This paper uses the Adomian Decomposition Method (ADM) to solve Boussinesq equations using Maple. The Boussinesq approximation for water waves is a weakly nonlinear and long-wave approximation in fluid dynamics. The approximation is named after Joseph Boussinesq, who developed it in response to John Scott Russell’s observation of a wave of translation (also known as solitary wave or soliton). Bossinesq’s article from 1872 introduced the equations that are now known as the Boussinesq equations. Numerical methods are commonly utilized to solve nonlinear equation systems. In this paper, we investigate a nonlinear singly perturbed advection-diffusion problem. Using the usual Adomian Decomposition Method, we formulate an approximate linear advection-diffusion problem and investigate several practical numerical approaches for solving it (ADM). The Adomian Decomposition Method (ADM) is a powerful tool for numerical simulations and approximation analytic solutions. The Adomian Decomposition Method (ADM) is used to solve nonlinear advection differential equations using Maple by illustrating numerous examples. The findings are presented in the form of tables and graphs for several examples. For various examples, the findings are presented in the form of tables and graphs. The difference between the precise and numerical solutions indicates the Maple program solution’s efficacy, as well as the ease and speed with which it was acquired.
References
[1]
Adomian, G. (1986) Nonlinear Stochastic Operator Equations, Academic Press, San Diego.
[2]
Adomian, G. (1988) A Review of the Decomposition Method in Applied Mathematics. Journal of Mathematical Analysis and Applications, 135, 501-544. https://doi.org/10.1016/0022-247X(88)90170-9
[3]
Adomian, G. (1989) Nonlinear Stochastic Systems Theory and Applications to Physics. Kluwer Academic, Boston.
[4]
Adomian, G. (1990) A Review of the Decomposition Method and Some Recent Results for Nonlinear Equation. Mathematical and Computer Modelling, 13, 17-43. https://doi.org/10.1016/0895-7177(90)90125-7
[5]
Adomian, G. and Rach, R. (1990) Equality of Partial Solutions in the Decomposition Method for Linear or Nonlinear Partial Differential Equations. Computers & Mathematics with Applications, 19, 9-12. https://doi.org/10.1016/0898-1221(90)90246-G
[6]
Adomian, G. (1991) A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations. Computers & Mathematics with Applications, 21, 101-127. https://doi.org/10.1016/0898-1221(91)90220-X
[7]
Adomian, G. (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston. https://doi.org/10.1007/978-94-015-8289-6
[8]
Adomian, G. (1994) Solution of Physical Problems by Decomposition. Computers & Mathematics with Applications, 27, 145-154. https://doi.org/10.1016/0898-1221(94)90132-5
[9]
Adomian, G. (1981) On Product Nonlinearities in Stochastic Differential Equations. Applied Mathematics and Computation, 8, 79-82. https://doi.org/10.1016/0096-3003(81)90033-3
[10]
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
[11]
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
[12]
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
[13]
Adomian, G. and Rach, R. (1984) Light Scattering in Crystals. Journal of Applied Physics, 56, 2592-2594. https://doi.org/10.1063/1.334291
[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) 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
[16]
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
[17]
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
[18]
Adomian, G. and Rach, R. (1992) Noise Terms in Decomposition Series Solution. Computers & Mathematics with Applications, 24, 61-64. https://doi.org/10.1016/0898-1221(92)90031-C
[19]
Wazwaz, A.M. (1997) Necessary Conditions for the Appearance of Noise Terms in Decomposition Series. Applied Mathematics and Computation, 81, 265-274. https://doi.org/10.1016/S0096-3003(95)00327-4
[20]
Luo, X.-G. (2005) A Two-Step Adomian Decomposition Method. Applied Mathematics and Computation, 170, 570-583. https://doi.org/10.1016/j.amc.2004.12.010
[21]
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
