The current manuscript makes
use of the prominent iterative procedure, called the Adomian Decomposition Method (ADM), to
tackle some important special differential
equations. The equations of curiosity in this study are the singular equations that arise in many physical
science applications. Thus, through the application of the ADM, a
generalized recursive scheme was successfully derived and further utilized to
obtain closed-form solutions for the models under
consideration. The method is, indeed, fascinating as respective exact analytical
solutions are accurately acquired with only a small number of iterations.
References
[1]
Alkarawi, A.H. and Al-Saiq, I.R. (2020) Applications of Modified Adomian Decomposition Method for Solving the Second-Order Ordinary Differential Equations. Journal of Physics: Conference Series, 1530, Article ID: 012155. https://doi.org/10.1088/1742-6596/1530/1/012155
[2]
Hasan, Y.Q. and Zhu, L.M. (2007) Solving Singular Initial Value Problems in the Second-Order Ordinary Differential Equations. Journal of Applied Sciences, 7, 2505-2508. https://doi.org/10.3923/jas.2007.2505.2508
[3]
Wazwaz, A. (2002) A New Method for Solving Singular Initial Value Problems in the Second-Order Ordinary Differential Equations. Applied Mathematics and Computation, 128, 45-57. https://doi.org/10.1016/S0096-3003(01)00021-2
[4]
Bildik, N. and Deniz, S. (2015) Modified Adomian Decomposition Method for Solving Riccati Differential Equations. Review of the Air Force Academy, 3, 21-26. https://doi.org/10.19062/1842-9238.2015.13.3.3
[5]
Liu, Y.C. (2012) Application of Legendre Polynomials in Adomian Decomposition Method. Proceedings of 2012 International Conference on Computer, Electrical, and Systems Sciences, Amsterdam, 13-14 May 2012, 567-571.
[6]
Hosseini, M.M. (2006) Adomian Decomposition Method with Chebyshev Polynomials. Applied Mathematics and Computation, 175, 1685-1693. https://doi.org/10.1016/j.amc.2005.09.014
[7]
Çenesiz, Y. and Kurnaz, A. (2011) Adomian Decomposition Method by Gegenbauer and Jacobi Polynomials. International Journal of Computer Mathematics, 88, 3666-3676. https://doi.org/10.1080/00207160.2011.611503
[8]
Mahmoudi, Y., Abdollahi, M., Karimian, N. and Khalili, H. (2012) Adomian Decomposition Method with Laguerre Polynomials for Solving Ordinary Differential Equation. Journal of Basic and Applied Scientific Research, 2, 2236-12241.
[9]
Bell, W.W. (2004) Special Functions for Scientists and Engineers. Courier Corporation, Chelmsford.
[10]
Adomian, G. (1990) A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations. Mathematical and Computer Modelling, 13, 17-43. https://doi.org/10.1016/0895-7177(90)90125-7
[11]
Adomian, G. (1994) Solving Frontier Problems of Physics: The Decomposition Method. Fundamental Theories of Physics. Springer, Berlin.
[12]
Wazwaz, A. (2002) A New Algorithm for Calculating Adomian Polynomials for Nonlinear Operators. Applied Mathematics and Computation, 111, 33-51. https://doi.org/10.1016/S0096-3003(99)00063-6
[13]
Wazwaz, A. and El-Sayed, S.M. (2001) A New Modification of the Adomian Decomposition Method for Linear and Nonlinear Operators. Applied Mathematics and Computation, 122, 393-405. https://doi.org/10.1016/S0096-3003(00)00060-6
[14]
Wazwaz, A. (1999) A Reliable Modification of Adomian Decomposition Method. Applied Mathematics and Computation, 102, 77-86. https://doi.org/10.1016/S0096-3003(98)10024-3
[15]
Liu, Y. (2009) Adomian Decomposition Method with Second Kind Chebyshev Polynomials. Proceedings of the Jangjeon Mathematical Society, 12, 57-67.
[16]
Tomaizeh, H. (2017) Modified Adomian Decomposition Method for Differential Equations. Hebron University, Hebron.
[17]
Hosseini, M.M. and Nasabzadeh, H. (2007) Modified Adomian Decomposition Method for Specific Second Order Ordinary Differential Equations. Applied Mathematics and Computation, 186, 17-123. https://doi.org/10.1016/j.amc.2006.07.094
[18]
Nuruddeen, R.I., Muhammad, L., Nass, A.M. and Sulaiman, T.A. (2018) A Review of the Integral Transforms-Based Decomposition Methods and Their Applications in Solving Nonlinear PDEs. Palestine Journal of Mathematics, 7, 262-280.
[19]
Bakodah, H.O., Banaja, M.A., Alrigi, B.A., Ebaid, A. and Rach, R. (2019) An Efficient Modification of the Decomposition Method with a Convergence Parameter for Solving Korteweg de Vries Equations. Journal of King Saud University-Science, 31, 1424-1430. https://doi.org/10.1016/j.jksus.2018.11.010
[20]
Rach, R., Baghdasarian, A. and Adomian, G. (1992) Differential Equations with Singular Coefficients. Applied Mathematics and Computation, 47, 179-184. https://doi.org/10.1016/0096-3003(92)90045-3
[21]
Bougoffa, L. (2009) On the Exact Solutions for Initial Value Problems of Second-Order Differential Equations. Applied Mathematics Letters, 22, 1248-1251. https://doi.org/10.1016/j.aml.2009.01.038
[22]
Dita, P. and Grama, N. (1997) On Adomian’s Decomposition Method for Solving Differential Equations. ArXiv: Solv-int/9705008.
[23]
Suresh, P.L., Krishna, G.V., Maheswari, K.U. and Ramanaiah, J.V. (2017) Solutions of Gauss’s Hypergeometric Equation, Laguerre’s Equation by Differential Transform Method. International Journal of Current Engineering and Scientific Research, 4, 32-38.
[24]
Mubeen, S., Naz, M., Rehman, A. and Rahman, G. (2014) Solutions of κ-Hypergeometric Differential Equations. Journal of Applied Mathematics, 2014, Article ID: 128787. https://doi.org/10.1155/2014/128787
[25]
Al-Mazmumy, M. and Alsulami, A.A. (2023) Solution of Laguerre’s Differential Equations via Modified Adomian Decomposition Method. Journal of Applied Mathematics and Physics, 11, 85-100. https://doi.org/10.4236/jamp.2023.111007
[26]
Al Mazmumy, M., Alsulami, A., Bakodah, H. and Alzaid, N. (2023) Adomian Modification Methods for the Solution of Chebyshev’s Differential Equations. Applied Mathematics, 14, 512-530.
