Analytical solution of the Hunter-Saxton equation using the reduced dierential transform method

Subject Areas : Applied Mathematics

H. Rouhparvar ¹

1 - Department of Mathematics, College of Technical and Engineering, Saveh Branch, Islamic Azad University, Saveh, Iran

Received: 2015-07-19 Accepted : 2015-10-27 Published : 2016-12-01

Keywords: Reduced differential transform method, Hunter-Saxton, equation, Taylor series,

Abstract :

In this paper, the reduced dierential transform method is investigated fora nonlinear partial dierential equation modeling nematic liquid crystals, itis called the Hunter-Saxton equation. The main advantage of this methodis that it can be applied directly to nonlinear dierential equations withoutrequiring linearization, discretization, or perturbation. It is a semi analytical-numerical method that formulizes Taylor series in a very dierent manner.The numerical results denote that reduced dierential transform method isecient and accurate for Hunter-Saxton equation.

References:

[1] I. H. Abdel-Halim Hassan, Vedat Suat Ertrk, Applying dierential
transformation method to the one-dimensional planar Bratu problem,
Int. J. Contemp. Math. Sciences, 2(30) (2007), 1493-1504.
[2] I. H. Abdel-Halim Hassan, Dierential transformation technique for
solving higher-order initial value problems, Appl. Math. Comput., 154
(2004), 299-311.
[3] I. H. Abdel-Halim Hassan, Application to dierential transformation
method for solving systems of dierential equations, Appl. Math.
Modell., 32(12) (2008), 2552-2559.
[4] A. Arikoglu and I. Ozkol, Solution of boundary value problems for
integro-dierential equations by using dierential transform method,
Appl. Math. Comput., 168 (2005), 1145-1158.
[5] F. Ayaz, Solutions of the systems of dierential equations by
dierential transform method, Appl. Math. Comput., 147 (2004), 547-
567.
[6] F. Ayaz, On the two-dimensional dierential transform method,
Appl. Math. Comput., 143 (2003) 361-374.
[7] R. Camassa and D. D. Holm, An integrable shallow water equation
with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
[8] C. K. Chen and S. H. Ho, Application of dierential transformation
to eigenvalue problems, Appl. Math. Comput., 79 (1996), 173-188.
[9] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal
shallow water equations, Acta Mathematics, 181 (1998), 229-243.

[10] J. K. Hunter and R. Saxton, Dynamics of director elds, SIAM J.
Appl. Math, 51 (1991), 1498-1521.
[11] M. J. Jang and C. L. Chen, Analysis of the response of a
strongly nonlinear damped system using a dierential transformation
technique, Appl. Math. Comput., 88 (1997), 137-151.
[12] M. J. Jang and C. K. Chen, Two-dimensional dierential
transformation method for partial dierantial equations, Appl. Math.
Copmut., 121 (2001), 261-270.
[13] R. S. Johnson and Camassa-Holm, Korteweg-de Vries and related
models for water waves, J. Fluid Mech., 455 (2002), 63-82.
[14] F. Kangalgil and F. Ayaz, Solitary wave solutions for the KdV and
mKdV equations by dierential transform method, Chaos Solitons &
Fractals, 41 (2009), 464-472.
[15] Y. Keskin and G. Oturanc, Reduced Dierential Transform Method
for Partial Dierential Equations, International Journal of Nonlinear
Sciences and Numerical Simulation, 10 (6) (2009), 741-749.
[16] Y. Keskin and G. Oturanc, Reduced dierential transform method
for solving linear and nonlinear wave equations, Iranian Journal of
Science & Technology, Transaction A, 34(A2) 2010, 113-122.
[17] Y. Keskin and G. Oturanc, Reduced Dierential Transform Method
for fractional partial dierential equations, Nonlinear Science Letters
A, 1(2) (2010), 61-72.
[18] Y. Keskin, Ph.D. Thesis, Selcuk University (to appear).
[19] H. Liu and Yongzhong Song, Dierential transform method
applied to high index dierential algebraic equations, Appl. Math.
Comput.,184 (2007), 748-753.
[20] J. K. Zhou, Dierential transformation and its application for
electrical circuits, Huarjung University PressWuuhahn, China (1986),
(in Chinese).