The Analytical Solution of Some Partial Differential Equations by the SBA Method

Yanick Alain Servais Wellot

doi:doi:10.11648/j.pamj.20221104.13

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The Analytical Solution of Some Partial Differential Equations by the SBA Method

Yanick Alain Servais Wellot

Published in Pure and Applied Mathematics Journal (Volume 11, Issue 4)

Received: 19 September 2022 Accepted: 4 October 2022 Published: 17 October 2022

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Abstract

Many phenomena in nature, especially in the current context of climate change, are modeled by nonlinear partial differential equations. Numerical methods exist to solve these equations numerically. But, the search for exact solutions, when they exist, is always necessary, in order to better explain the modeled phenomenon. The interest of the search for the exact solution results in the advantage of avoiding to analyze again the margins of errors, which sometimes, require the minimization. Thus, several methods are implemented to search for possible exact solutions. Despite the existence of various methods, difficulties have always surfaced. The case considered is that of strongly nonlinear partial differential equations. Thus, in the literature approached a new method called, the SBA method. In this paper, is used an analytical method called SOME-BLAISE-ABBO method (SBA method), to solve nonlinear partial differential equations. It is a method that is exclusively presented for the solution of exclusively nonlinear partial differential equations. The fundamental objective of this work is to show the effectiveness of the method for nonlinear problems. To this end, a reaction-convection-diffusion problem, a biological population model and a system of coupled Burgers’ equations are chosen to demonstrate the effectiveness, accuracy and efficiency of the said method. The easy obtaining of the exact solutions of these three chosen nonlinear problems allowed us to affirm the effectiveness of the method.

Published in	Pure and Applied Mathematics Journal (Volume 11, Issue 4)
DOI	10.11648/j.pamj.20221104.13
Page(s)	70-77
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Nonlinear PDEs, Reaction-Convection-Diffusion Problem, Biological Population Model, Coupled Burgers’ Equations, SBA Method

References

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[2]	Ahmad, H., Khan, T. A., & Cesarano, C. (2019). Numerical Solutions of Coupled Burgers’equations. Axioms, 8 (4), 1-16.
[3]	Alomari, A. K. (2020). Homotopy-Sumudu transforms for Solving system of fractional partial differential equations. Advances in Difference Equations, (1), 1-16.
[4]	Bonazebi-Yindoula, J., Pare, Y., Bissanga, G., Bassono, F., Some, B. (2014). Application of the Adomian Decomposition Method (ADM) and the SOME BLAISE ABBO (SBA) method to solving the diffusion-reaction equations. Advances in theoretical and Applied Mathematics, 9 (2), 97-104.
[5]	Bonazebi-Yindoula, J., Pare, Y., Bassono, F., & Bissanga, G. (2017). Solving a linear convection-diffusion problem of Cauchy kind by Laplace-Adomian method. Far East Journal of Mathematical Sciences, 101 (3), 517–527.
[6]	Cheng, R. J., Zhang, L. W., & Liew, K. M. (2014). Modeling of biological population problems using the element-free Kp-Ritz method. Applied Mathematics and Computation, Vol. 227, 274–290.
[7]	Garzón-Alvarado, D. A., Galeano, C. H., & Mantilla, J. M. (2012). Computational examples of reaction-convection-diffusion equations solution under the influence of fluid flow: first example. Applied Mathematical Modelling, (10), 5029–5045.
[8]	Khalouta, A., & Kadem, A. (2020). New analytical method for solving nonlinear time-fractional reaction-diffusion-convection problems. Revista Colombia de Matemáticas, 54 (1), 1–11.
[9]	Kappor, M., & Joshi, V. (2021). A new technique for numerical solution of 1D and 2D non-linear coupled Burgers’equations by using Cubic Uniform Algebraic Trigonometric (UAT) tension B-spline based differential quadrature method. Ain Shams Engineering Journal, Vol. 12, 3947-3965.
[10]	Kim A. S. (2020). Complete analytic Solutions for convection-diffusion-reaction-source equations without using and inverse Laplace transform. Scientific reports. 10 (1), 1-13.
[11]	Nebie, A. W., Bere, F., Abbo, B., & Pare, Y. (2021). Solving Some Derivate Equations Fractional Order Nonlinear Partials Using the Some Blaise Abbo Method. Journal of Mathematics Research, ISSN 1916–9795, E–ISSN 1916–9809. Pp. 101–115.
[12]	Nkaya, G. D., Bassono, F., Bonazebi-Yindoula, J., & Bissanga, G. (2018). Application of the SBA Method for Solving the Partial Differential Equation. Journal of Mathematics Research, 10 (6), 98–107.
[13]	Nkaya, G. D., Mouyedo-Loufouilou, J., Wellot, Y. A. S., Bonazebi-Yindoula, J., & Bissanga, G. (2018). SBA Method in solving the Schrödinger equation. International J. Functional Analysis, Operator Theory and Applications, 10 (1), 1-10.
[14]	Rashid, A., & Ismail, A. I. B. M. (2009 A fourier pseudospectral method for solving coupled viscous Burgers’ equations. Computational Methods in Applied Mathematics, 9 (4), 412-420.
[15]	Roul, P. (2010). Application of homotopy perturbation method to biological population model. Applications and Applied Mathematics: An international Journal (AAM), 5 (2), 1-10.
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Yanick Alain Servais Wellot. (2022). The Analytical Solution of Some Partial Differential Equations by the SBA Method. Pure and Applied Mathematics Journal, 11(4), 70-77. https://doi.org/10.11648/j.pamj.20221104.13

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ACS Style

Yanick Alain Servais Wellot. The Analytical Solution of Some Partial Differential Equations by the SBA Method. Pure Appl. Math. J. 2022, 11(4), 70-77. doi: 10.11648/j.pamj.20221104.13

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AMA Style

Yanick Alain Servais Wellot. The Analytical Solution of Some Partial Differential Equations by the SBA Method. Pure Appl Math J. 2022;11(4):70-77. doi: 10.11648/j.pamj.20221104.13

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@article{10.11648/j.pamj.20221104.13,
  author = {Yanick Alain Servais Wellot},
  title = {The Analytical Solution of Some Partial Differential Equations by the SBA Method},
  journal = {Pure and Applied Mathematics Journal},
  volume = {11},
  number = {4},
  pages = {70-77},
  doi = {10.11648/j.pamj.20221104.13},
  url = {https://doi.org/10.11648/j.pamj.20221104.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20221104.13},
  abstract = {Many phenomena in nature, especially in the current context of climate change, are modeled by nonlinear partial differential equations. Numerical methods exist to solve these equations numerically. But, the search for exact solutions, when they exist, is always necessary, in order to better explain the modeled phenomenon. The interest of the search for the exact solution results in the advantage of avoiding to analyze again the margins of errors, which sometimes, require the minimization. Thus, several methods are implemented to search for possible exact solutions. Despite the existence of various methods, difficulties have always surfaced. The case considered is that of strongly nonlinear partial differential equations. Thus, in the literature approached a new method called, the SBA method. In this paper, is used an analytical method called SOME-BLAISE-ABBO method (SBA method), to solve nonlinear partial differential equations. It is a method that is exclusively presented for the solution of exclusively nonlinear partial differential equations. The fundamental objective of this work is to show the effectiveness of the method for nonlinear problems. To this end, a reaction-convection-diffusion problem, a biological population model and a system of coupled Burgers’ equations are chosen to demonstrate the effectiveness, accuracy and efficiency of the said method. The easy obtaining of the exact solutions of these three chosen nonlinear problems allowed us to affirm the effectiveness of the method.},
 year = {2022}
}

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TY - JOUR
T1 - The Analytical Solution of Some Partial Differential Equations by the SBA Method
AU - Yanick Alain Servais Wellot
Y1 - 2022/10/17
PY - 2022
N1 - https://doi.org/10.11648/j.pamj.20221104.13
DO - 10.11648/j.pamj.20221104.13
T2 - Pure and Applied Mathematics Journal
JF - Pure and Applied Mathematics Journal
JO - Pure and Applied Mathematics Journal
SP - 70
EP - 77
PB - Science Publishing Group
SN - 2326-9812
UR - https://doi.org/10.11648/j.pamj.20221104.13
AB - Many phenomena in nature, especially in the current context of climate change, are modeled by nonlinear partial differential equations. Numerical methods exist to solve these equations numerically. But, the search for exact solutions, when they exist, is always necessary, in order to better explain the modeled phenomenon. The interest of the search for the exact solution results in the advantage of avoiding to analyze again the margins of errors, which sometimes, require the minimization. Thus, several methods are implemented to search for possible exact solutions. Despite the existence of various methods, difficulties have always surfaced. The case considered is that of strongly nonlinear partial differential equations. Thus, in the literature approached a new method called, the SBA method. In this paper, is used an analytical method called SOME-BLAISE-ABBO method (SBA method), to solve nonlinear partial differential equations. It is a method that is exclusively presented for the solution of exclusively nonlinear partial differential equations. The fundamental objective of this work is to show the effectiveness of the method for nonlinear problems. To this end, a reaction-convection-diffusion problem, a biological population model and a system of coupled Burgers’ equations are chosen to demonstrate the effectiveness, accuracy and efficiency of the said method. The easy obtaining of the exact solutions of these three chosen nonlinear problems allowed us to affirm the effectiveness of the method.
VL - 11
IS - 4
ER -

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Author Information

Yanick Alain Servais Wellot

Department of Exacts Sciences, Teachers Training College, Marien Ngouabi University, Brazzaville, Congo

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APA Style

Yanick Alain Servais Wellot. (2022). The Analytical Solution of Some Partial Differential Equations by the SBA Method. Pure and Applied Mathematics Journal, 11(4), 70-77. https://doi.org/10.11648/j.pamj.20221104.13

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ACS Style

Yanick Alain Servais Wellot. The Analytical Solution of Some Partial Differential Equations by the SBA Method. Pure Appl. Math. J. 2022, 11(4), 70-77. doi: 10.11648/j.pamj.20221104.13

Copy | Download

AMA Style

Yanick Alain Servais Wellot. The Analytical Solution of Some Partial Differential Equations by the SBA Method. Pure Appl Math J. 2022;11(4):70-77. doi: 10.11648/j.pamj.20221104.13

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@article{10.11648/j.pamj.20221104.13,
  author = {Yanick Alain Servais Wellot},
  title = {The Analytical Solution of Some Partial Differential Equations by the SBA Method},
  journal = {Pure and Applied Mathematics Journal},
  volume = {11},
  number = {4},
  pages = {70-77},
  doi = {10.11648/j.pamj.20221104.13},
  url = {https://doi.org/10.11648/j.pamj.20221104.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20221104.13},
  abstract = {Many phenomena in nature, especially in the current context of climate change, are modeled by nonlinear partial differential equations. Numerical methods exist to solve these equations numerically. But, the search for exact solutions, when they exist, is always necessary, in order to better explain the modeled phenomenon. The interest of the search for the exact solution results in the advantage of avoiding to analyze again the margins of errors, which sometimes, require the minimization. Thus, several methods are implemented to search for possible exact solutions. Despite the existence of various methods, difficulties have always surfaced. The case considered is that of strongly nonlinear partial differential equations. Thus, in the literature approached a new method called, the SBA method. In this paper, is used an analytical method called SOME-BLAISE-ABBO method (SBA method), to solve nonlinear partial differential equations. It is a method that is exclusively presented for the solution of exclusively nonlinear partial differential equations. The fundamental objective of this work is to show the effectiveness of the method for nonlinear problems. To this end, a reaction-convection-diffusion problem, a biological population model and a system of coupled Burgers’ equations are chosen to demonstrate the effectiveness, accuracy and efficiency of the said method. The easy obtaining of the exact solutions of these three chosen nonlinear problems allowed us to affirm the effectiveness of the method.},
 year = {2022}
}

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