New Integral Transform for Solving Nonlinear Partial Dierential Equations of fractional order

Neamaty, A.; Agheli, B.; Darzi, R.

New Integral Transform for Solving Nonlinear Partial Dierential Equations of fractional order

Document Type : Research Articles

Authors

¹ Department of Mathematics, University of Mazandaran, Babolsar, Iran

² Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

³ Department of Mathematics, Neka Branch, Islamic Azad University, Neka, Iran

Abstract

In this work, we have applied Elzaki transform and He's homotopy perturbation method to solve
partial dierential equation (PDEs) with time-fractional derivative. With help He's homotopy per-
turbation, we can handle the nonlinear terms. Further, we have applied this suggested He's homotopy
perturbation method in order to reformulate initial value problem. Some illustrative examples are
given in order to show the ability and simplicity of the approach. All numerical calculations in this
manuscript were performed on a PC applying some programs written in Maple.

References

[1] J.H. He, Homotopy perturbation technique, Computer Methods in Applied
Mechanics and Engineering, 178 (3-4) (1999), pp. 257{262.
[2] E. Babolian, A. Azizi, J. Saeidian , Some notes on using the homotopy
perturbation method for solving time-dependent dierential equations,
Mathematical and Computer Modelling, 50 ( 2009), pp. 213{224.
[3] S. Abbasbandy, Iterated He's homotopy perturbation method for quadratic
Riccati dierential equation, Journal of Computational and Applied
Mathematics, 175 (2006), pp. 581{589.
[4] Z. Odibat, Shaher Momani, The variational iteration method: an ecient
scheme for handling fractional partial dierential equations in uid
mechanics, Computers and Mathematics with Applications, 58 (2009),
pp. 2199{2208.
[5] G.M. Mophou, Existence and uniqueness of mild solutions to impulsive
fractional dierential equations, Nonlinear Analysis: Theory, Methods and
Applications, 72 (2010), pp. 1604{1615.
[6] F. Huang, F. Liu, The time fractional diusion and fractional advection-
dispersion equation, ANZIAM, 46 (2005), pp. 1{14.
[7] D. Taka^ci, A. Taka^ci, M.^Strboja, On the character of operational solutions
of the time-fractional diusion equation, Nonlinear Analysis: Theory,
Methods and Applications, 72 (2010), pp. 2367{2374.

[8] C. Xue, J. Nie, W. Tan, An exact solution of start-up ow for the fractional
generalized Burgers' uid in a porous half-space, Nonlinear Analysis:
Theory, Methods and Applications, 69 (2008), pp. 2086{2094.
[9] S. Guo, L. Mei, Y. Fang, Z. Qiu, Compacton and solitary pattern solutions
for nonlinear dispersive KdV-type equations involving Jumarie's fractional
derivative, Physics Letters A, 376 (2012), pp. 158{164.
[10] S. Guo, L. Mei, Y. Ling, Y. Sun, The improved fractional sub-equation
method and its applications to the space-time fractional dierential equations
in uid mechanics, Physics Letters A, 376 (2012), pp. 407{411.
[11] Y. Liu, Variational homotopy perturbation method for solving fractional
initial boundary value problems, Abstract and Applied Analysis, 2012
(2012), http://dx.doi.org/10.1155/2012/727031.
[12] T. M. Elzaki and S. M. Elzaki, Application of New Transform "Elzaki
Transform" to Partial Dierential Equations, Global Journal of Pure and
Applied Mathematics, 1 (2011), pp. 65{70.
[13] J. Zhang, A Sumudu based algorithm for solving dierential equations,
Computational Science Journal Moldova, 15(3), pp. 303 { 313.
[14] H. Eltayeb and A. kilicman, A Note on the Sumudu Transforms and
dierential Equations, Applied Mathematical Sciences, 4 (2010), pp. 1089{
1098.
[15] T. Elzaki, S. M. Elzaki, and E. M. A. Hilal, Elzaki and Sumudu Transforms
for Solving Some Dierential Equations, Global Journal of Pure and
Applied Mathematics, 8 (2012), pp. 167{173.
[16] I. Podlubny, Fractional dierential equations, Academic Press, San Diego,
CA, (1999).
[17] Tarig M. Elzaki, Salih M. Elzaki, and Eman M. A. Hilal, Elzaki and
Sumudu Transforms for Solving Some Dierential Equations, Global
Journal of Pure and Applied Mathematics, ISSN 0973-1768, Volume 8,
Number., 2 (2012), pp. 167{173.
[18] Z. Odibat, S. Momani, Numerical methods for nonlinear partial dierential
equations of fractional order, Applied Mathematical Modelling, 32 (2008),
pp. 28{39.