The
immense quest for proficient numerical schemes for the solution of mathematical
models featuring nonlinear differential equations led to the realization of the
Adomian decomposition method (ADM) in the 80th. Undoubtedly, the
solution of nonlinear differential equations using ADM is presided over by the
acquisition of Adomian polynomials, which are not always easy to find. Thus,
the present study proposes easy-to-implement Maple programs for the computation of Adomian polynomials. In fact,
the proposed algorithms performed remarkably on several test functions,
consisting of one- and multi-variable nonlinearities. Moreover, the introduced
programs are advantageous in terms of simplicity; coupled with the requirement
of less computational time in comparison with what is known in the literature.
Adomian, G. (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic, Dordrecht. https://doi.org/10.1007/978-94-015-8289-6
Wazwaz, A.M. (2011) Linear and Nonlinear Integral Equations: Methods and Applications. Higher Education Press, Beijing.
[5]
Wazwaz, A.M. (2000) The Modified Adomian Decomposition Method for Solving Linear and Nonlinear Boundary Value Problems of Tenth-Order and Twelfth-Order. International Journal of Nonlinear Sciences and Numerical Simulation, 1, 17-24. https://doi.org/10.1515/IJNSNS.2000.1.1.17
[6]
Ebaid, A. (2011) A New Analytical and Numerical Treatment for Singular Two-Point Boundary Value Problems via the Adomian Decomposition Method. Journal of Computational and Applied Mathematics, 235, 1914-1924. https://doi.org/10.1016/j.cam.2010.09.007
[7]
Bakodah, H.O., Al-Zaid, N.A., Al-Mazmumy, M. and Biswas, A. (2017) An Efficient Numerical Algorithm for Solving Dirichlet Initial-Boundary Value Problems. Mathematics in Engineering, Science and Aerospace, 8, 489-510.
[8]
Bakodah, H.O., Al-Mazmumy, M. and Almuhalbedi, S.O. (2019) Solving System of Integro Differential Equations Using Discrete Adomian Decomposition Method. Journal of Taibah University for Science, 13, 805-812. https://doi.org/10.1080/16583655.2019.1625189
[9]
Alzhrani, M.A., Bakodah, H.O. and Al-Mazmumy, M. (2019) A 3/8 Simpson’s Numerical Scheme for the Classes of Volterra Integral Equations of First Kind. Nonlinear Analysis and Differential Equations, 7, 99-113. https://doi.org/10.12988/nade.2019.9812
[10]
Al-Mazmumy, M., Alsulami, A., Bakodah, H.O. and Alzaid, N. (2022) Modified Adomian Method for the Generalized Inhomogeneous Lane-Emden-Type Equations. Nonlinear Analysis and Differential Equations, 10, 15-35. https://doi.org/10.12988/nade.2022.91142
[11]
AL-Zaid, N., AL-Refaidi, A., Bakodah, H.O. and Al-Mazmumy, M. (2022) Solution of Second- and Higher-Order Nonlinear Two-Point Boundary-Value Problems Using Double Decomposition Method. Mathematics, 10, Article 3519. https://doi.org/10.3390/math10193519
[12]
Adomian, G. (1981) On Product Nonlinearities in Stochastic Differential Equations. Applied Mathematics and Computation, 8, 35-49. https://doi.org/10.1016/0096-3003(81)90035-7
[13]
Adomian, G. and Rach, R. (1985) Nonlinear Plasma Response. Journal of Mathematical Analysis and Applications, 111, 114-118. https://doi.org/10.1016/0022-247X(85)90204-5
[14]
Adomian, G. and Rach, R. (1986) On Composite Nonlinearities and the Decomposition Method. Journal of Mathematical Analysis and Applications, 113, 504-509. https://doi.org/10.1016/0022-247X(86)90321-5
[15]
Adomian, G. and Rach, R. (1985) Polynomial Nonlinearities in Differential Equations. Journal of Mathematical Analysis and Applications, 109, 90-95. https://doi.org/10.1016/0022-247X(85)90178-7
[16]
Adomian, G. and Rach, R. (1983) Anharmonic Oscillator Systems. Journal of Mathematical Analysis and Applications, 91, 229-236. https://doi.org/10.1016/0022-247X(83)90101-4
[17]
Adomian, G. and Rach, R. (1985) Nonlinear Differential Equations with Negative Power Nonlinearities. Journal of Mathematical Analysis and Applications, 112, 497-501. https://doi.org/10.1016/0022-247X(85)90258-6
[18]
Adomian, G., Rach, R. and Sarafyan, D. (1985) On the Solution of Equations Containing Radicals by the Decomposition Method. Journal of Mathematical Analysis and Applications, 111, 423-426. https://doi.org/10.1016/0022-247X(85)90226-4
[19]
Adomian, G. and Rach, R. (1984) Light Scattering in Crystals. Journal of Applied Physics, 56, 2592-2594. https://doi.org/10.1063/1.334291
[20]
Adomian, G. and Rach, R. (1986) Solving Nonlinear Differential Equations with Decimal Power Nonlinearities. Journal of Mathematical Analysis and Applications, 114, 423-425. https://doi.org/10.1016/0022-247X(86)90094-6
[21]
Cherruault, Y. (1990) Convergence of Adomian’s Method. Mathematical and Computer Modelling, 14, 83-86. https://doi.org/10.1016/0895-7177(90)90152-D
[22]
Abbaoui, K. and Cherruault, Y. (1995) New Ideas for Proving Convergence of Decomposition Methods. Computers & Mathematics with Applications, 29, 103-108. https://doi.org/10.1016/0898-1221(95)00022-Q
[23]
Hosseini, M.M. and Nasabzadeh, H. (2006) On the Convergence of Adomian Decomposition Method. Applied Mathematics and Computation, 182, 536-543. https://doi.org/10.1016/j.amc.2006.04.015
[24]
Adomian, G. and Rach, R. (1992) Generalization of Adomian Polynomials to Functions of Several Variables. Computers & Mathematics with Applications, 24, 11-24. https://doi.org/10.1016/0898-1221(92)90037-I
[25]
Wazwaz, A.M. (2000) 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
[26]
Abdelwahid, F. (2003) A Mathematical Model of Adomian Polynomials. Applied Mathematics and Computation, 141, 447-453. https://doi.org/10.1016/S0096-3003(02)00266-7
[27]
Rach, R. (2008) A New Definition of the Adomian Polynomials. Kybernetes, 37, 910-955. https://doi.org/10.1108/03684920810884342
[28]
Biazar, J., Babolian, E., Kember, G., Nouri, A. and Islam, R. (2003) An Alternate Algorithm for Computing Adomian Polynomials in Special Cases. Applied Mathematics and Computation, 138, 523-529. https://doi.org/10.1016/S0096-3003(02)00174-1
[29]
Abbaoui, K., Cherruault, Y. and Seng, V. (1995) Practical Formulae for the Calculus of Multivariable Adomian Polynomials. Mathematical and Computer Modelling, 22, 89-93. https://doi.org/10.1016/0895-7177(95)00103-9
[30]
Zhu, Y., Chang, Q. and Wu, S. (2005) A New Algorithm for Calculating Adomian Polynomials. Applied Mathematics and Computation, 169, 402-416. https://doi.org/10.1016/j.amc.2004.09.082
[31]
Biazar, J., Ilie, M. and Khoshkenar, A. (2006) An Improvement to an Alternate Algorithm for Computing Adomian Polynomials in Special Cases. Applied Mathematics and Computation, 173, 582-592. https://doi.org/10.1016/j.amc.2005.04.052
[32]
Azreg-Aïnou, M. (2009) A Developed New Algorithm for Evaluating Adomian Polynomials. CMES-Computer Modeling in Engineering & Sciences, 42, 1-18.
[33]
Duan, J.S. (2010) Recurrence Triangle for Adomian Polynomials. Applied Mathematics and Computation, 216, 1235-1241. https://doi.org/10.1016/j.amc.2010.02.015
[34]
Duan, J.S. (2010) An Efficient Algorithm for the Multivariable Adomian Polynomials. Applied Mathematics and Computation, 217, 2456-2467. https://doi.org/10.1016/j.amc.2010.07.046
[35]
Duan, J.S. (2011) Convenient Analytic Recurrence Algorithms for the Adomian Polynomials. Applied Mathematics and Computation, 217, 6337-6348. https://doi.org/10.1016/j.amc.2011.01.007
[36]
Choi, H.W. and Shin, J.G. (2003) Symbolic Implementation of the Algorithm for Calculating Adomian Polynomials. Applied Mathematics and Computation, 146, 257-271. https://doi.org/10.1016/S0096-3003(02)00541-6
[37]
Chen, W. and Lu, Z. (2004) An Algorithm for Adomian Decomposition Method. Applied Mathematics and Computation, 159, 221-235. https://doi.org/10.1016/j.amc.2003.10.037
[38]
Pourdarvish, A. (2006) A Reliable Symbolic Implementation of Algorithm for Calculating Adomian Polynomials. Applied Mathematics and Computation, 172, 545-550. https://doi.org/10.1016/j.amc.2005.02.040
[39]
Duan, J.S. and Guo, A.P. (2010) Reduced Polynomials and Their Generation in Adomian Decomposition Methods. CMES-Computer Modeling in Engineering & Sciences, 60, 139-150.
[40]
Duan, J.S. (2010) An Efficient Algorithm for the Multivariable Adomian Polynomials. Applied Mathematics and Computation, 217, 2456-2467. https://doi.org/10.1016/j.amc.2010.07.046
[41]
Hendi, F.A., Bakodah, H.O., Almazmumy, M. and Alzumi, H. (2012) A Simple Program for Solving Nonlinear Initial Value Problem Using Adomian Decomposition Method. International Journal of Research and Reviews in Applied Sciences, 12, 397-406.
[42]
Bairagi, M. (2023) A New Algorithm to Determine Adomian Polynomials for Nonlinear Polynomial Functions. Asian Research Journal of Mathematics, 19, 1-12. https://doi.org/10.2139/ssrn.4419957
[43]
Fatoorehchi, H. and Abolghasemi, H. (2011) On Calculation of Adomian Polynomials by MATLAB. Journal of Applied Computer Science & Mathematics, 5, 85-88.
