ORIGINAL_ARTICLE
A High Order Approximation of the Two Dimensional Acoustic Wave Equation with Discontinuous Coefficients
This paper concerns with the modeling and construction of a fifth order method for two dimensional acoustic wave equation in heterogenous media. The method is based on a standard discretization of the problem on smooth regions and a nonstandard method for nonsmooth regions. The construction of the nonstandard method is based on the special treatment of the interface using suitable jump conditions. We derive the required linear systems for evaluation of the coefficients of such a nonstandard method. The given novel modeling provides an overall fifth order numerical model for two dimensional acoustic wave equation with discontinuous coefficients.
http://cmde.tabrizu.ac.ir/article_231_e22984f025d7dacb0d33f0f8384d3d84.pdf
2013-12-01T11:23:20
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1
15
Interface methods
two dimensional acoustic wave equation
high order methods
Lax-Wendroff method
WENO
discontinuous coefficients
Jump conditions
Javad
Farzi
true
1
Sahand University Of Technology
Sahand University Of Technology
Sahand University Of Technology
LEAD_AUTHOR
[1] M. Dehghan and A. Mohebbi, High Order Implicit Collocation Method for the Solution of Two-Dimensional Linear Hyperbolic Equation , Numerical Methods for Partial Differential Equations, Vol. 25, Issue 1, 232-243, 2009.
1
[2] J. Farzi and S. M. Hosseini, A High Order Method for the Solution of One Way Wave Equation in Heterogenous Media, Far East J. Appl. Math., Vol. 36, No. 3, 317-330, 2009.
2
[3] B. Gustafsson and P. Wahlund Time Compact High Order Difference Methods for Wave Propagation, 2D, . Sci. Comput., Vol. 25, pp. 195-211, 2005.
3
[4] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems., Cambridge University Press, Cambridge, 2004.
4
[5] R. J. LeVeque, Wave Propagation Algorithms for Multidimensional Hyperbolic systems., J. comp. phys. 131,pp. 327-353, 1997.
5
[6] R. J. LeVeque and C. Zhang, The Immersed interface methods for wave equations with discontinuous coefficients., Wave Motion, 25, PP. 237-263, 1997.
6
[7] J. Qiu and C.-W. Shu, Finite difference WENO schemes with Lax-Wendroff type time discretizations., SIAM J. Sci. Comput. 24, pp. 2185-2198, 2003.
7
[8] C.-W. Shu, Efficient Implementation of Weighted ENO Schemes., J. Comput. Phys., 126, pp. 202-228, 1996.
8
[9] C. Zhang and W.W. Symes, A Forth Order Method for Acoustic Waves in Heterogenous Media, Proceedings of International Conference on Mathematical and Numerical Aspects of Wave Propagation, 1998.
9
ORIGINAL_ARTICLE
Chebyshev Spectral Collocation Method for Computing Numerical Solution of Telegraph Equation
In this paper, the Chebyshev spectral collocation method(CSCM) for one-dimensional linear hyperbolic telegraph equation is presented. Chebyshev spectral collocation method have become very useful in providing highly accurate solutions to partial differential equations. A straightforward implementation of these methods involves the use of spectral differentiation matrices. Firstly, we transform telegraph equation to system of partial differential equations with initial condition. Using Chebyshev differentiation matrices yields a system of ordinary differential equations. Secondly, we apply fourth order Runge-Kutta formula for the numerical integration of the system of ODEs. Numerical results verified the high accuracy of the new method, and its competitive ability compared with other newly appeared methods.
http://cmde.tabrizu.ac.ir/article_242_25ad79d0795b8c4807a3d86288473135.pdf
2013-12-20T11:23:20
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29
Chebyshev spectral collocation method
telegraph equation
numerical results
Runge-Kutta formula
M.
Javidi
mo_javidi@yahoo.com
true
1
University of Tabriz
University of Tabriz
University of Tabriz
LEAD_AUTHOR
[1] A. Mohebbi, M. Dehaghan, High order compact solution of the one dimensional lin-
1
ear hyperbolic equation, Numerical method for partial differential equations, 24 (2008)
2
[2] F. Gao, C. Chi, Unconditionally stable difference scheme for a one-space dimensional
3
linear hyperbolic equation, Applied Mathematics and Computation 187 (2007) 12721276.
4
[3] A. Saadatmandi, M. Dehghan, Numerical solution of hyperbolic telegraph equation
5
using the Chebyshev Tau method, Numer. Methods Partial Differential Equations 26
6
(1) (2010) 239-252.
7
[4] S.A. Yousefi, Legendre multi wavelet Galerkin method for solving the hyperbolic
8
telegraph equation, Numerical Method for Partial Differential Equations, (2008).
9
doi:10.1002/num.
10
[5] M. Dehghan, A. Ghesmati, Solution of the second-order one-dimensional hyperbolic
11
telegraph equation by using the dual reciprocity boundary integral equation (DRBIE)
12
method, Engineering Analysis with Boundary Elements 34 (2010) 5159.
13
[6] S. Das, P.K. Gupta, Homotopy analysis method for solving fractional hyperbolic par-
14
tial differential equations, International Journal of Computer Mathematics 88 (2011)
15
[7] M.A. Abdou, Adomian decomposition method for solving the telegraph equation in
16
charged particle transport, J. Quant. Spectrosc. Radiat. Transfer 95 (2005) 407-414.
17
[8] M. Lakestani, B. N. Saray, Numerical solution of telegraph equation using interpolating
18
scaling functions, Computers Mathematics with Applications, 60(2010) 1964-1972.
19
[9] R.K. Mohanty, An unconditionally stable difference scheme for the one-space dimen-
20
sional linear hyperbolic equation, Appl. Math. Lett. 17 (2004) 101-105.
21
[10] R.K. Mohanty, An unconditionally stable finite difference formula for a linear second
22
order one space dimensional hyperbolic equation with variable coefficients, Appl. Math.
23
Comput. 165 (2005) 229-236.
24
[11] L. Lapidus, G.F. Pinder, Numerical Solution of Partial Differential Equations in Science
25
and Engineering, Wiley, New York, 1982.
26
[12] I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs,
27
Communications in Nonlinear Science and Numerical Simulation 14 (2009) 674684.
28
[13] A. Borhanifar, Reza Abazari, An unconditionally stable parallel difference scheme for
29
telegraph equation scheme for telegraph equation, Math. Probl. Eng. (2009) Article ID
30
[14] M. Dehghan, A. Shokri, A numerical method for solving the hyperbolic telegraph equa-
31
tion, Numer. Methods Partial Differential Equations 24 (2008) 10801093.
32
[15] M. Dehghan, M. Lakestani, The use of Chebyshev cardinal functions for solution of the
33
second-order one-dimensional telegraph equation, Numer. Methods Partial Differential
34
Equations 25 (2009) 931938.
35
[16] J. Biazar, M. Eslami, Analytic solution for Telegraph equation by differential transform
36
method, Physics Letters A, 374(29)(2010) 2904-2906.
37
[17] L.N. Trefethen, Spectral methods in MATLAB, SIAM, Philadelphia(2000).
38
[18] W.S. Don and A. Solomonoff, Accuracy and speed in computing the Chebyshev collo-
39
cation derivative, SIAM J. of Sci. Coput., 16 No. 4(1995) 1253-1268.
40
[19] C. Canuto ,A. Quarteroni, M.Y. Hussaini and T. Zang, Spectral method in fluied me-
41
chanics, Springer-Verlag, New York (1988).
42
[20] J.P. Boyd, Chebyshev and Fourier spectral methods, Lecture notes in engineering, 49,
43
Springer-verlag, Berlin(1989).
44
[21] R. Baltensperger and M.R. Trummer, Spectral differencing with a twist, SIAM J. of
45
Sci. Comp., 24,no. 5(2003),1465-1487.
46
[22] R. Baltensperger and J.P. Berrut, The errors in calculating the pseudospectral differen-
47
tiation matrices for Chebyshev-Gauss-Lobatto point, Comput. Math. Appl., 37(1999),41-48.
48
ORIGINAL_ARTICLE
2-stage explicit total variation diminishing preserving Runge-Kutta methods
In this paper, we investigate the total variation diminishing property for a class of 2-stage explicit Rung-Kutta methods of order two (RK2) when applied to the numerical solution of special nonlinear initial value problems (IVPs) for (ODEs). Schemes preserving the essential physical property of diminishing total variation are of great importance in practice. Such schemes are free of spurious oscillations around discontinuities.
http://cmde.tabrizu.ac.ir/article_259_f275211af12c25479a94ac0787dc3e03.pdf
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30
38
Initial value problem
Method of line
Total-variation-diminishing
Rung-Kutta methods
M.
Mehdizadeh Khalsaraei
muhammad.mehdizadeh@gmail.com
true
1
University of Maragheh
University of Maragheh
University of Maragheh
LEAD_AUTHOR
F.
Khodadosti
fayyaz64dr@gmail.com
true
2
University of Maragheh
University of Maragheh
University of Maragheh
AUTHOR
[1] R. Anguelov, Total variation diminishing nonstandard finite difference schemes for conservation laws, J. Math. Comput 51 (2010), 160-166.
1
[2] M. Mehdizadeh Khalsaraei, An improvement on the positivity results for 2-stage explicit Runge-Kutta methods, J. Comput. Appl. Math 235 (2010), 137-143.
2
[3] B. Koren, A robust upwind discretization for advection, diffusion and source terms. In: Numerical Methods for Advection-Diffusion Problems, Notes on Numerical Fluid Mechanics 45 (1993), 117-138.
3
[4] C.W. Shu, Total-variation-diminishing time discretizations, SIAM J. Sci. Statist. Comput 9 (1988), 1073-1084.
4
[5] A. Harten, High resolution schemes for hyperbolic conservation laws, SJournal of Computational Physics 49 (1983), 357-393.
5
[6] W. Hundsdorfer, J. G. Verwer Numerical Solution of Time-Dependent Advection Diffusion-Reaction Equation, Springer (2003)
6
ORIGINAL_ARTICLE
Existence and multiplicity of positive solutions for a coupled system of perturbed nonlinear fractional differential equations
In this paper, we consider a coupled system of nonlinear fractional differential equations (FDEs), such that both equations have a particular perturbed terms. Using emph{Leray-Schauder} fixed point theorem, we investigate the existence and multiplicity of positive solutions for this system.
http://cmde.tabrizu.ac.ir/article_260_01737ac7f4a2458cfc9c83932cc6ebdb.pdf
2013-12-20T11:23:20
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39
54
Kazem
Ghanbari
true
1
Sahand University of
Technology
Sahand University of
Technology
Sahand University of
Technology
AUTHOR
Yousef
Gholami
true
2
Sahand University of
Technology
Sahand University of
Technology
Sahand University of
Technology
AUTHOR
[1] B. Ahmad, Juan J. Nieto, Existence results for a coupled system of nonlinear fractional
1
differential equations with three point boundary conditions, Computers and Mathemat-
2
ics with Applications 58 (2009), 1838-1843.
3
[2] Z. Bai, Existence of solutions for some third-order boundary-value problems, Electronic
4
Journal of Differential Equations No. 25 (2008) 1-6.
5
54 KAZEM GHANBARI AND YOUSEF GHOLAMI
6
[3] K. Ghanbari, Y. Gholami, Existence and multiplicity of positive solutions for m-point
7
nonlinear fractional differential equations on the half line, Electronic Journal of Differ-
8
ential equations No. 238 (2012) 1-15.
9
[4] K. Ghanbari, Y. Gholami, Existence and nonexistence results of positive solutions for
10
nonlinear fractional eigenvalue problem, Journal of Fractional Calculus and Applications
11
Vol.4 No.2 (2013) 1-12.
12
[5] K. Ghanbari, Y. Gholami, H. Mirzaei, Existence and multiplicity results of positive
13
solutions for boundary value problems of nonlinear fractional differential equations,
14
Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical
15
Analysis 20 (2013) 543-558.
16
[6] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of fractional
17
Differential Equations, North-Holland mathematics studies, Elsvier science 204 (2006).
18
[7] K. Q. Lan, Multiple positive solutions of semilinear differential equations with singu-
19
larities, J. Lond. Math. Soc. 63 (2001) 690-704.
20
[8] K. S. Miller, B. Ross, An Introduction to fractional calculus and fractioal differential
21
equation, John Wiley, New York (1993).
22
[9] I. Poudlobny, Fractional Differential Equations, Mathematics in Science and Applica-
23
tions Academic Press 19 (1999).
24
[10] M. ur Rehman, R. A. Khan, A note on boundary value problems for a coupled system
25
of fractional differential equations, Computers and Mathematics with Applications 61(2011) 2630-2637.
26
[11] S. Zhang, Positive solutions for boundary value problem of nonlinear fractional differ-
27
ential equations, Electronic Journal of Differential Equations 36 (2006) 1-12.
28
[12] S. Liang, J. Zhang, Existence of multiple positive solutions for m-point fractional bound-
29
ary value problems on an infinite interval, Mathematical and Computer modelling 54
30
(2011) 1334-1346.
31
[13] S. Zhang, Unbounded solutions to a boundary value problem of fractional order on the
32
half-line, Computers and mathematics with Applications 61 (2011) 1079-1087.
33
[14] X. Zhao, W. Ge, Unbounde solutions for a fractional boundary value problem on the
34
infinite interval, Acta Appl Math 109 (2010) 495-505.
35
[15] S. Zhang, Existence of positive solutions for some class of fractional differential equa-
36
tions, J. Math. Anal. Appl 278 (2003) 136-148.
37
[16] S. Zhang, G. Han, The existence of a positive solution for a nonlinear fractional differ-
38
ential equation, J. Math. Anal. Appl 252 (2000) 804-812.
39
[17] X. Zhang, L. Liu, Y. Wu, Multiple positive solutions of a singular fractional differential
40
equation with negatively perturbed term, Mathematical and Computer Modelling 55
41
(2012) 1263-1274.
42
[18] Y. Zhang, Z. Bai, T. Feng, Existence results for a coupled system of nonlinear fractional
43
three-point boundary value problems at resonance, Computers and Mathematics with
44
Applications 61 (2011) 1032-1047.
45
ORIGINAL_ARTICLE
Parameter determination in a parabolic inverse problem in general dimensions
It is well known that the parabolic partial differential equations in two or more space dimensions with overspecified boundary data, feature in the mathematical modeling of many phenomena. In this article, an inverse problem of determining an unknown time-dependent source term of a parabolic equation in general dimensions is considered. Employing some transformations, we change the inverse problem to a Volterra integral equation of convolution-type. By using an explicit procedure based on Sinc function properties, the resulting integral equation is replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number and the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some numerical examples are given to demonstrate the computational efficiency of the method.
http://cmde.tabrizu.ac.ir/article_277_64be06a89a3e7813e006630e885ee04c.pdf
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55
70
Reza
Zolfaghari
rzolfaghari@iust.ac.ir
true
1
Salman Farsi University of Kazerun
Salman Farsi University of Kazerun
Salman Farsi University of Kazerun
AUTHOR
[1] P. Amore, A variational Sinc collocation method for Strong-Coupling problems, Journal
1
of Physics A, 39 (22) (2006) 349-355.
2
[2] J.R. Cannon, Y. Lin, Determination of parameter p(t) in Holder classes for some semi-
3
linear parabolic equations, Inverse Problems 4, (1988) 595-606.
4
[3] J.R. Cannon, Y. Lin, An inverse problem of nding a parameter in a semi-linear heat
5
equation, Journal of Mathematical Analysis and Applications, 145(2) (1990) 470-484.
6
[4] J.R. Cannon, Y. Lin, S. Wang, Determination of source parameter in parabolic equa-
7
tions, Meccanica 27, (1992) 85-94.
8
[5] J.R. Cannon, Y. Lin, Determination of a parameter p(t) in some quasilinear parabolic
9
dierential equations, Inverse Problems 4, (1988) 35-45.
10
[6] J.R. Cannon, The one dimensional heat equation, 1984 (Reading, MA: Addison-Wesley).
11
[7] M. Dehghan, M. Tatari, Determination of a control parameter in a one-dimensional par-
12
abolic equation using the method of radial basis functions, Mathematical and Computer
13
Modelling 44, (2006) 1160-1168.
14
[8] M. Dehghan, An inverse problem of nding a source parameter in a semilinear parabolic
15
equation, Applied Mathematical Modelling 25, (2001) 743-754.
16
[9] M. Dehghan, A. Saadatmandi, A tau method for the one-dimensional parabolic inverse
17
problem subject to temperature overspecication, Computational Mathematics with Ap-
18
plications, 52 (2006) 933-940.
19
[10] M. Dehghan, Finding a control parameter in one-dimensional parabolic equations, Ap-
20
plied Mathematics and Computation, 135 (2003) 491-503.
21
[11] M. Dehghan, Numerical solution of one-dimensional parabolic inverse problem, Applied
22
Mathematics and Computation, 136 (2003) 333-344.
23
[12] M. Dehghan, Determination of a control function in three-dimensional parabolic equa-
24
tions, Mathematics and Computers in Simulation, 61 (2003) 89-100.
25
[13] M. Dehghan, Determination of a control parameter in the two-dimensional diusion
26
equation,Applied Numerical Mathematics, 37 (2001) 489-502.
27
[14] M. Dehghan, Fourth order techniques for identing a control parameter in the parabolic
28
equations, International Journal of Engineering Science, 40 (2002) 433-447.
29
[15] M. Dehghan, Method of lines solutions of the parabolic inverse problem with an over-
30
specication at apoint, Numerical Algorithms, 50 (2009) 417-437.
31
[16] M. Dehghan, Finite dierence schemes for two-dimensional parabolic inverse problem
32
with temperature overspecication, International Journal of Computer Mathematics,
33
75 (3) (2000) 339-349.
34
[17] F. Li, Z. Wu, Ch. Ye, A nite dierence solution to a two-dimensional parabolic inverse
35
problem,Applied Mathematical Modelling, 36 (2012) 2303-2313.
36
[18] Y. Lin, An inverse problem for a cleass of quasilinear parabolic equations, SIAM Journal
37
on Mathematical Analysis, 22(1) (1991) 146-156.
38
[19] J. Lund, K. Bowers, Sinc methods for quadrature and dierential equations,SIAM,
39
Philadelphia, 1992.
40
[20] J. Lund, C. Vogel, A Fully-Galerkin method for the solution of an inverse problem in a
41
parabolic partial dierential equation, Inverse Problems, 6 (1990) 205-217.
42
[21] A.I. Prilepko, D.G. Orlovskii, Determination of the evolution parameter of an equation
43
and inverse problems of mathematical physics, Part I. Journal of Dierential Equations,
44
21 (1985) 119-129 [and part II, 21 (1985) 694-701].
45
[22] A.I. Prilepko, V.V. Soloev, Solvability of the inverse boundary value problem of nd-
46
ing a coecient of a lower order term in a parabolic equation. Journal of Dierential
47
Equations, 23(1) (1987) 136-143.
48
[23] W. Rundell, Determination of an unknownnon-homogeneous term in a linear partial
49
dierential equation from overspecied boundary data, Applicable Analysis, 10 (1980)
50
[24] A. Shidfar, R. Zolfaghari, J. Damirchi, Application of Sinc-collocation method for solv-
51
ing an inverse problem, Journal of Computational and Applied Mathematics, 233 (2009)
52
[25] A. Shidfar, R. Zolfaghari, Determination of an unknown function in a parabolic in-
53
verse problem by Sinc-collocation method,Numerical Methods for Partial Dierential
54
Equations, 27 (6) (2011) 1584-1598.
55
[26] R. Smith, K. Bowers, A Sinc-Galerkin estimation of diusivity in parabolic problems,
56
Inverse Problems, 9 (1993).
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[27] F. Stenger, Numerical methods based on Sinc and analytic functions, Springer, New
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York, 1993.
59
[28] S. Wang, Y. Lin, A nite dierence solution to an invese problem determining a control
60
function in a parabolic partial dierential equation, Inverse Problems, 5 (1989) 631-640.
61
[29] S. A. Youse, M. Dehghan, Legendre multiscaling functions for solving the one-
62
dimensional parabolic inverse problem, Numerical Methods for Partial Dierential
63
Equations, 25 (2009) 1502-1510.
64
ORIGINAL_ARTICLE
The modified simplest equation method and its application
In this paper, the modified simplest equation method is successfully implemented to find travelling wave solutions of the generalized forms $B(n,1)$ and $B(-n,1)$ of Burgers equation. This method is direct, effective and easy to calculate, and it is a powerful mathematical tool for obtaining exact travelling wave solutions of the generalized forms $B(n,1)$ and $B(-n,1)$ of Burgers equation and can be used to solve other nonlinear partial differential equations in mathematical physics.
http://cmde.tabrizu.ac.ir/article_306_fcbed350396f3b4bf56c373881e123f7.pdf
2013-12-20T11:23:20
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77
The modified simplest equation method
traveling wave solutions
homogeneous balance
Solitary wave solutions
The generalized forms $B(n
1)$ and $B(-n
1)$ of Burgers equation
M.
Akbari
true
1
University of Guilan
University of Guilan
University of Guilan
AUTHOR
[1] E. Fan, Extended tanh-function method and its applications to nonlinear equations,
1
Physics Letters A, 277(4-5) (2000) 212-218.
2
[2] E. G. Fan, Extended tanh-function method and its applications to nonlinear equations,
3
Phys. Lett. A, 277 (2000) 212-218.
4
[3] E. Fan and H. Zhang, A note on the homogeneous balance method, Phys. Lett. A, 246
5
(1998) 403-406.
6
[4] J. H. He and X.H. Wu, Exp-function method and for nonlinear wave equations. Chaos,
7
Solitons and Fractals, 30 (2006) 700-708.
8
[5] A. J. M. Jawad, M. D. Petkovic and A. Biswas, Modied simple equation method for
9
nonlinear evolution equations, Appl. Math. Comput., 217 (2010) 869-877.
10
[6] S. K. Liu, Z. T. Fu, S. D. Liu and Q. Zhao, Jacobi elliptic function expansion method
11
and periodic wave solutions of nonlinear wave equatins, Phys. Lett. A, 289 (2001) 72-76.
12
[7] N. K. Vitanov, Z. I. Dimitrova and H. Kantz, Modied method simplest equation and
13
application to nonlinear PDFs, Appl. Math. Comput., 216 (2010) 2587-2595.
14
[8] N. K. Vitanov, Modied method simplest equation poerful tool for obtaining exact
15
and approximate traveling-wave solutions of nonlinear PDFs, Commun Nonlinear. Sci.
16
Numer. Simulat., 16 (2011) 1179-1185.
17
[9] N. K. Vitanov and Z. I. Dimitrova, Application of the method of simplest equation for
18
obtaining exact traveling-wave solutions for two classes of model PDFs from ecoloy and
19
population dynamics. Commun. Nonlinear Sci. Numer. Simulat., 15 (2010) 2836-2845.
20
[10] M.Wang, X. Li and J. Zhang, The (G′
21
G )-expansion method and travelling wave solutions
22
of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008) 417-
23
[11] M. L. Wang and X. Z. Li, Applications of F-expansion to periodic wave solutions for a
24
new Hamiltonian amplitude equation, Chaos Soliton Fract., 24 (2005) 1257-1268.
25
[12] E. M. E. Zayed, A note on the modied simple equation method applied to Sharma-
26
Tasso-Olver equation, Appl. Math. Comput., 218 (2011) 3962-3964.
27
[13] E. M. E. Zayed and S.A.H. Ibrahim, Exact solutions of nonlinear evolution equations
28
in mathematical physics using the modied simple equation method, chinese physics
29
Letters, 29(6) (2012), Article ID 060201.
30
[14] H. Zhang, New application of the (G′,G )-expansion method, Commun. Nonlinear Sci.
31
Numer. Simul., 14 (2009) 3220-3225.
32