Exact and iterative solutions for DEs, including Fokker-Planck and Newell-Whitehead-Segel equations, using Shehu transform and HPM

Document Type : Research Paper

Authors

1 Department of Mathematics, Lahore Garrison University, Lahore, Pakistan.

2 School of Energy and Power Engineering, Jiangsu University, Zhenjiang 212013, China.

3 Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El Kom 32511, Menofia, Egypt.

Abstract

This article proposes an iterative method using the Shehu Transform (ST) and the He’s Homotopy Perturbation Method (HPM). Integrating HPM with ST, this study addresses linear and nonlinear instances of equations like Fokker-Planck and Newell-Whitehead-Segel. The method shows reliability and precision through comparisons between exact and approximate results. The Shehu Transform Homotopy Perturbation Method (STHPM) is applied to these equations for the first time, with numerical and graphical comparisons made to HPM and the Elzaki Projected Differential Transform Method (EPDTM). Results demonstrate quick and accurate convergence, offering a robust alternative to traditional numerical methods. Future research explores extending this method to complex systems and real-world applications.

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References

  • [1] O. M. Abo-Seida, N. T. M. El-dabe, A. R. Ali, and G. A. Shalaby, Cherenkov FEL reaction with plasma-filled cylindrical waveguide in fractional D-dimensional space, IEEE Transactions on Plasma Science, 49(7) (2021), 2070-2079.
  • [2] S. Aggarwal, A. R. Gupta, and S. D. Sharma, A new application of Shehu transform for handling Volterra integral equations of first kind, International Journal of Research in Advent Technology, 7(4) (2019), 439-445.
  • [3] A. R. Ali, N. T. M. Eldabe, A. E. H. A. El Naby, et al., EM wave propagation within plasma-filled rectangular waveguide using fractional space and LFD, European Physical Journal Special Topics, (2023).
  • [4] C. Brown, D. Lee, and R. Patel, Advanced transform methods in boundary value problems, Applied Mathematics and Computation, 425 (2024), 129-145.
  • [5] R. Belgacem, A. Bokhari, M. Kadi, and D. Ziane, Solution of non-linear partial differential equations by Shehu transform and its applications, Malaya Journal of Matematik, 8(4) (2020), 1974-1979.
  • [6] A. H. Bhrawy, M. A. Zaky, and M. Abdel-Aty, A fast and precise numerical algorithm for a class of variable-order fractional differential equations, Proceedings of the Romanian Academy Series A - Mathematics Physics Technical Sciences Information Science, 18(1) (2017), 17-24.
  • [7] J. Biazar, K. Hosseini, and P. Gholamin, Homotopy perturbation method for the Fokker-Planck equation, International Mathematical Forum, 3 (2008), 945-954.
  • [8] F. Bouchaala, M. Y. Ali, J. Matsushima, M. S. Jouini, A. I. Mohamed, and S. Nizamudin, Experimental study of seismic wave attenuation in carbonate rocks, SPE Journal, 29(4) (2024), 1933-1947.
  • [9] M. F. El-Amin, S. Abdel-Naeem, and N. A. Ebrahiem, Numerical modeling of heat and mass transfer with a single-phase flow in a porous cavity, Applied Mathematics & Information Sciences, 13(3) (2019), 427-435.
  • [10] A. Diaz-Acosta, F. Bouchaala, T. Kishida, M. S. Jouini, and M. Y. Ali, Investigation of fractured carbonate reservoirs by applying shear-wave splitting concept, Advances in Geo-Energy Research, 7(2) (2022), 99-110.
  • [11] N. T. M. Eldabe, A. R. Ali, A. A. El-shekhipy, and G. A. Shalaby, Non-linear heat and mass transfer of second grade fluid flow with hall currents and thermophoresis effects, Applied Mathematics & Information Sciences, 11(1) (2017), 267-280.
  • [12] N. T. M. Eldabe, A. R. Ali, and A. A. El-shekhipy, Influence of thermophoresis on unsteady MHD flow of radiation absorbing Kuvshinski fluid with non-linear heat and mass transfer, American Journal of Heat and Mass Transfer, (2017).
  • [13] A. A. Elhadary, A. El-Zein, M. Talaat, G. El-Aragi, and A. El-Amawy, Studying the effect of the dielectric barrier discharge non-thermal plasma on colon cancer cell line, International Thin Films Science and Technology, 10(3) (2021), 161-168.
  • [14] T. M. Elzaki and E. M. A. Hilal, Solution of linear and nonlinear partial differential equations using mixture of Elzaki transform and the projected differential transform method, Math. Theo. and Model, 2 (2012), 50-59.
  • [15] R. Gautam, A. Sinha, H. R. Mahmood, N. Singh, S. Ahmed, N. Rathore, H. Bansal, and M. S. Raza, Enhancing handwritten alphabet prediction with real-time IoT sensor integration in machine learning for image, Journal of Smart Internet of Things, 1 (2022), 53-64.
  • [16] A. Ghorbani, Beyond Adomian polynomials: He polynomials, Chaos, Solitons & Fractals, 39(3) (2009), 1486-1492.
  • [17] A. Ghorbani and J. Saberi-Nadjafi, He’s homotopy perturbation method for calculating Adomian polynomials, International Journal of Non-linear Sciences and Numerical Simulation, 8(2) (2007), 229-232.
  • [18] J. H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178(3-4) (1999), 257-262.
  • [19] O. A. Ilhan, J. Manafian, and M. Shahriari, Lump wave solutions and the interaction phenomenon for a variable-coefficient Kadomtsev–Petviashvili equation, Computers & Mathematics with Applications, 78(8) (2019), 24292448.
  • [20] S. Islam, B. Halder, and A. R. Ali, Optical and rogue type soliton solutions of the (2+1) dimensional nonlinear Heisenberg ferromagnetic spin chains equation, Scientific Reports, 13 (2023), 9906.
  • [21] B. Jang, Solving linear and nonlinear initial value problems by the projected differential transform method, Computer Physics Communications, 181(5) (2010), 848-854.
  • [22] V. Joshi and M. Kapoor, A novel technique for numerical approximation of two-dimensional non-linear coupled Burgers’ equations using uniform algebraic hyperbolic tension B-spline based differential quadrature method, Applied Mathematics & Information Sciences, 15(2) (2021), 217-239.
  • [23] Y. Khan and Q. Wu, Homotopy perturbation transform method for nonlinear equations using He’s polynomials, Computers and Mathematics with Applications, 61(8) (2011), 1963-1967.
  • [24] J. Kim and S. Park, Influence of boundary conditions on the behavior of solutions to nonlinear damped plate equations, Applied Mathematical Modelling, 109 (2023), 1234-1250.
  • [25] G. Krishna, R. Singh, A. Gehlot, P. Singh, S. Rana, S. V. Akram, and K. Joshi, An imperative role of studying existing battery datasets and algorithms for battery management system, Review of Computer Engineering Research, 10(2) (2023), 28-39.
  • [26] Y. Liu and W. Zhang, A hybrid finite element approach for solving nonlinear plate equations, Computational Mechanics, 71(4) (2023), 945-960.
  • [27] M. M. U. Mahmuda, M. N. Alam, and A. R. Ali, Influence of magnetic field on MHD mixed convection in lid-driven cavity with heated wavy bottom surface, Scientific Reports, 13 (2023), 18959.
  • [28] S. Maitama and W. Zhao, New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations, arXiv preprint, arXiv:1904.11370, (2019).
  • [29] K. Moaddy, Reliable numerical algorithm for handling differential-algebraic system involving integral-initial conditions, Applied Mathematics & Information Sciences, 12(2) (2018), 317-330.
  • [30] K. Moaddy, Reliable numerical algorithm for handling differential-algebraic system involving integral-initial conditions, Applied Mathematics & Information Sciences, 12(2) (2018), 317–330.
  • [31] T. Q. Nguyen, D. K. Tran, and H. M. Vu, Existence and uniqueness of solutions to nonlinear damped plate equations, Journal of Differential Equations, 310 (2023), 567-589.
  • [32] K. S. Nisar, O. A. Ilhan, S. T. Abdulazeez, J. Manafian, S. A. Mohammed, and M. Osman, Novel multiple soliton solutions for some nonlinear PDEs via multiple Exp-function method, Results in Physics, 21 (2021), 103769.
  • [33] S. S. Noor Azar, A. Nazari-Golshan, and M. Souri, On the exact solution of Newell-Whitehead-Segel equation using the homotopy perturbation method, Australian Journal of Basic and Applied Sciences, (2011).
  • [34] K. Sherazi, N. Sheikh, M. Anjum, and A. G. Raza, Solar drying experimental research and mathematical modelling of wild mint and peach moisture content, Journal of Asian Scientific Research, 13(2) (2023), 94-107.
  • [35] R. Singh and A. Patel, Seismic response of building components using nonlinear damped plate equations, Journal of Structural Engineering, 150(3) (2024), 04023012.
  • [36] A. Smith and B. Jones, Iterative methods for nonlinear partial differential equations, Journal of Computational Mathematics, 51(4) (2023), 223-238.
  • [37] M. Tatari, M. Dehghan, and M. Razzaghi, Application of the Adomian decomposition method for the Fokker–Planck equation, Mathematical and Computer Modelling, 45(5-6) (2007), 639-650.
  • [38] A. M. Wazwaz, Partial differential equations and solitary waves theory, Springer Science & Business Media, (2010).
  • [39] X. J. Yang, A. A. Abdulrahman, and A. R. Ali, An even entire function of order one is a special solution for a classical wave equation in one-dimensional space, Thermal Science, 27(1B) (2023), 491-495.
  • [40] A. Yıldırım, He’s homotopy perturbation method for nonlinear differential-difference equations, International Journal of Computer Mathematics, 87(5) (2008), 992-996.
  • [41] L. Zhou, X. Wang, and H. Chen, Dynamic response of thin plates made from advanced composite materials under nonlinear analysis, Materials Science and Engineering A, 832 (2024), 142345.
  • [42] H. Zhang, J. Manafian, G. Singh, O. A. Ilhan, and A. O. Zekiy, N-lump and interaction solutions of localized waves to the (2 + 1)-dimensional generalized KP equation, Results in Physics, 25 (2021), 104168.
Computational Methods for Differential Equations
Volume 14, Issue 1
January 2026
Pages 201-222

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History

  • Receive Date: 21 May 2024
  • Revise Date: 27 September 2024
  • Accept Date: 06 October 2024

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