Shehu Variational Iteration Method For Solve Some Fractional Differential Equations

Authors

DOI:

https://doi.org/10.26713/cma.v13i3.1906

Keywords:

Caputo fractional derivate, Variational iteration method, Shehu transform, Fokker-Planck equation, Telegraph equation of space

Abstract

The idea proposed in the work is to extend the Shehu transform method to resolve the nonlinear fractional partial differential equations by combining them with the variational iteration method (VIM). We apply this technique to solve nonlinear fractional equations as nonlinear time fractional Fokker Planck equation.

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References

G. E. Andrews, R. Askey and R. Roy, Special functions, in: Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, (1999), DOI: 10.1017/CBO9781107325937.

R. Belgacem, D. Baleanu and A. Bokhari, Shehu transform and applications to Caputo fractional differential equations, International Journal of Analysis and Applications 17(6) (2019), 917 – 927, URL: http://etamaths.com/index.php/ijaa/article/view/1958.

K. Diethelm, The Analysis Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer-Verlag, Berlin — Heidelberg (2010), DOI: 10.1007/978-3-642-14574-2.

T. M. Elzaki, The new integral transform “Elzaki transform”, Global Journal of Pure and Applied Mathematics 7(1) (2011), 57 – 64, URL: https://www.kau.edu.sa/Files/0057821/Researches/60337_31161.pdf.

J.-H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering 167 (1998), 57 – 68, DOI: 10.1016/S0045-7825(98)00108-X.

J.-H. He, Variational iteration method – a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanics 34 (1999), 699 – 708, DOI: 10.1016/S0020-7462(98)00048-1.

M. Inokuti, H. Sekine and T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: Variational Methods in the Mechanics of Solids, S. Nemat-Nasser (ed.), pp. 156 – 162, Pergamon Press, Oxford, UK (1980), DOI: 10.1016/B978-0-08-024728-1.50027-6.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V, Amsterdam (2006), URL: https://shop.elsevier.com/books/theory-and-applications-of-fractional-differential-equations/kilbas/978-0-444-51832-3#full-description.

S. Maitama and W. Zhao, New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations, International Journal of Analysis and Applications 17(2) (2019), 167 – 190, URL: https://www.etamaths.com/index.php/ijaa/article/view/1771.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Songs, Inc., New York (1993), URL: https://www.gbv.de/dms/ilmenau/toc/122837029.PDF.

S. T. Mohyud-Din, W. Sikander, U. Khan and N. Ahmed, Optimal variational iteration method for nonlinear problems, Journal of the Association of Arab Universities for Basic and Applied Sciences 24 (2017), 191 – 197, DOI: 10.1016/j.jaubas.2016.09.004.

M. A. Noor and S. T. Mohyud-Din, Variational iteration method for solving initial and boundary value problems of Bratu-type, Applications and Applied Mathematics: An International Journal 3(1) (2008), 89 – 99, URL: https://digitalcommons.pvamu.edu/aam/vol3/iss1/8.

M. D. Ortigueira and J. A. T. Machado, What is a fractional derivative?, Journal of Computational Physics 293 (2015), 4 – 13, DOI: 10.1016/j.jcp.2014.07.019.

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 1st edition, Academic Press, San Diego — California, USA (1998), URL: https://www.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9.

M. Safari, D. D. Ganji and M. Moslemi, Application of He’s variational iteration method and Adomian’s decomposition method to the fractional KdV–Burgers–Kuramoto equation, Computers & Mathematics with Applications 58 (2009), 2091 – 2097, DOI: 10.1016/j.camwa.2009.03.043.

M. Safari, Application of He’s variational iteration method for the analytical solution of space fractional diffusion equation, Applied Mathematics 2 (2011), 1091 – 1095, DOI: 10.4236/am.2011.29150.

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives – Theory and Applications, Gordon and Breach, Amsterdam, (1993).

G. K. Watugala, Sumudu transform: a new integral transform to solve differential equations and control engineering problems, International Journal of Mathematical Education in Science and Technology 24(1) (1993), 35 – 43, DOI: 10.1080/0020739930240105.

G.-C. Wu, Challenge in the variational iteration method – A new approach to identification of the Lagrange multipliers, Journal of King Saud University - Sciences 25 (2013), 175 – 178, DOI: 10.1016/j.jksus.2012.12.002.

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Published

29-11-2022
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How to Cite

Boulekhras, A., & Belghaba, K. (2022). Shehu Variational Iteration Method For Solve Some Fractional Differential Equations. Communications in Mathematics and Applications, 13(3), 1207–1219. https://doi.org/10.26713/cma.v13i3.1906

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Research Article