Variational iteration method combined with new transform to solve fractional partial differential equations

Year 2018, , 113 - 120, 26.06.2018

Mountassir Hamdi Cherif , Djelloul Ziane

https://doi.org/10.32323/ujma.396941

Cited By: 8

Abstract

The aim of this paper is to combined the variational iteration method with Aboodh transform method to solve linear and nonlinear fractional partial differential equations. Some illustrative examples are given as the linear and nonlinear fractional Klein-Gordon equations and the time fractional diffusion equation. The results reveal that this method is very effective, simple and can be applied to other physical differential equations with fractional order. The fractional derivative is taken in the Caputo sense.

  

Keywords

Caputo fractional derivative, variational iteration method, Aboodh transform, Klein-Gordon equation, diffusion equation.

References

• [1] A. K. Hassan Sedeeg and M. M. Abdelrahim Mahgob, Comparison of New Integral Transform Aboodh Transform and Adomain Decomposition Method, Int. J. Math. And its Appl. 4 (2)-B (2016), 127-135.
• [2] A. Kiliçman and H. Eltayeb, On a New Integral Transform and Differential Equations, Math. Problems in Eng. A. ID 463579, (2010), 13 pp.
• [3] A. S. Abedl-Rady, S. Z. Rida, A. A. M. Arafa and H. R. Abedl-Rahim,Variational Iteration Sumudu Transform Method for Solving Fractional Nonlinear Gas Dynamics Equation, Int. J. Res. Stu. Sci. Eng. Tech. 1 (2014), 82-90.
• [4] A. S. Arife and A. Yildirim, New Modified Variational Iteration Transform Method (MVITM) for solving eighth-order Boundary value problems in one Step, W. Appl. Sci. J. 13 (2011), 2186 -2190.
• [5] D. Kumar, J. Singh and S. Rathore, Sumudu Decomposition Method for Nonlinear Equations, Int. Math. For. 7 (2012), 515-521.
• [6] D. Ziane and M. Hamdi Cherif, Modified homotopy analysis method for nonlinear fractional partial differential equations, Int. J. Anal. Appl. 14 (1), (2017), 77-87.
• [7] G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer Academic Publishers, Netherlands, 1989.
• [8] G. Adomian, Solution of physical problems by decomposition, Comput. Math. Appl. 27 (1994), 145-154.
• [9] G. Wu and E. W. M. Lee, Fractional variational iteration method and its application, Phys. Letters A. 374 (2010), 2506-2509.
• [10] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999, 1-365.
• [11] I. Podlubny, The Laplace Transform Method for Linear Differential Equations of the Fractional Order, Slovak Acad. Sci. Inst. Exp. Phys, 1997.
• [12] J. H. He, A new approach to nonlinear partial differential equations, Comm. Nonlinear. Sci. Numer. Simul. 2 (1997), 203-205.
• [13] J. H. He, A new perturbation technique which is also valid for large parameters, J. Sound Vib. 229 (2000), 1257-1263.
• [14] J. H. He, A variational iteration approach to nonlinear problems and its applications, Mech. Appl. 20 (1), (1998), 30-31.
• [15] J. H. He, Homotopy perturbation technique, Comput. Meth. Appl. Mech. Eng. 178 (1999), 257-262.
• [16] J. Singh, D. Kumar and Sushila,Homotopy Perturbation Sumudu Transform Method for Nonlinear Equations, Adv. Theor. Appl. Mech. 4 (2011), 165-175.
• [17] K. Diethelm, The Analysis Fractional Differential Equations, Springer-Verlag Berlin Heidelberg, 2010, 1-262.
• [18] K. S. Aboodh, A. Idris and R. I. Nuruddeen, On the Aboodh Transform Connections with Some Famous Integral Transforms, Int. J. Engin. Info. Syst. 1(9), (2017), 143-151.
• [19] K. S. Aboodh, Solving Fourth Order Parabolic PDE with Variable Coefficients Using Aboodh Transform Homotopy Perturbation Method, Pure. Appl. Math. J. 4(5), (2015), 219-224.
• [20] K. S. Aboodh, The New Integral Transform ”AboodhTransform, Glob. J. Pure. Appl. Math. 9 (1), (2013), 35-43.
• [21] K. Wang and S. Liu, Application of new iterative transform method and modified fractional homotopy analysis transform method for fractional Fornberg-Whitham equation, J. Nonlinear Sci. Appl. 9 (2016), 2419-2433.
• [22] M. Hamdi Cherif and D. Ziane, A New Numerical Technique for Solving Systems of Nonlinear Fractional Partial Differential Equations, Int. J. Anal. Appl. 15(2), (2017), 188-197.
• [23] M. Khalid, M. Sultana, F. Zaidi and A. Uroosa,Solving linear and nonlinear Klein-Gordan equations by new perturbation iteration transform method, J. App. Eng. Math. 6(1), (2016), 115-125.
• [24] M. Tatari and M. Dehghan, On the convergence of He’s variational iteration method, J. Comp. Appl. Math. 207(1), (2007), 121-128.
• [25] M. Zurigat, Solving Fractional Oscillators Using Laplace Homotopy Analysis Method, Annals of the University of Craiova, Math. Comp. Sci. Series. 38 (2011), 1-11.
• [26] R. I. Nuruddeen and A. M Nass, Aboodh Decomposition Method and its Application in Solving Linear and Nonlinear Heat Equations, Europ. J. Advances in Engin. Techn. 3(7), (2016), 34-37.
• [27] S. A. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, J. Math. Annl. Appl. 4 (2001), 141-155.
• [28] S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004), 499-513.
• [29] S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
• [30] S. Kumara, A. Yildirim, Y. Khan and L. Weid, A fractional model of the diffusion equation and its analytical solution using Laplace transform, Scientia Iranica B. 19 (2012), 1117-1123.
• [31] S. Rathore, D. Kumar, J. Singh and S. Gupta, Homotopy Analysis Sumudu Transform Method for Nonlinear Equations, Int. J. Industrial Math. 4 (2012), 301-314.