Fractional natural decomposition method for solving a certain class of nonlinear time-fractional wave-like equations with variable coefficients

Ali Khalouta; Abdelouahab Kadem

doi:10.2478/ausm-2019-0009

Abstract

In this paper, we propose a new approximate method, namely fractional natural decomposition method (FNDM) in order to solve a certain class of nonlinear time-fractional wave-like equations with variable coefficients. The fractional natural decomposition method is a combined form of the natural transform method and the Adomian decomposition method. The nonlinear term can easily be handled with the help of Adomian polynomials which is considered to be a clear advantage of this technique over the decomposition method. Some examples are given to illustrate the applicability and the easiness of this approach.

References

[1] S. Abbasbandy, Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomians decomposition method, Appl. Math. Comput., 172 (1) (2006), 485–490.10.1016/j.amc.2005.02.014
Search in Google Scholar Back to article
[2] F. B. M. Belgacem, R. Silambarasan, Theory of natural transform, Math. Eng. Sci. Aerosp., 3 (1) (2012), 105–135.10.1063/1.4765477
Search in Google Scholar Back to article
[3] M. Ganjiani, Solution of nonlinear fractional differential equations using homotopy analysis method, Appl. Math. Model., 34 (2010), 1634–1641.10.1016/j.apm.2009.09.011
Search in Google Scholar Back to article
[4] M. Ghoreishi, A. I. B. Ismail, N. H. M. Ali, Adomain decomposition method for nonlinear wave-like equation with variable coefficients, Appl. Math. Sci., 4 (49) (2010), 2431–2444.
Search in Google Scholar Back to article
[5] V. G. Gupta, S. Gupta, Homotopy perturbation transform method for solving nonlinear wave-like equations of variable coefficients, J. Inf. Comput. Sci., 8 (3) (2013), 163-172.
Search in Google Scholar Back to article
[6] A. A Kilbas, H. M Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
Search in Google Scholar Back to article
[7] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
Search in Google Scholar Back to article
[8] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
Search in Google Scholar Back to article
[9] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367–386.
Search in Google Scholar Back to article
[10] Z. Odibat, S. Momani, A generalized differential transform method for linear partial differential equations of fractional order, Appl. Math. Lett., 21 (2008), 194–199.10.1016/j.aml.2007.02.022
Search in Google Scholar Back to article
[11] Z. Odibat, S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals, 36 (1) (2008), 167–174.10.1016/j.chaos.2006.06.041
Search in Google Scholar Back to article
[12] Y. Zhang, Time-Fractional Generalized Equal Width Wave Equations: Formulation and Solution via Variational Methods, Nonlinear Dyn. Syst. Theory, 14 (4) (2014), 410–425.10.1186/1687-1847-2014-65
Search in Google Scholar Back to article
[13] Y. Zhu, Q. Chang, S. Wu, A new algorithm for calculating Adomian polynomials, Appl. Math. Comput., 169 (2005), 402–416.10.1016/j.amc.2004.09.082
Search in Google Scholar Back to article

Fractional natural decomposition method for solving a certain class of nonlinear time-fractional wave-like equations with variable coefficients

Abstract

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