Utilizing the Hybrid approach of the Ramadan group transform and accelerated Adomian method for solving nonlinear integro-differential equations

Document Type : Research Article

Authors

• M.A. Ramadan ¹
• M.M.A. Mansour ²
• H.S. Osheba ²

¹ Mathematics and Computer Science Department, Faculty of Science, Menoufia University, Egypt.

² Department of basic science, Modern Academy of Computer Science and Management Technology in Maadi, Egypt.

10.22067/ijnao.2025.93657.1653

Abstract

In this paper, we investigate the application of the combination of the Ramadan group transform and the accelerated Adomian polynomial method for solving integro-differential equations. Integro-differential equations arise in various fields such as physics, engineering, and biology, often modeling complex phenomena. The Ramadan group transform, known for its transformation properties and its ability to simplify computational complexities, is coupled with the accelerated Adomian polynomial method, which is an effective series expansion technique. This combination enhances the convergence and efficiency of solving nonlinear integro-differential equations that are difficult to handle using traditional methods. The paper demonstrates the utility of this hybrid approach through several test cases, comparing it with existing methods in terms of accuracy, computational efficiency, and convergence rate. 

Keywords

• Ramadan group transform (RGT)
• Adomian polynomials
• accelerated Adomian
• Integro-differential equations
• Accuracy

Main Subjects

• Numerical methods for integral equations and its applications

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References

[1] Adomian, G. Nonlinear stochastic systems theory and applications to physics, Kluwer Academic Publishers, Dordrecht, 1989.

[2] Al-Hayani, W. Combined Laplace transform-homotopy perturbation method for Sine-Gordon equation, Appl. Math. Inf. Sci. 10(5) (2016), 1–6.
[3] Al-Hayani, W. and Younis, M.T. The homotopy perturbation method for solving nonlocal initial-boundary value problems for parabolic and hyperbolic partial differential equations, Eur. J. Pure Appl. Math. 16(3) (2023), 155–156.
[4] Almousa, M., Saidat, S., Al-Hammouri, A., Alsaadi, S. and Banihani, G. Solutions of nonlinear integro-differential equations using a hybrid Adomian decomposition method with modified Bernstein polynomials, Int. J. Fuzzy Log. Intell. Syst. 24(3) (2024) 271–279.
[5] Avudainayagam, A. and Vani, C. Wavelet–Galerkin method for integro-differential equations, Appl. Math. Comput. 32 (2000), 247–254. 
[6] Bahuguna, D., Ujlayan, A. and Pandey, D.N. A comparative study of numerical methods for solving an integro-differential equation, Comput. Math. Appl. 57 (2009), 1485–1493.
[7] Behzadi, S., Abbasbandy, S., Allahviranloo, T. and Yildirim, A. Application of homotopy analysis method for solving a class of nonlinear Volterra-Fredholm integro-differential equations, J. Appl. Anal. Comput. 2(2) (2012), 127–136.
[8] Biazar, J., Babolian, E. and Islam, R. Solution of a system of Volterra integral equations of the first kind by Adomian method, Appl. Math. Comput. 139 (2003), 249–258.
[9] Biazar, J., Ghazvini, H. and Eslami, M. He’s homotopy perturbation method for systems of integro-differential equations, Chaos Solitons Frac-tals 39 (2007), 1253–1258.
[10] Duan, J., Rach, R., Baleanu, D. and Wazwaz, A. A review of the Adomian decomposition method and its applications to fractional differential equations, Commun. Frac. Calc. 3(2) (2012), 73–99.

[11] El-Kalla, I.L. Error analysis of Adomian series solution to a class of nonlinear differential equations, Appl. Math. E-Notes 7 (2007), 214–221.
[12] El-Kalla, I.L. New results on the analytic summation of Adomian series for some classes of differential and integral equations, Appl. Math. Comput. 217 (2010), 3756–3763.
[13] El-Kalla, I.L. Piece-wise continuous solution to a class of nonlinear boundary value problem, Ain Shams Eng. J. 4 (2013), 325–331.
[14] Golberg, A.M. Solution methods for Integral Equations: Theory and Applications, Plenum Publishing Corporation, New York, 1979.
[15] He, J.H. A coupling method of a homotopy technique and a perturbation technique for nonlinear problems, Int. J. Non-Linear Mech. 35(1) (2000), 37–43.
[16] Jerri, A.J. Introduction to integral equations with applications, Marcel Dekker, New York, 1999.
[17] Khanlari, N. and Paripour, M. Solving nonlinear integro-differential equations using the combined homotopy analysis transform method with Adomian polynomials, RGN Publ. 9(4) (2018), 637–650.
[18] Maleknejad, K., Mirzaee, F. and Abbasbandy, S. Solving linear integro-differential equations system by using rationalized Haar functions method, Appl. Math. Comput. 155 (2004), 317–328.
[19] Maleknejad, K. and Tavassoli Kajani, M. Solving linear integro-differential equation system by Galerkin methods with hybrid functions, Appl. Math. Comput. 159 (2004), 603–612.
[20] Ramadan, M.A. The convolution for Ramadan group integral transform: Theory and applications, J. Adv. Trends Basic Appl. Sci. (2017), 191–197.
[21] Ramadan, M.A., Mansour, M.M.A., El-Shazly, N.M. and Osheba, H.S. A blended numerical procedure for quadratic Riccati differential equations utilizing Ramadan group transform and variations of Adomian decomposition, Eng. Appl. Sci. Lett. 7(3) (2024), 11–25.

[22] Ramadan, M.A., Mansour, M.M.A., El-Shazly, N.M. and Osheba, H.S. The double Ramadan group accelerated Adomian decomposition method for solving nonlinear partial differential equations, Comput. Methods Differ. Equ. (2025).
[23] Ramadan, M.A., Mansour, M.M.A. and El-Shazly, N.M. Semi‑analytic solution of the nonlinear Sharma‑Tasso‑Olver equation via Ramadan group integral transform and accelerated Adomian decomposition, J. Umm Al-Qura Univ. Appl. Sci. (2025) 1–14.
[24] Ramadan, M.A. and Mesrega, A.K. Solution of partial and integro-differential equations using the convolution of Ramadan group transform, Asian Res. J. Math. 11(3) (2018), 1–15.
[25] Rani, D. and Mishra, V. Modification of Laplace Adomian decomposition method for solving nonlinear Volterra integral and integro-differential equations based on Newton Raphson Formula, Eur. J. Pure Appl. Math. 11(1) (2018), 202–214.
[26] Sadeghi Goghary, H., Javadi, Sh. and Babolian, E. Restarted Adomian method for system of nonlinear Volterra integral equations, Appl. Math. Comput. 161 (2005), 745–751.
[27] Sayed, A.Y., Sayed, E.A., Rashwan, M.H. and El-Kalla, I.L. Using an accelerated technique of the Laplace Adomian decomposition method in solving a class of nonlinear integro-differential equations, Eng. Res. J. 175 (2022), 371–385.
[28] Sharif, A.A., Hamoud, A.A. and Ghadle, K.P. Solving nonlinear integro-differential equations by using numerical technique, Acta Univ. Apulensis 61 (2019), 45–53.
[29] Wang, S.Q. and He, J.H. Variational iteration method for solving integro-differential equations, Phys. Lett. A 367 (2007), 188–191. 
[30] Younis, M.T. and Al-Hayani, W. Solving fuzzy system of Volterra integro-differential equations by using Adomian decomposition method, Eur. J. Pure Appl. Math. 15(1) (2022), 290–313.

[31] Younis, M.T. and Al-Hayani, W. A numerical study for solving the systems of fuzzy Fredholm integral equations of the second kind using the Adomian decomposition method, Iraqi J. Sci. 64(7) (2023), 4407–4430.
[32] Yusufoglu, E. (Agadjanov) An efficient algorithm for solving integro-differential equations system, Appl. Math. Comput. 192 (2007), 51–55. 
[33] Zhao, J. and Corless, R.M. Compact finite difference method for integro-differential equations, Appl. Math. Comput. 1(6) (2006), 271–288.