001, Gujarat, India

Bibliografia

• [1] Kim, H. (2017). On the form and properties of an integral transform with strength in integral transforms. Far East Journal of Mathematical Sciences, 102(11), 2831-2844.
• [2] Watugala, G.K. (1993). Sumudu transform: a new integral transform to solve differential equations and control engineering problems. Integrated Education, 24(1), 35-43.
• [3] Mahgoub, M.M.A., & Mohand, M. (2019). The new integral transform “Sawi Transform”. Advances in Theoretical and Applied Mathematics, 14(1), 81-87.
• [4] Elzaki, T.M. (2011). The new integral transform Elzaki transform. Global Journal of Pure and Applied Mathematics, 7(1), 57-64.
• [5] Ahmadi, S.A.P., Hosseinzadeh, H., & Cherati, A.Y. (2019). A new integral transform for solving higher order linear ordinary Laguerre and Hermite differential equations. International Journal of Applied and Computational Mathematics, 5(5), 1-7.
• [6] Khan, Z.H., & Khan, W.A. (2008). N-transform properties and applications. NUST Journal of Engineering Sciences, 1(1), 127-133.
• [7] Mohand, M., & Mahgoub, A. (2017). The new integral transform “Mohand Transform”. Advances in Theoretical and Applied Mathematics, 12(2), 113-120.
• [8] Aboodh, K.S. (2013). The new integral transform “Aboodh Transform”. Global Journal of Pure and Applied Mathematics, 9(1), 35-43.
• [9] Kamal, H., & Sedeeg, A. (2016). The new integral transform Kamal transform. Advances in Theoretical and Applied Mathematics, 11(4), 451-458.
• [10] Jafari, H. (2021). A new general integral transform for solving integral equations. Journal of Advanced Research, 32, 133-138.
• [11] Qureshi, S. (2021). Fox H-functions as exact solutions for Caputo type mass spring damper system under Sumudu transform. Journal of Applied Mathematics and Computational Mechanics, 20(1), 83-89.
• [12] Qureshi, S., & Kumar, P. (2019). Using Shehu integral transform to solve fractional order Caputo type initial value problems. Journal of Applied Mathematics and Computational Mechanics, 18(2), 75-83.
• [13] Kumar, P., & Qureshi, S. (2020). Laplace-Carson integral transform for exact solutions of non-integer order initial value problems with Caputo operator. Journal of Applied Mathematics and Computational Mechanics, 19(1), 57-66.
• [14] Shaikh, A.A., & Qureshi, S. (2022). Comparative analysis of Riemann-Liouville, Caputo-Fabrizio, and Atangana-Baleanu integrals. Journal of Applied Mathematics and Computational Mechanics, 21(1), 91-101.
• [15] Qureshi, S., Yusuf, A., & Aziz, S. (2020). On the use of Mohand integral transform for solving fractional-order classical Caputo differential equations. Journal of Applied Mathematics and Computational Mechanics, 19(3), 99-109.
• [16] Richard, M., & Zhao, W. (2021). Padé-Sumudu-Adomian decomposition method for nonlinear Schrödinger equation. Journal of Applied Mathematics, 2021, 1-19.
• [17] Bahgat, M.S.M., & Sebaq, A.M. (2021). An analytical computational algorithm for solving multipantograph DDEs using Laplace variational iteration. Advances in Astronomy, 2021, 1 16.
• [18] Khirsariya, S.R., Rao, S.B., & Chauhan, J.P. (2022). Analytic solution of time-fractional Korteweg-de Vries equation using residual power series method. Results in Nonlinear Analysis, 5(3), 222-234.
• [19] Zhang, J., Wei, Z., Li, L., & Zhou, C. (2019). Least-squares residual power series method for the time-fractional differential equations. Complexity, 13(4), 1-15.
• [20] Khirsariya, S.R., Rao, S.B., & Chauhan, J.P. (2022). A novel hybrid technique to obtain the solution of generalized fractional-order differential equations. Mathematics and Computers in Simulation, 205, 272-290.
• [21] Mishra, H.K., & Pandey, R.K. (2021). The numerical solution of time fractional Kuramoto-Sivashinsky equations via homotopy analysis fractional Sumudu transform method. Journal – MESA, 12(3), 863-882.
• [22] He, J.H. (1999). Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178(3), 257-262.
• [23] He, J.H. (2000). A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-linear Mechanics, 35(1), 37- 43.
• [24] Podlubny, I. (1998). Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, and some of their applications. Mathematical Sciences and Engineering, 198, 340.
• [25] Chakraverty, S., Mahato, N., Karunakar, P., & Rao, T.D. (2019). Advanced numerical and semi-analytical methods for differential equations. John Wiley & Sons, 256.
• [26] Argyros, I.K. (2008). Convergence and applications of Newton-type iterations. Springer Science & Business Media, 506.
• [27] Nieto, J.J. (2022). Solution of a fractional logistic ordinary differential equation. Applied Mathematics Letters, 123, 107-568.
• [28] Gorenflo, R., Kilbas, A.A., Mainardi, F., & Rogosin, S.V. (2020). Mittag-Leffler Functions, Related Topics and Applications. Berlin: Springer, 540.
• [29] Saxena, R.K., Chauhan J.P., Jana R.K., & Shukla A.K. (2015). Further results on the generalized Mittag-Leffler function operator. Journal of Inequalities and Applications, 75(1), 1-12.
• [30] Gupta, P.K., & Singh, M. (2011). Homotopy perturbation method for fractional Fornberg-Whitham equation. Computers and Mathematics with Applications, 61(2), 250-254.

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).