Semi-Analytical Method for Solving Coupled Nonlinear Partial Differential Equations Using Hybrid Iterative Method
Keywords:
Nonlinear Partial Differential Equations , Sumudu Integral Transform, Shehu Integral Transform, Variational Iteration MethodAbstract
This article aims to propose a new efficient hybrid method for solving different types of nonlinear differential equations. The method combines mixed Shehu and Sumudu integral transforms with the variational iteration method. The proposed method is termed the multiple transform iteration method to solve different nonlinear partial differential equations, by engaging a new time and frequency domain. The outcomes that arise from this method show the efficiency, accuracy, and simplicity of applying the approach.
References
Abdou, M.A.; Soliman, A.A.; "Variational iteration method for solving Burgers' and coupled Burgers' equations". J. Comput. Appl. Math, 181: 245-251, 2009.
Adomian, G.; "Solving Frontier Problems of Physics, The decomposition method". Int. Comput. Cent. Bull, 60: 354, 1994.
Ahmed, H.; Khan T.A.; Cesarano, C.; "Numerical solutions of coupled Burger's equation". Axioms, 8, 2019.
Al-Fayadh, A.; Hasan, A. K.; "Variational iteration transform method for solving Burger and coupled Burger's equations". ARPN J. Eng. Appl. Sci, 12(23): 6926-6932, 2017.
Al-Tai, M. H.; Al-Fayadh, A.; "Solving two dimensional coupled Burger's equations and Sine Gordon equation using El-Zaki transform-variational iteration method". Al-Nahrain J. Sci, 24(2): 41-47, 2021.
Arun, K.; Pankaj, R. D; "Solitary wave solutions of Schrödinger equation by Laplace-Adomian decomposition method". Phys. Rev. Res. Int, 3(4): 702-712, 2013.
Belgacem, F.BM.; Karballi, A. A.; "Sumudu transform fundamental properties investigations and applications". J. Appl. Math. Stoch. Anal, 91083: 1-23, 2005.
Bermudez, R.F.; "Coupled system of nonlinear Schrӧdinger and Kortweg-de Vries equations". Univ. Carlos III de Madrid, Depart. Math, Leganes, 2016.
Emad, K. J.; Omar, A.; Mohammad, A.; Ala’a, M. Al-Faqih; "An approximate analytical solution of the nonlinear Schrödinger equation with harmonic oscillator using homotopy perturbation method and Laplace-Adomian decomposition method". Adv. Math. Phys, 6765021, 11(5): 2018.
He, JH.; "Homotopy perturbation method: a new nonlinear analytical technique". App. Math. Comput, 135: 73–79, 2003.
He, JH.; "Variational iteration method—a kind of nonlinear analytical technique: some examples". Int. J. Nonlinear Mech, 34 (4): 699–708, 1999.
Heris, J.M.; Bagheri, M.; "Exact solutions for the modified KdV and the generalized KdV equations via Exp-function method". J. Math. Ext, 4 (2): 77–98, 2010.
Huda, J. K.; Al-Fayadh, A.; "An efficient variational homotopy transform method for solving Schrödinger equation". J. Phys. Conf. Ser, 2322012044, 2022.
Huda, J. K.; Al-Fayadh, A.; "Solving Schrödinger equations using Kashuri and Fundo transform decomposition method". Al-Nahrain J. Sci, 25 (2): 25-28, 2022.
Hussain, A. K.; Fadhel, F. S.; Yahya, Z. R.; Rusli, N.; "Variational Iteration Method (VIM) for Solving Partial integro-differential Equations", J. Theo. Appl. Info. Tech, 88(2): 2016.
Inokuti, M.; Sekine, H.; Mura, T.; "General use of the Lagrange multiplier in nonlinear mathematical physics". Pergamon Press, Oxford,156-162: 1978.
Jaradat, E.K.; Alomari, O.; Abudayah, M.; Al-Faqih, A.A.M.; "An approximate analytical solution of the nonlinear Schrödinger equation with harmonic oscillator using homotopy perturbation method and Laplace-Adomian decomposition method". Adv. Math. Phys, (10) 1155, 2018.
Kashuri, A.; Fundo A.; Kreku, M.; "Mixture of a new integral transform and homotopy perturbation method for solving nonlinear partial differential equations". Adv. Pure Math, 3: 317-323, 2013.
Maitama, S.; Zhao, W.; "New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations". Int. J. Anal. Appl, 17(2): 167-190, 2019.
Marwa H. Al-Tai ; Al-Fayadh, A.; "Solving two dimensional coupled Burger's equations using transform variational iteration method". AIP Conf. Proc, 2414, 040030, 2023.
Mohanned, F. K.; Al-Fayadh, A.; "Mixed Homotopy integral transform method for solving non-linear integro-differential equation". Al-Nahrain J. Sci, 25(1): 35-40, 2022.
Mohanned, F. K.; Al-Fayadh, A.; "Solving Fredholm integro-differential equation of fractional order by using Sawi homotopy perturbation method". J. Phys. Conf. Ser, 2322012056, 2022.
Mustafa, A.A.; Al-Hayani, W.; "Solving the coupled Schrӧdinger-Kortweg-de Veries system by modified variational iteration method with genetic algorithm". Wasit J. Comput. Math. Sci, 2 (2): 97-108, 2023.
Mustafa, H.; Al-Fayadh, A.; "Efficient Iterative Transform Method for Solving the Fokker-Planck Equation". AIP Conf. Proc, 3036, 040027, 2024.
Nadeem, M.; Fengquan, Li; "Modified Laplace variational iteration method for analytical approach of Klein–Gordon and Sine-Gordon equations". Iran. J. Sci. Technol, 43(4):1933-1940, 2019.
Nuruddeen, R. I., "Elzaki decomposition method and its applications in solving linear and nonlinear Schrödinger equations," Sohag J. Math, 4 (2): 1-5, 2017.
Ruaa, S. I.; Al-Fayadh, A.; "Mixed transform iterative method for solving wave-like equations". Al-Nahrain J. Sci., 26 (2): 40-46, 2023.
Sahib, A. A. A.; Fadhel, F. S.; "Solution of fractional order random integro-differential equations using Homotopy Perturbation Method", AIP Conf. Proc, 2457: 1, 2023.
Singh, G.; Singh, I.; "New Laplace variational iterative method for solving 3D Schrödinger equations". J. Math. Comput. Sci, 10(5): 2015-2024, 2020.
Soliman, A. A.; Raslan, K. R.; Abdallah, A. M.; "On some modified methods on fractional delay and nonlinear integro-differential equation". Sound Vib, 55(4): 263-279, 2021.
Soltanalizadeh, B.; "Application of differential transformation method for solving fourth-order parabolic partial differential equations". Int. J. Pure Appl. Math, 78(3): 299–308, 2012.
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Nisreen S. Hasan, Ali Al-Fayadh
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg)
