Numerical solution of fuzzy Hunter-Saxton equation by using Adomian decomposition and Homotopy analysis methods

Subject Areas : Applied Mathematics

Sh Sadigh behzadi ¹

1 - Department of Mathematics, Islamic Azad University, Qazvin Branch, Qazvin, Iran

Received: 2015-09-21 Accepted : 2015-09-21 Published : 2015-12-01

Keywords: Homotopy Analysis Method, fuzzy number, Generalized dierentiability, Adomian decomposition method, Hunter-Saxton equation, Hukuhara- dierence,

Abstract :

In this paper, a fuzzy Hunter-Saxton equation is solved by using the Adomian'sdecomposition method (ADM) and homotopy analysis method (HAM). Theapproximation solution of this equation is calculated in the form of series whichits components are computed by applying a recursive relation. The existenceand uniqueness of the solution and the convergence of the proposed methodsare proved. A numerical example is studied to demonstrate the accuracy ofthe presented methods.In this paper, a fuzzy Hunter-Saxton equation is solved by using the Adomian'sdecomposition method (ADM) and homotopy analysis method (HAM). Theapproximation solution of this equation is calculated in the form of series whichits components are computed by applying a recursive relation. The existenceand uniqueness of the solution and the convergence of the proposed methodsare proved. A numerical example is studied to demonstrate the accuracy ofthe presented methods.

References:

[1] Allahviranloo. T, Ahmady. N, Ahmady. E (2009). A method for
solving n-th order fuzzy linear dierential equations: Comput. Math.
Appl 86: 730-742.
[2] Abbasbandy. S, Allahviranloo. T (2002). Numerical solutions of
fuzzy dierential equations by taylor method: Comput. Methods
Appl. Math. 2: 113-124.
[3] Abbasbany. S (2008). Homptopy analysis method for generalized
Benjamin-Bona-Mahony equation: Zeitschri fur angewandte
Mathematik und Physik ( ZAMP). 59: 51-62.
[4] Abbasbany. S (2010). Homptopy analysis method for the Kawahara
equation: Nonlinear Analysis: Real Wrorld Applications. 11: 307-
312.
[5] Bressan. A, Constantin. A (2005). Global solutions of the Hunter-
Saxton equation: SIAM J. Math. Anal. 37: 996-1026.
[6] Beals. R, Sattinger. D, Szmigielski. J (2001). Inverse scattering
solutions of the Hunter-Saxton equation: Applicable Analysis. 78
: 255-269.
[7] Behriy.S.H, Hashish.H, E-Kalla.I.L , A.Elsaid (2007). A new
algorithm for the decomposition solution of nonlinear dierential
equations: Appl. Math. Comput. 54: 459-466.
[8] El-KallaI.L (2008). Convergence of the Adomian method applied
to a class of nonlinear integral equations: Appl.Math.Comput. 21:
372-376.
[9] Fariborzi Araghi M.A, Sadigh Behzadi.Sh (2009). Solving nonlinear
Volterra-Fredholm integral dierential equations using the modied
Adomian decomposition method: Comput. Methods in Appl. Math.
9: 1-11.
[10] Fariborzi Araghi.M.A ,
Sadigh Behzadi.Sh (2010). Numerical solution of nonlinear Volterra-
Fredholm integro-dierential equations using Homotopy analysis
method: Journal of Applied Mathematics and Computing, DOI:
10.1080/00207161003770394.

[11] Guan.C (2012). Global weak solutions for a periodic two-
component PERIODIC Hunter-Saxton system: Quarterly of
Applied Mathematics. 2: 285-297.
[12] Hunter.J.K, Saxton.R (1991). Dynamics of director elds: SIAM J.
Appl. Math. 51: 1498-1521.
[13] Li.J, Zhang.K (2011). Global existence of dissipative solutions to
the Hunter-Saxton equation via vanishing viscosity: J. Dierential
Equations. 250: 1427-1447.
[14] Khesin.B, Lenells.J, Misiolek.G (2013). Generalized HunterSaxton
equation and the geometry of the group of circle dieomorphisms:
Math. Ann, DOI 10.1007/s00208-008-0250-3.
[15] Lenells.J (2008). Poisson structure of a modied Hunter-Saxton
equation: J. Phys. A: Math. Theor. 41: 1-9.
[16] Lenells.J (2007). The Hunter-Saxton equation describes the geodesic
ow on a sphere: Journal of Geometry and Physics. 57: 2049-2064.
[17] LiaoS.J (2003). Beyond Perturbation: Introduction to the Homotopy
Analysis Method: Chapman and Hall/CRC Press,Boca Raton.
[18] LiaoS.J (2009). Notes on the homotopy analysis method: some
denitions and theorems: Communication in Nonlinear Science and
Numerical Simulation. 14:983-997.
[19] Nadjakhah.M, Ahangari.F (2013). Symmetry Analysis and
Conservation Laws for the Hunter-Saxton Equation: Commun.
Theor. Phys, doi:10.1088/0253-6102/59/3/16.
[20] PenskoiA. V (2002). Lagrangian time-discretization of the Hunter-
Saxton equation: Physics Letters A. 304: 157-167.
[21] BehzadiSh.S (2010). The convergence of homotopy methods
for nonlinear Klein-Gordon equation: J.Appl.Math.Informatics.
28:1227-1237.
[22] behzadiSh.S , Fariborzi Araghi.M.A (2011).
The use of iterative methods for solving Naveir-Stokes equation:
J.Appl.Math.Informatics. 29: 1-15.

3] BehzadiSh.S, Fariborzi AraghiM.A (2011). Numerical solution
for solving Burger's-Fisher equation by using Iterative Methods:
Mathematical and Computational Applications. 16:443-455.
[24] BehzadiSh.S (2011). Numerical solution of fuzzy Camassa-Holm
equation by using homotopy analysis method: Joournal of Applied
Analysis and Computations. 1:1-9.
[25] BehzadiSh.S (2011). Numerical solution
of Hunter-Saxton equation by using iterative methods: Journal of
Information and Mathematical Sciences. 3: 127-143.
[26] BehzadiSh.S (2011). Solving Schrodinger equation by using modied
variational iteration and homotopy analysis methods: Journal of
Applied Analysis and Computations. 4: 427-437.
[27] Wazwaz.A.M (2001). Construction of solitary wave solution and
rational solutions for the KdV equation by ADM.: Chaos,Solution
and fractals. 12: 2283-2293.
[28] YinZ.Y (2004). On the structure of solutions to the periodic Hunter-
Saxton equation: SIAM J. Math. Anal. 36: 272-283.