Elzaki transform combined with variational iteration method for partial differential equations of fractional order

Year 2018, Volume: 1 Issue: 1, 102 - 108, 30.06.2018

Djelloul Ziane Tarig M Elzaki Mountassir Hamdi Cherif

https://doi.org/10.33401/fujma.415892

Cited By: 4

Abstract

The idea, which will be communicated through this paper is to make a change to the proposed method by Tarig M. Elzaki [6] and we extend it to solve nonlinear partial differential equations with time-fractional derivative. This document also includes illustrative examples show us how to apply this method, we also show the interest of combining these two methods is the speed of the calculates the terms, and not calculating the Lagrange multipliers.

Keywords

Caputo fractional derivative, Elzaki transform, Analytical solution, Variational iteration method

References

• [1] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
• [2] A. Neamaty, B. Agheli and R. Darzi, New Integral Transform for Solving Nonlinear Partial Differential Equations of fractional order, Theor. Appr. Appl. 10(1), (2014), 69-86., [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] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
• [7] J. H. He and X. H. Wu, Variational iteration method: new development and applications, Comput. Math. Appl. 54(7/8), (2007), 881-894.
• [8] J. H. He, A variational iteration approach to nonlinear problems and its applications, Mech. Appl. 20(1), (1998), 30-31.
• [9] J. H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput. 114(2/3), (2000), 115-123.
• [10] J. Singh, D. Kumar and Sushila, Homotopy Perturbation Sumudu Transform Method for Nonlinear Equations, Adv. Theor. Appl. Mech. 4 (2011), 165-175.
• [11] 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.
• [12] P. K. G. Bhadane and V. H. Pradhan, Elzaki Trnsform Homotopy Perturbation Method for Solving Porous Medium Equation, Int. J. Res. Eng. Tech. 2(12), (2013), 116-119.
• [13] M. Khalid, M. Sultana, F. Zaidi and U. Arshad, An Elzaki Transform Decomposition Algorithm Applied to a Class of Non-Linear Differential Equations, J. of Natural Sci. Res. 5 (2015), 48-55.
• [14] M. S. Rawashdeh S. Maitama, Solving Coupled System of Nonliear PDEs Using the Naturel Decomposition Method, Int. J. of Pure. Appl. Math. 92(5), (2014), 757-776.
• [15] M. Zurigat, Solving Fractional Oscillators Using Laplace Homotopy Analysis Method, Annals of the Univ. Craiova. Math. Comp. Sci. Series. 38 (2011), 1-11.
• [16] S. A. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, J. Math. Annl. Appl. 4 (2001), 141-155.
• [17] S. Kumar, 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.
• [18] 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.
• [19] T. M. Elzaki and S. M. Elzaki, On the Connections Between Laplace and ELzaki Transforms, Adv. Theor. Appl. Math. 6(1), (2014), 1-10.
• [20] T. M. Elzaki, S. M. Elzaki and E. A. Elnour, On the New Integral Transform “ELzaki Transform” Fundamental Properties Investigations and Applications, Glo. J. Math. Sci. 4 (2012), 1-13.
• [21] T. M. Elzaki, The Solution of Radial Diffusivity and Shock Wave Equations by Elzaki Variational Iteration Method, Int. J. Math. Analy. 9(22), (2015), 1065-1071.