Efficient Modified Adomian Decomposition Method for Solving Nonlinear Fractional Differential Equations

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Mariam Al-Mazmumy, Maryam Ahmed Alyami, Mona Alsulami, Asrar Saleh Alsulami


Based on the application of the standard Adomian method, the current paper proposes two modification approaches for the classes of the fractional differential equations and the system of fractional differential equations, which are featured through initial-value problems. Certainly, the constructed iterative schemes for the two classes are shown to be reliable, considering a number of test problems for demonstration, and upon deploying other existing numerical approaches for contrasting. In fact, the proposed schemes are found to portray less error, rapidity, accuracy and consume less computational time among others.

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  1. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993
  2. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
  3. J.H. He, Fractal Calculus and Its Geometrical Explanation, Results Phys. 10 (2018), 272–276. https://doi.org/10.1016/j.rinp.2018.06.011.
  4. Q. Wang, X. Shi, J.H. He, Z.B. Li, Fractal Calculus and Its Application to Explanation of Biomechanism of Polar Bear Hairs, Fractals. 26 (2018), 1850086. https://doi.org/10.1142/s0218348x1850086x.
  5. S.R. Saratha, M. Bagyalakshmi, G. Sai Sundara Krishnan, Fractional generalised homotopy analysis method for solving nonlinear fractional differential equations, Comput. Appl. Math. 39 (2020), 112. https://doi.org/10.1007/s40314-020-1133-9.
  6. S. Javeed, D. Baleanu, A. Waheed, M. Shaukat Khan, H. Affan, Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations, Mathematics 7 (2019), 40. https://doi.org/10.3390/math7010040.
  7. B.R. Sontakke, A.S. Shelke, A.S. Shaikh, Solution of Non-Linear Fractional Differential Equations by Variational Iteration Method and Applications, Far East J. Math. Sci. 110 (2019), 113–129. https://doi.org/10.17654/ms110010113.
  8. S. D. Oloniiju, S.P. Goqo, P. Sibanda, A Chebyshev Pseudo-Spectral Method for the Numerical Solutions of Distributed Order Fractional Ordinary Differential Equations, Appl. Math. E-Notes, 22 (2022), 132–141.
  9. S.S. Ezz-Eldien, E.H. Doha, D. Baleanu, A.H. Bhrawy, A Numerical Approach Based on Legendre Orthonormal Polynomials for Numerical Solutions of Fractional Optimal Control Problems, J. Vibration Control, 23 (2017), 16–30. https://doi.org/10.1177/1077546315573916.
  10. K. Parmikant, E. Rusyaman, Grundwald-Letnikov Operator and Its Role in Solving Fractional Differential Equations, EKSAKTA: Berkala Ilmiah Bidang MIPA, 23 (2022), 223–230. https://doi.org/10.24036/eksakta/vol23-iss03/331.
  11. I. Ameen, P. Novati, The Solution of Fractional Order Epidemic Model by Implicit Adams Methods, Appl. Math. Model. 43 (2017), 78–84. https://doi.org/10.1016/j.apm.2016.10.054.
  12. R. Garrappa, Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial, Mathematics. 6 (2018), 16. https://doi.org/10.3390/math6020016.
  13. R. Amin, B. Alshahrani, M. Mahmoud, A.-H. Abdel-Aty, K. Shah, W. Deebani, Haar Wavelet Method for Solution of Distributed Order Time-Fractional Differential Equations, Alexandria Eng. J. 60 (2021), 3295–3303. https://doi.org/10.1016/j.aej.2021.01.039.
  14. G. Adomian, Nonlinear Stochastic System Theory and Applications to Physics, Kluwer Academic Publishers, Dordrecht, 1989.
  15. K. Abbaoui, Y. Cherruault, Convergence of Adomian’s Method Applied to Differential Equations, Computers Math. Appl. 28 (1994), 103–109. https://doi.org/10.1016/0898-1221(94)00144-8.
  16. J.S. Duan, R. Rach, D. Baleanu, A.M.Wazwaz, A Review of the Adomian DecompositionMethod and Its Application to Fractional Differential Equations, Commun. Frac. Calc. 3 (2012), 73–99.
  17. A. Sadeghinia, P. Kumar, One Solution of Multi-Term Fractional Differential Equations by Adomian Decomposition Method, Int. J. Sci. Innov. Math. Res. 3 (2015), 14–21.
  18. P. Guo, The Adomian Decomposition Method for a Type of Fractional Differential Equations, J. Appl. Math. Phys. 07 (2019), 2459–2466. https://doi.org/10.4236/jamp.2019.710166.
  19. A. Afreen, A. Raheem, Study of a Nonlinear System of Fractional Differential Equations with Deviated Arguments Via Adomian Decomposition Method, Int. J. Appl. Comput. Math. 8 (2022), 269. https://doi.org/10.1007/s40819-022-01464-5.
  20. H. Thabet, S. Kendre, New Modification of Adomian Decomposition Method for Solving a System of Nonlinear Fractional Partial Differential Equations, Int. J. Adv. Appl. Math. Mech. 6 (2019), 1–13.
  21. M. Botros, E. Ziada, I. El-Kalla, Solutions of Fractional Differential Equations With Some Modifications of Adomian Decomposition Method, Delta Univ. Sci. J. 6 (2023), 292–299. https://doi.org/10.21608/dusj.2023.291073.
  22. J. Mulenga, P.A. Phiri, Solving Different Types of Differential Equations Using Modified and New Modified Adomian Decomposition Methods, J. Appl. Math. Phys. 11 (2023), 1656–1676. https://doi.org/10.4236/jamp.2023.116108.
  23. H.O. Bakodah, M. Al-Mazmumy, S.O. Almuhalbedi, An Efficient Modification of the Adomian decomposition Method for Solving Integro-Differential Equations, Math. Sci. Lett. 6 (2017), 15–21. https://doi.org/10.18576/msl/060103.
  24. J. Biazar, K. Hosseini, A Modified Adomian Decomposition Method for Singular Initial Value Emden-Fowler Type Equations, Int. J. Appl. Math. Res. 5 (2016), 69–72. https://doi.org/10.14419/ijamr.v5i1.5666.
  25. A. Abdullah Alderremy, T.M. Elzaki, M. Chamekh, Modified Adomian Decomposition Method to Solve Generalized Emden–Fowler Systems for Singular IVP, Math. Probl. Eng. 2019 (2019), 6097095. https://doi.org/10.1155/2019/6097095.
  26. S.C. Shiralashetti, A.B. Deshi, An Efficient Haar Wavelet Collocation Method for the Numerical Solution of Multi-Term Fractional Differential Equations, Nonlinear Dyn. 83 (2015), 293–303. https://doi.org/10.1007/s11071-015-2326-4.
  27. N.J. Ford, J.A. Connolly, Systems-Based Decomposition Schemes for the Approximate Solution of Multi-Term Fractional Differential Equations, J. Comput. Appl. Math. 229 (2009), 382–391. https://doi.org/10.1016/j.cam.2008.04.003.
  28. A.M.A. El-Sayed, I.L. El-Kalla, E.A.A. Ziada, Analytical and Numerical Solutions of Multi-Term Nonlinear Fractional Orders Differential Equations, Appl. Numer. Math. 60 (2010), 788–797. https://doi.org/10.1016/j.apnum.2010.02.007.
  29. Y. Yang, Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method, Appl. Math. 04 (2013), 113–118. https://doi.org/10.4236/am.2013.41020.
  30. V. Daftardar-Gejji, H. Jafari, Solving a Multi-Order Fractional Differential Equation Using Adomian Decomposition, Appl. Math. Comput. 189 (2007), 541–548. https://doi.org/10.1016/j.amc.2006.11.129.