Document Type: Research Articles

**Author**

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

**Abstract**

In this paper, a fuzzy Hunter-Saxton equation is solved by using the Adomian's

decomposition method (ADM) and homotopy analysis method (HAM). The

approximation solution of this equation is calculated in the form of series which

its components are computed by applying a recursive relation. The existence

and uniqueness of the solution and the convergence of the proposed methods

are proved. A numerical example is studied to demonstrate the accuracy of

the presented methods.

In this paper, a fuzzy Hunter-Saxton equation is solved by using the Adomian's

decomposition method (ADM) and homotopy analysis method (HAM). The

approximation solution of this equation is calculated in the form of series which

its components are computed by applying a recursive relation. The existence

and uniqueness of the solution and the convergence of the proposed methods

are proved. A numerical example is studied to demonstrate the accuracy of

the presented methods.

decomposition method (ADM) and homotopy analysis method (HAM). The

approximation solution of this equation is calculated in the form of series which

its components are computed by applying a recursive relation. The existence

and uniqueness of the solution and the convergence of the proposed methods

are proved. A numerical example is studied to demonstrate the accuracy of

the presented methods.

In this paper, a fuzzy Hunter-Saxton equation is solved by using the Adomian's

decomposition method (ADM) and homotopy analysis method (HAM). The

approximation solution of this equation is calculated in the form of series which

its components are computed by applying a recursive relation. The existence

and uniqueness of the solution and the convergence of the proposed methods

are proved. A numerical example is studied to demonstrate the accuracy of

the presented methods.

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fuzzy dierential equations by taylor method: Comput. Methods

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[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

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[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 modied

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.

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 modied

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 modied 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

denitions and theorems: Communication in Nonlinear Science and

Numerical Simulation. 14:983-997.

[19] Nadjakhah.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.

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 modied 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

denitions and theorems: Communication in Nonlinear Science and

Numerical Simulation. 14:983-997.

[19] Nadjakhah.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 modied

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.

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 modied

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.

Volume 9, Issue 2

Summer and Autumn 2013

Pages 115-133