Research Article
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A Comparison Between Analytical and Numerical Solutions for Time-Fractional Coupled Dispersive Long-Wave Equations

Year 2021, Volume: 2 Issue: 1, 8 - 29, 29.01.2021

Abstract

In this article, a technique namely Tanh method is applied to obtain some traveling wave
solutions for coupled Dispersive Long-Wave equations, and by using LADM we obtain an approximate
solution to TFDLW (time-fractional DLW equations). A comparison between the traveling wave solution
and the approximate one of the DLW system indicates that Laplace Adomian Decomposition Method
(LADM) is highly accurate and can be considered a very useful and valuable method. The availability of
computer systems like Mathematica 11 software facilitates the tedious algebraic calculations and plots
of surfaces of solutions. The methods which we will propose in this paper are also standard, direct and
computerizable methods, which allows us to do complicated and tedious algebraic calculation.

Supporting Institution

Laboratory ACEDP, Djillali Liabes University, 22000 SIDI-BEL-ABBES, Algeria

References

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  • [2] Adomian G., A review of the decomposition method in applied mathematics, Journal of Mathematical Analysis and Applications, 135, 501544, 1988.
  • [3] Adomian G., System of nonlinear partial differential equations, Journal of Mathematical Analysis and Applications, 115(1), 235-238, 1986.
  • [4] Adomian G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publication, 1994.
  • [5] Akbar M.A., Mohd. Ali N.H., Exp-function method for duffing equation and new solutions of (2+1) dimensional dispersive long wave equations, Progress in Applied Mathematics, 1(2), 3042, 2011.
  • [6] Babolian E., Biazar J., Vahidi A.R., A new computational method for Laplace transforms by decomposition method, Applied Mathematics and Computation, 150, 841846, 2004.
  • [7] Babolian E., Javadi S., New method for calculating Adomian polynomials, Applied Mathematics and Computation, 153, 253259, 2004.
  • [8] Boitit M., Leon J.J., Pempinell F., Integrable two-dimensional generalization of the sine-Gordon and sinh-Gordon equations, Inverse Problems, 3, 37-49, 1987.
  • [9] Cherruault Y., Adomian G., Decomposition methods: A new proof of convergence, Mathematical and Computer Modelling, 18(12), 103-106, 1993.
  • [10] El-Danaf T.S., Ramadan M.A., Abd Alaal F.E.I., The use of adomian decomposition method for solving the regularized long-wave equation, Chaos, Solitons and Fractals, 26, 747757, 2005.
  • [11] Fan E.G., Traveling wave solutions for nonlinear equations using symbolic computation, Computers and Mathematics with Applications, 42(4), 671-680, 2002.
  • [12] Fadaei J., Application of Laplace-Adomian decomposition method on linear and nonlinear system of PDEs, Applied Mathematical Sciences, 5(27), 1307-1315, 2011.
  • [13] Helal M.A., Mehanna M.S., The tanh method and Adomian decomposition method for solving the foam drainage equation, Applied Mathematics and Computation, 190, 599-609, 2007.
  • [14] Jaradat K., ALoqali D., Alhabashene W., Using Laplace decomposition method to solve nonlin- ear Klien-Gordan equation, University Politehnica of Bucharest Scienti c Bulletin, Series D, 80(2), 213222, 2018.
  • [15] Jie-Fang Z., Guan-Ping G., Feng-Min W., New multi-soliton solutions and travelling wave solutions of the dispersive long-wave equations, Chinese Physics, 11(6), 533-536, 2002.
  • [16] Jie-Fang Z., Multiple soliton solutions of the dispersive long-wave equations, Chinese Physics Letters, 16(1), 4-5, 1999.
  • [17] Khan K., Ali Akbar M., Arnous A.H., Exact traveling wave solutions forsystem ofnonlinear evolution equations, Springer Plus, 5(663), 2016.
  • [18] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Di erential Equa- tions, Elsevier, 2006.
  • [19] Luo X.G., A two-step Adomian decomposition method, Applied Mathematics and Computation, 170, 570-583, 2005.
  • [20] Malfliet W., The tanh method: A tool for solving certain classes of nonlinear evolution and wave equations, Journal of Computational and Applied Mathematics, 164-165, 529-541, 2004.
  • [21] Malfliet W., The tanh method: A tool for solving certain classes of non-linear PDEs, Mathematical Methods in the Applied Sciences, 28, 20312035, 2005.
  • [22] Odibat Z.M., Momani S., Approximate solutions for boundary value problems of time-fractional wave equation, Applied Mathematics and Computation, 181, 767-774, 2006.
  • [23] Podlubny I., Fractional Di erential Equations, Academic Press, 1999.
  • [24] Qingling G., Exact Solutions of the mBBM Equation, Applied Mathematical Sciences, 5(25), 1209- 1215, 2011.
  • [25] Ray S.S., Bera R.K., Analytical solution of a fractional di usion equation by Adomian decomposition method, Applied Mathematics and Computation, 174, 329-336, 2006.
  • [26] Schiff J.L., The Laplace Tranform, Theory and Applications, Springer-Verlag, 1999.
  • [27] Sen-Yue L., Similarity solutions of dispersive long-wave equations in two space dimensions, Mathe- matical Methods in the Applied Science, 18(6), 789-802, 1995.
  • [28] Spiegel M.R., Laplace Tranforms, McGraw-Hill, 1965.
  • [29] Sumbal Shaikh T., Ahmed N., Shahid N., Iqbal Z., Solution of the Zabolotskaya-Khokholov equation by Laplace decomposition method, International Journal of Scienti fic & Engineering Research, 9(2), 18111816, 2018.
  • [30] Wazwaz A.M., Mehanna M.S., The combined Laplace-Adomian method for handling singular integral equation of heat transfer, International Journal of Nonlinear Science, 10(2), 248-252, 2010.
  • [31] Wazwaz A.M., The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations, Applied Mathematics and Computation, 216(4), 1304-1309, 2010.
  • [32] Wazwaz A.M., Partial Differential Equations and Solitary Waves Theory, Higher Education Press, 2009.
  • [33] Yan L., Numerical solutions of fractional Fokker-Planck equations using iterative Laplace transform method, Abstract and Applied Analysis, Article ID 465160, 2013.
  • [34] Zarea S.A., The tanh method: A tool for solving some mathematical models, Chaos, Solitons and Fractals, 41, 979988, 2009.
Year 2021, Volume: 2 Issue: 1, 8 - 29, 29.01.2021

Abstract

References

  • [1] Abdelrazec A., Pelinovsky D., Convergence of the adomian decomposition method for initial-value problems, Numerical Methods for Partial Differential Equations, 27, 749766, 2011.
  • [2] Adomian G., A review of the decomposition method in applied mathematics, Journal of Mathematical Analysis and Applications, 135, 501544, 1988.
  • [3] Adomian G., System of nonlinear partial differential equations, Journal of Mathematical Analysis and Applications, 115(1), 235-238, 1986.
  • [4] Adomian G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publication, 1994.
  • [5] Akbar M.A., Mohd. Ali N.H., Exp-function method for duffing equation and new solutions of (2+1) dimensional dispersive long wave equations, Progress in Applied Mathematics, 1(2), 3042, 2011.
  • [6] Babolian E., Biazar J., Vahidi A.R., A new computational method for Laplace transforms by decomposition method, Applied Mathematics and Computation, 150, 841846, 2004.
  • [7] Babolian E., Javadi S., New method for calculating Adomian polynomials, Applied Mathematics and Computation, 153, 253259, 2004.
  • [8] Boitit M., Leon J.J., Pempinell F., Integrable two-dimensional generalization of the sine-Gordon and sinh-Gordon equations, Inverse Problems, 3, 37-49, 1987.
  • [9] Cherruault Y., Adomian G., Decomposition methods: A new proof of convergence, Mathematical and Computer Modelling, 18(12), 103-106, 1993.
  • [10] El-Danaf T.S., Ramadan M.A., Abd Alaal F.E.I., The use of adomian decomposition method for solving the regularized long-wave equation, Chaos, Solitons and Fractals, 26, 747757, 2005.
  • [11] Fan E.G., Traveling wave solutions for nonlinear equations using symbolic computation, Computers and Mathematics with Applications, 42(4), 671-680, 2002.
  • [12] Fadaei J., Application of Laplace-Adomian decomposition method on linear and nonlinear system of PDEs, Applied Mathematical Sciences, 5(27), 1307-1315, 2011.
  • [13] Helal M.A., Mehanna M.S., The tanh method and Adomian decomposition method for solving the foam drainage equation, Applied Mathematics and Computation, 190, 599-609, 2007.
  • [14] Jaradat K., ALoqali D., Alhabashene W., Using Laplace decomposition method to solve nonlin- ear Klien-Gordan equation, University Politehnica of Bucharest Scienti c Bulletin, Series D, 80(2), 213222, 2018.
  • [15] Jie-Fang Z., Guan-Ping G., Feng-Min W., New multi-soliton solutions and travelling wave solutions of the dispersive long-wave equations, Chinese Physics, 11(6), 533-536, 2002.
  • [16] Jie-Fang Z., Multiple soliton solutions of the dispersive long-wave equations, Chinese Physics Letters, 16(1), 4-5, 1999.
  • [17] Khan K., Ali Akbar M., Arnous A.H., Exact traveling wave solutions forsystem ofnonlinear evolution equations, Springer Plus, 5(663), 2016.
  • [18] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Di erential Equa- tions, Elsevier, 2006.
  • [19] Luo X.G., A two-step Adomian decomposition method, Applied Mathematics and Computation, 170, 570-583, 2005.
  • [20] Malfliet W., The tanh method: A tool for solving certain classes of nonlinear evolution and wave equations, Journal of Computational and Applied Mathematics, 164-165, 529-541, 2004.
  • [21] Malfliet W., The tanh method: A tool for solving certain classes of non-linear PDEs, Mathematical Methods in the Applied Sciences, 28, 20312035, 2005.
  • [22] Odibat Z.M., Momani S., Approximate solutions for boundary value problems of time-fractional wave equation, Applied Mathematics and Computation, 181, 767-774, 2006.
  • [23] Podlubny I., Fractional Di erential Equations, Academic Press, 1999.
  • [24] Qingling G., Exact Solutions of the mBBM Equation, Applied Mathematical Sciences, 5(25), 1209- 1215, 2011.
  • [25] Ray S.S., Bera R.K., Analytical solution of a fractional di usion equation by Adomian decomposition method, Applied Mathematics and Computation, 174, 329-336, 2006.
  • [26] Schiff J.L., The Laplace Tranform, Theory and Applications, Springer-Verlag, 1999.
  • [27] Sen-Yue L., Similarity solutions of dispersive long-wave equations in two space dimensions, Mathe- matical Methods in the Applied Science, 18(6), 789-802, 1995.
  • [28] Spiegel M.R., Laplace Tranforms, McGraw-Hill, 1965.
  • [29] Sumbal Shaikh T., Ahmed N., Shahid N., Iqbal Z., Solution of the Zabolotskaya-Khokholov equation by Laplace decomposition method, International Journal of Scienti fic & Engineering Research, 9(2), 18111816, 2018.
  • [30] Wazwaz A.M., Mehanna M.S., The combined Laplace-Adomian method for handling singular integral equation of heat transfer, International Journal of Nonlinear Science, 10(2), 248-252, 2010.
  • [31] Wazwaz A.M., The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations, Applied Mathematics and Computation, 216(4), 1304-1309, 2010.
  • [32] Wazwaz A.M., Partial Differential Equations and Solitary Waves Theory, Higher Education Press, 2009.
  • [33] Yan L., Numerical solutions of fractional Fokker-Planck equations using iterative Laplace transform method, Abstract and Applied Analysis, Article ID 465160, 2013.
  • [34] Zarea S.A., The tanh method: A tool for solving some mathematical models, Chaos, Solitons and Fractals, 41, 979988, 2009.
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Medjahed Djilali 0000-0002-4390-9404

Hakem Alı 0000-0001-6145-4514

Benali Abdelkader 0000-0001-6205-3499

Publication Date January 29, 2021
Published in Issue Year 2021 Volume: 2 Issue: 1

Cite

19113 FCMS is licensed under the Creative Commons Attribution 4.0 International Public License.