Laplace Substitution – Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives

Ali Al-Fayadh; Dina Saad Faraj

doi:doi:10.11648/j.ajmcm.20190401.12

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Laplace Substitution – Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives

Ali Al-Fayadh, Dina Saad Faraj

Published in American Journal of Mathematical and Computer Modelling (Volume 4, Issue 1)

Received: 26 February 2019 Accepted: 4 April 2019 Published: 10 May 2019

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Abstract

This paper will investigate a method to achieve the exact solution of special type of nonlinear partial differential equations (NLPDEs) involving mixed partial derivatives. This proposed method named as Laplace substitution - Variation iteration method (LS-VIM). The method exploits the properties of Laplace substitution method and the Variational iteration method to find the exact solution for Goursat problem involving mixed partial derivatives. In addition, this paper emphasizes the effectiveness of the LS-VIM by solving two examples. The results show that the exact solution can be achieved from a single iteration of the propose method.

Published in	American Journal of Mathematical and Computer Modelling (Volume 4, Issue 1)
DOI	10.11648/j.ajmcm.20190401.12
Page(s)	16-20
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

Laplace Transforms, Laplace Substitution Method, Variation Iteration Method, Mixed Partial Derivatives, Goursat Problems

References

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[2]	A. AL-Fayadh and H. Khawwan, “Variation Iteration Transform Method For Solving Burger and Coupled Burgers eqution, ” ARPN J. Eng. Appl. Sci., vol. 12, no. 23, pp. 6926–6932, 2017.
[3]	Evans D., Sanusi B. Numerical Solution of the Goursat Problem By A Nonlinear Trapezoidal Forula. Appl Math Lett. 1988; I (3): 221-223.
[4]	Handibag S., Karande BD. An Application for Nonlinear Partial Differential Equations Involving Mixed Partial Derivatives by Laplace Substitution Method. In: 10th International Conference on Mathematical Problems in Engineering, Aerospace and Science. AIP Conf. Proc. Vol 1637; 384-394, (2014).
[5]	J.T. Day, A Runge–Kutta method for the numerical solution of the Goursat problem in hyperbolic partial differential equations.
[6]	Muhammad Aslam Noor ,Syed Tauseef Mohyud-Din.Modified Variational Iteration Method for Goursat and Laplace Problems.Appl Sciences Journal 4 (4): 487-498, 2008.
[7]	SharafMohmoud, Mohamed Gubara. Reduced Differential Transform Method.Appl Math, 2016, 7, 1049-1056
[8]	Taghvafard H, Erjaee GH. Two-dimensional Differential Transform Method for Solving Linear and Non-linear Goursat Problem. Int J Math Comput Phys Electr Comput Eng. 2010; 4 (3):432-435.
[9]	Usman M, Zubair T, Ali U, Mohyud-din ST. On Goursat Problems. Int J Mod Math Sci. 2012; 3 (3): 63-76.
[10]	Wazwaz A. The variational iteration method for a reliable treatment of the linear and the nonlinear Goursat problem. Appl Math Comput. 2007; 193: 455-462
[11]	Handibag S, Karande BD. Laplace Substitution Method for Solving Partial Differential Equations Involving Mixed Partial Derivatives. Int J Comput Eng Res. 2012; 2 (4): 1049-1052.
[12]	Handibag S, Karande B. LAPLACE SUBSTITUTION METHOD FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS INVOLVING MIXED PARTIAL DERIVATIVES. Int J Pure Appl Math. 2012; 78 (7): 973-979.
[13]	Handibag S., Karande B. Laplace Substitution Method for n th Order Linear and Non-Linear PDE ’ s Involving Mixed Partial Derivatives. Int Res J Eng Technol. 2015; 2 (9): 378-388.
[14]	Pandey PK. A Finite Difference Method for Numerical Solution of Goursat Problem of Partial Differential Equation. Open Access Libr J. 2014; 1: 1-6.
[15]	Ramadan MA, Raslan KR, Hadhoud AR, Mesrega AK. A Substitution Method for Partial Differential Equations Using Ramadan Group Integral Transform. Asian Res J Math. 2017; 7 (4): 1-10.

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Ali Al-Fayadh, Dina Saad Faraj. (2019). Laplace Substitution – Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives. American Journal of Mathematical and Computer Modelling, 4(1), 16-20. https://doi.org/10.11648/j.ajmcm.20190401.12

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ACS Style

Ali Al-Fayadh; Dina Saad Faraj. Laplace Substitution – Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives. Am. J. Math. Comput. Model. 2019, 4(1), 16-20. doi: 10.11648/j.ajmcm.20190401.12

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AMA Style

Ali Al-Fayadh, Dina Saad Faraj. Laplace Substitution – Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives. Am J Math Comput Model. 2019;4(1):16-20. doi: 10.11648/j.ajmcm.20190401.12

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@article{10.11648/j.ajmcm.20190401.12,
  author = {Ali Al-Fayadh and Dina Saad Faraj},
  title = {Laplace Substitution – Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {4},
  number = {1},
  pages = {16-20},
  doi = {10.11648/j.ajmcm.20190401.12},
  url = {https://doi.org/10.11648/j.ajmcm.20190401.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20190401.12},
  abstract = {This paper will investigate a method to achieve the exact solution of special type of nonlinear partial differential equations (NLPDEs) involving mixed partial derivatives. This proposed method named as Laplace substitution - Variation iteration method (LS-VIM). The method exploits the properties of Laplace substitution method and the Variational iteration method to find the exact solution for Goursat problem involving mixed partial derivatives. In addition, this paper emphasizes the effectiveness of the LS-VIM by solving two examples. The results show that the exact solution can be achieved from a single iteration of the propose method.},
 year = {2019}
}

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TY  - JOUR
T1  - Laplace Substitution – Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives
AU  - Ali Al-Fayadh
AU  - Dina Saad Faraj
Y1  - 2019/05/10
PY  - 2019
N1  - https://doi.org/10.11648/j.ajmcm.20190401.12
DO  - 10.11648/j.ajmcm.20190401.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 16
EP  - 20
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20190401.12
AB  - This paper will investigate a method to achieve the exact solution of special type of nonlinear partial differential equations (NLPDEs) involving mixed partial derivatives. This proposed method named as Laplace substitution - Variation iteration method (LS-VIM). The method exploits the properties of Laplace substitution method and the Variational iteration method to find the exact solution for Goursat problem involving mixed partial derivatives. In addition, this paper emphasizes the effectiveness of the LS-VIM by solving two examples. The results show that the exact solution can be achieved from a single iteration of the propose method.
VL  - 4
IS  - 1
ER  -

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Author Information

Ali Al-Fayadh

Department of Mathematics and Computer Applications, College of Science, Al-Nahrain University, Baghdad, Iraq
Dina Saad Faraj

Department of Mathematics and Computer Applications, College of Science, Al-Nahrain University, Baghdad, Iraq

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APA Style

Ali Al-Fayadh, Dina Saad Faraj. (2019). Laplace Substitution – Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives. American Journal of Mathematical and Computer Modelling, 4(1), 16-20. https://doi.org/10.11648/j.ajmcm.20190401.12

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ACS Style

Ali Al-Fayadh; Dina Saad Faraj. Laplace Substitution – Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives. Am. J. Math. Comput. Model. 2019, 4(1), 16-20. doi: 10.11648/j.ajmcm.20190401.12

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AMA Style

Ali Al-Fayadh, Dina Saad Faraj. Laplace Substitution – Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives. Am J Math Comput Model. 2019;4(1):16-20. doi: 10.11648/j.ajmcm.20190401.12

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@article{10.11648/j.ajmcm.20190401.12,
  author = {Ali Al-Fayadh and Dina Saad Faraj},
  title = {Laplace Substitution – Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {4},
  number = {1},
  pages = {16-20},
  doi = {10.11648/j.ajmcm.20190401.12},
  url = {https://doi.org/10.11648/j.ajmcm.20190401.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20190401.12},
  abstract = {This paper will investigate a method to achieve the exact solution of special type of nonlinear partial differential equations (NLPDEs) involving mixed partial derivatives. This proposed method named as Laplace substitution - Variation iteration method (LS-VIM). The method exploits the properties of Laplace substitution method and the Variational iteration method to find the exact solution for Goursat problem involving mixed partial derivatives. In addition, this paper emphasizes the effectiveness of the LS-VIM by solving two examples. The results show that the exact solution can be achieved from a single iteration of the propose method.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Laplace Substitution – Variational Iteration Method for Solving Goursat Problems Involving Mixed Partial Derivatives
AU  - Ali Al-Fayadh
AU  - Dina Saad Faraj
Y1  - 2019/05/10
PY  - 2019
N1  - https://doi.org/10.11648/j.ajmcm.20190401.12
DO  - 10.11648/j.ajmcm.20190401.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 16
EP  - 20
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20190401.12
AB  - This paper will investigate a method to achieve the exact solution of special type of nonlinear partial differential equations (NLPDEs) involving mixed partial derivatives. This proposed method named as Laplace substitution - Variation iteration method (LS-VIM). The method exploits the properties of Laplace substitution method and the Variational iteration method to find the exact solution for Goursat problem involving mixed partial derivatives. In addition, this paper emphasizes the effectiveness of the LS-VIM by solving two examples. The results show that the exact solution can be achieved from a single iteration of the propose method.
VL  - 4
IS  - 1
ER  -

Copy | Download