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Combination of Reduced Differential Transformation Method and Picard’s Principle

Received: 30 June 2022     Accepted: 15 July 2022     Published: 22 July 2022
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Abstract

In this paper, the combination of methods is used for the search for exact solutions, when they exist of mixed and non-mixed nonlinear partial differential equations. it is the combinate of reduced differential transform method and Picard principle. This combination gave us an algorithm that promotes the rapid convergence of the problem given the exact solution desired. Some complex physical behaviors can be described by mathematical expressions. These expressions can be nonlinear partial differential equations and sometimes mixed. For a better understanding of the physical phenomena associated with such partial differential equations, the exact solution, when it exists, is better indicated. However, by using classical analytical methods, the access or the obtaining of the exact solution is not always obvious. With some hybrid algorithms, the difficulties of accessing this exact solution can be difficult or almost impossible. Hence the coupling of some algorithms to reach the desired result. The objective of our work is the search for exact solutions when they exist of mixed and unmixed nonlinear partial differential equations. Although the reduced transform method has presented several interesting results, the difficulties of obtaining exact solutions have also been encountered. Thus, in this paper, a combination is used to find exact solutions, when they exist, of these types of partial differential equations. It is the combination of the reduced transform method and Picard's principle. This Picard principle, which uses the Adomian decomposition method, works as a method of successive approximations, approaching the problem by an iterative scheme. This combination gave us an algorithm that favors the fast convergence of the problem. Thus, the exact solutions of the selected problems are obtained.

Published in Applied and Computational Mathematics (Volume 11, Issue 4)
DOI 10.11648/j.acm.20221104.11
Page(s) 87-94
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), 2022. Published by Science Publishing Group

Keywords

Nonlinear PDEs, Reduced Differential Transform Method (RDTM), SBA Method, Picard Principle

References
[1] Afzal Soomro M., & Hussain J. (2019). On Study of Generalized Novikov Equation by Reduced Differential Transform Method. Indian Journal of Sciences and Technology, Vol. 12. DOI: 10.17 485/ijst/2019, ISSN (online): 0974-5645, pp: 1-6.
[2] Al-Sawoor, Ann J., Al-Amr Mohammed O. (2013). Reduced Differential Transform Method for the Generalized Ito System. International Journal of Enhanced Research in Science Technology & Engineering. ISSN: 2319-7463, Vol. 2, Issue 11, novembre-2013, pp: 135-145.
[3] Arslan D. (2021). The Approximate Solution of Singularly Perturbed Burgers-Huxley Equation with RDTM. BEU Journal of Science. 10 (3), pp: 703-709.
[4] Az-Zo’bi E., A. (2014). On the Reduced Differential Transform Method and it’s application to the Generalized Burgers-Huxley Equation. Applied Mathematical Sciences. Vol. 8, N° 177, pp: 8823-8831.
[5] Behammouda B., Vazquez-Leal H. and Sarmiento-Reyes. (2014). Modified Reduced Differential Transform Method for Partial Differential-Algebraic Equations. Journal of Applied Mathematics. Vol 2014, Article ID 279481, pp: 1-9.
[6] Gashaw Belayeh, W., Obsie Mussa, Y., and Kebede Gizaw, A. (2020). Approximate Analytic Solutions of Two-Dimensional Nonlinear Klein-Gordon Equation by Using the Reduced Differential Transform Method. Mathematical Problems in Engineering. Vol. Article ID 5753974, pp: 1-12.
[7] Günerhan H. (2018). Analytical Solutions of One-Dimensional Convection-Diffusion Probems. Turkish Journal of Analysis and Number Theory. Vol. 6, N°6, pp: 152-154.
[8] Hesam, S., Nazemi, A., Haghbin A. (2012). Reduced Differential Transform Method for Solving the Fornberg-Witham Type Equation. International Journal of Nonlinear Science. Vol. 13, N°.2, pp: 158-162.
[9] Jafari H., Jassim, H., K., Moshokoa S., P., Ariyan V., Tcchier F., (2016). Reduced differential transform method for partial differential equations within local fractional derivative operators. Advances in Mechanical Engineering. Vol. 8 (4). pp: 1–6.
[10] Khalouta A. (2019). Résolution des équations aux dérivées partielles linéaires et non-linéaires moyennant des approaches analytiques. Extension aux cas d’EDP d’ordre fractionnaire. PhD diss., Université de Sétif 1-Ferhat Abbas.
[11] Karbalaie, A., Muhammed, H., H., Shabani, M., & Montazeri, M., M. (2014). Exact solution of partial differential equation using homo-separation of variables. International Journal of Nonlinear Science, ISSN 1749-3889 (print), 1749-3897, Vol. 17, N°.1, pp: 84-90.
[12] Keskin Y., Oturanc G., (2009). Reduced Differential Transform Method for Partial Differential Equations. International Journal of Nonlinear Sciences and Numerical Simulation. 10 (6), pp: 741-749.
[13] Keskin Y., (2010). Application of Reduced Differential Transformation Method for Solving Gas Dynamics Equation. Int. J. Contemp. Math. Sciences, Vol. 5, N°22, pp: 1091-1096.
[14] Keskin Y., Oturanc G. (2010). Reduced Differential Transform Method for solving linear and Nonlinear wave Equations. Iranian Journal of Science & Technology. Transaction A, Vol. 34, N°A2, pp: 113-122.
[15] Mikamona M., S., Bonazebi-Yindoula J., Paré, Y. & Bissanga G. (2019). Application of SBA method to solve the nonlinear Biological population models. European Journal of Pure and Applied Mathematics. 12 (3), 1096-1105.
[16] Mirzaee, F. (2011). Differential Transform Method for Solving Linear and Nonlinear Systems of ordinary Differential Equations. Applied Mathematical Sciences. Vol. 5, n°.70, pp: 3465-3472.
[17] Mohamed S. Mohamed, Khaled A. Gepreel. (2017). Reduced Differential Transform Method for Nonlinear integral member of Kadomtsev-Petviashvili hierarchy differential equations. Journal of the Egyptian Mathematical Society 25. pp: 1-7.
[18] 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. Vol. 10, n°1, pp: 1-10.
[19] Moosavi Noori S. R. & Taghizadeh N. (2020). Study on Solving Two-dimensional Linear and Nonlinear Volterra Partial Integro-differential Equations by Reduced Differential Transform Method. Applications and Applied Mathematics (AAM). Vol. 15 Issue 1, (June 2020), pp: 394-407.
[20] Paré Y., Migoungou Y. & Wassiha Nébié A. (2019). Resolution of nonlinear convection-diffusion-reaction equations of Cauchy’s kind by the Laplace SBA method. European Journal of Pure and Applied Mathematics. 12 (3), 771-789.
[21] Rawashdeh M. (2013) Improved Approximate solutions for Nonlinear Evolutions Equations in Mathematical Physics Using the Reduced Differential Transform Method. Journal of Applied Mathematics & Bioinformatics, vol. 3, no. 2, pp: 1-14. ISSN: 1792-6602.
[22] Rawashdeh M., & Obeidat N. A. (2014). On finding Exact and Approximate Solutions to Some PDEs Using the Reduced Differential Transform Method. Appl. Math. Inf. Sci. Vol. 8, No. 5, pp: 2171-2176.
[23] Soltanalizadeh B. (2011). Application of Differential Transformation Method for Numerical Analysis of Kawahara Equation. Australian Journal of Basic and Applied Sciences, 5 (12). pp: 490-495. ISSN 1991-8178.
[24] Taghizadeh, N., & Moosavi oori, S. R. (2016). Exact solutions of Cubic Nonlinear Schrödinger Equation with a Trapping Potential by Reduced Differential Transform Method. Math. Sci. Lett. 5, N°3, pp: 1-5.
[25] Yildirim, K. (2012). A solution Method for Solving Systems of Nonlinear PDEs. World Applied Sciences Journal, 18 (11), 1527-1532.
[26] Zuriqat, M. (2018). The homo separation analysis method for solving the partial differential equation. Italian Journal of Pure and Applied Mathematics, n°40, pp: 535-543.
Cite This Article
  • APA Style

    Yanick Alain Servais Wellot, Gires Dimitri Nkaya. (2022). Combination of Reduced Differential Transformation Method and Picard’s Principle. Applied and Computational Mathematics, 11(4), 87-94. https://doi.org/10.11648/j.acm.20221104.11

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    ACS Style

    Yanick Alain Servais Wellot; Gires Dimitri Nkaya. Combination of Reduced Differential Transformation Method and Picard’s Principle. Appl. Comput. Math. 2022, 11(4), 87-94. doi: 10.11648/j.acm.20221104.11

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    AMA Style

    Yanick Alain Servais Wellot, Gires Dimitri Nkaya. Combination of Reduced Differential Transformation Method and Picard’s Principle. Appl Comput Math. 2022;11(4):87-94. doi: 10.11648/j.acm.20221104.11

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  • @article{10.11648/j.acm.20221104.11,
      author = {Yanick Alain Servais Wellot and Gires Dimitri Nkaya},
      title = {Combination of Reduced Differential Transformation Method and Picard’s Principle},
      journal = {Applied and Computational Mathematics},
      volume = {11},
      number = {4},
      pages = {87-94},
      doi = {10.11648/j.acm.20221104.11},
      url = {https://doi.org/10.11648/j.acm.20221104.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221104.11},
      abstract = {In this paper, the combination of methods is used for the search for exact solutions, when they exist of mixed and non-mixed nonlinear partial differential equations. it is the combinate of reduced differential transform method and Picard principle. This combination gave us an algorithm that promotes the rapid convergence of the problem given the exact solution desired. Some complex physical behaviors can be described by mathematical expressions. These expressions can be nonlinear partial differential equations and sometimes mixed. For a better understanding of the physical phenomena associated with such partial differential equations, the exact solution, when it exists, is better indicated. However, by using classical analytical methods, the access or the obtaining of the exact solution is not always obvious. With some hybrid algorithms, the difficulties of accessing this exact solution can be difficult or almost impossible. Hence the coupling of some algorithms to reach the desired result. The objective of our work is the search for exact solutions when they exist of mixed and unmixed nonlinear partial differential equations. Although the reduced transform method has presented several interesting results, the difficulties of obtaining exact solutions have also been encountered. Thus, in this paper, a combination is used to find exact solutions, when they exist, of these types of partial differential equations. It is the combination of the reduced transform method and Picard's principle. This Picard principle, which uses the Adomian decomposition method, works as a method of successive approximations, approaching the problem by an iterative scheme. This combination gave us an algorithm that favors the fast convergence of the problem. Thus, the exact solutions of the selected problems are obtained.},
     year = {2022}
    }
    

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    AU  - Yanick Alain Servais Wellot
    AU  - Gires Dimitri Nkaya
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    UR  - https://doi.org/10.11648/j.acm.20221104.11
    AB  - In this paper, the combination of methods is used for the search for exact solutions, when they exist of mixed and non-mixed nonlinear partial differential equations. it is the combinate of reduced differential transform method and Picard principle. This combination gave us an algorithm that promotes the rapid convergence of the problem given the exact solution desired. Some complex physical behaviors can be described by mathematical expressions. These expressions can be nonlinear partial differential equations and sometimes mixed. For a better understanding of the physical phenomena associated with such partial differential equations, the exact solution, when it exists, is better indicated. However, by using classical analytical methods, the access or the obtaining of the exact solution is not always obvious. With some hybrid algorithms, the difficulties of accessing this exact solution can be difficult or almost impossible. Hence the coupling of some algorithms to reach the desired result. The objective of our work is the search for exact solutions when they exist of mixed and unmixed nonlinear partial differential equations. Although the reduced transform method has presented several interesting results, the difficulties of obtaining exact solutions have also been encountered. Thus, in this paper, a combination is used to find exact solutions, when they exist, of these types of partial differential equations. It is the combination of the reduced transform method and Picard's principle. This Picard principle, which uses the Adomian decomposition method, works as a method of successive approximations, approaching the problem by an iterative scheme. This combination gave us an algorithm that favors the fast convergence of the problem. Thus, the exact solutions of the selected problems are obtained.
    VL  - 11
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Author Information
  • Department of Exacts Sciences, Teachers Training College, Marien Ngouabi University, Brazzaville, Congo

  • Department of Mathematics, Faculty of Sciences and Technology, Marien Ngouabi University, Brazzaville, Congo

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