ORIGINAL_ARTICLE
Numerical solution of delay differential equations via operational matrices of hybrid of block-pulse functions and Bernstein polynomials
In this paper, we introduce hybrid of block-pulse functions and Bernstein polynomials and derive operational matrices of integration, dual, differentiation, product and delay of these hybrid functions by a general procedure that can be used for other polynomials or orthogonal functions. Then, we utilize them to solve delay differential equations and time-delay system. The method is based upon expanding various time-varying functions as their truncated hybrid functions. Illustrative examples are included to demonstrate the validity, efficiency and applicability of the method.
https://cmde.tabrizu.ac.ir/article_307_17289efa2e1cc599ee284fe876cc5c65.pdf
2013-10-01
78
95
Delay differential equation
Bernstein polynomial
Hybrid of block-pulse function
Operational matrix
M.
Behroozifar
m_behroozifar@nit.ac.ir
1
Babol University of Technology
AUTHOR
S. A.
Yousefi
s-yousefi@sbu.ac.ir
2
Shahid Beheshti University
AUTHOR
[1] M. I. Bhatti and P. Bracken, Solutions of dierential equations in a Bernstein polynomial
1
basis, Journal of Computational and Applied Mathematics, 205, (2007), 272 − 280.
2
[2] R.Y. Chang, M.L. Wang, Shifted Legendre direct method for variational problems, J.
3
Optim. Theory Appl. 30, (1983), 299-307.
4
[3] C.F. Chen, C.H. Hsiao, A Walsh series direct method for solving variational problems,
5
J. Franklin Inst. 300, (1975), 265-280.
6
[4] H.Y. Chung, Y.Y. Sun, Analysis of time-delay systems using an alternative method,
7
Int. J. Control 46, (1987), 1621-1631.
8
[5] K.B. Datta, B.M. Mohan, Orthogonal Functions in Systems and Control, World Scien-
9
tic, Singapore, 1995.
10
[6] J.S. Gu, W.S. Jiang, The Haar wavelets operational matrix of integration, Int. J. Syst.
11
Sci. 27, (1996), 623-628.
12
[7] I.R. Horng, J.H. Chou, Shifted Chebyshev direct method for solving variational prob-
13
lems, Int. J. Syst. Sci. 16, (1985), 855-861.
14
[8] C. Hwang, M.Y. Chen, Analysis of time-delay systems using the Galerkin method, Int.
15
J. Control 44, (1986), 847-866.
16
[9] C. Hwang, Y.P. Shih, Laguerre series direct method for variational problems, J. Optim.
17
Theory Appl. 39, (1983), 143-149.
18
[10] M. Jamshidi, C.M.Wang, A computational algorithm for large-scale nonlinear time-
19
delays systems, IEEE Trans. Syst. Man. Cybernetics SMC-14, (1984), 2-9.
20
[11] K. Maleknejad, Y. Mahmoudi, Numerical solution of linear Fredholm integral equation
21
by using hybrid Taylor and block-pulse functions, Appl. Math. Comput. 149, (2004),
22
[12] H.R. Marzban, H.R. Tabrizidooz, M. Razzaghi, Solution of variational problems via
23
hybrid of block-pulse and Lagrange interpolating, IET Control Theory Appl. 3,(2009),
24
1363-1369.
25
[13] H.R. Marzban, M. Razzaghi, Solution of time-varying delay systems by hybrid functions,
26
Math. and Com. in Sim. 64, (2004), 597-607.
27
[14] H.R. Marzban, M. Razzaghi, Optimal control of linear delay systems via hybrid of
28
block-pulse and Legendre polynomials, J. Franklin Inst. 34, (2004), 279-293.
29
[15] H.R. Marzban, M. Razzaghi, Analysis of Time-delay Systems via Hybrid of Block-pulse
30
Functions and Taylor Series, J. Vibration and Con. 11, (2005), 1455-1468.
31
[16] P.N. Paraskevopoulos, P. Sklavounos, and G.CH. Georgiou The operation matrix of
32
integration for Bessel functions, Journal of the Franklin Institute, 327, (1990), 329-341.
33
[17] M. Razzaghi, S. Youse, The Legendre wavelets operational matrix of integration, Int.
34
J. Syst. Sci. 32 (4),(2001), 495-502.
35
[18] M. Razzaghi, S. Youse, Sine-cosine wavelets operational matrix of integration and its
36
applications in the calculus of variations, Int. J. Syst. Sci. vol. 33, no. 10,(2002), 805-810.
37
[19] M. Razzaghi, M. Razzaghi, Fourier series direct method for variational problems, Int.
38
J. Control 48, (1988), 887-895.
39
[20] M. Razzaghi, H.R. Marzban, Direct method for variational problems via hybrid of
40
block-pulse and Chebyshev functions, Math. Probl. Eng. 6, (2000), 85-97.
41
[21] X.T. Wang, Numerical solution of delay systems containing inverse time by hybrid
42
functions, Appl. Math. Comput. 173, (2006), 535-546.
43
[22] X.T. Wang, Numerical solutions of optimal control for time delay systems by hybrid
44
of block-pulse functions and Legendre polynomials, Appl. Math. Comput. 184, (2007),
45
[23] S. A. Youse, M. Behroozifar, Operational matrices of Bernstein polynomials and their
46
applications, Int. J. Syst. Sci. 41, (2010), 709-716.
47
[24] S. A. Youse, M. Behroozifar, Mehdi Dehghan, The operational matrices of Bernstein
48
polynomials for solving the parabolic equation subject to specication of mass, Journal
49
of Computational and Applied Mathematics 235, (2011), 5272-5283.
50
[25] S. A. Youse, M. Behroozifar, Mehdi Dehghan, Numerical solution of the nonlinear
51
age-structured population models by using the operational matrices of Bernstein poly-
52
nomials, Applied Mathematical Modelling 36, (2012), 945-963.
53
ORIGINAL_ARTICLE
A fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations
In this paper we introduce a type of fractional-order polynomials based on the classical Chebyshev polynomials of the second kind (FCSs). Also we construct the operational matrix of fractional derivative of order $ gamma $ in the Caputo for FCSs and show that this matrix with the Tau method are utilized to reduce the solution of some fractional-order differential equations.
https://cmde.tabrizu.ac.ir/article_598_0ef9db406547966aff664044ad1a9c85.pdf
2013-10-01
96
107
Chebyshev polynomials
orthogonal system
fractional differential equation
fractional-order Chebyshev functions
Operational matrix
Mohammadreza
Ahmadi Darani
ahmadi.darani@sci.sku.ac.ir
1
Shahrekord University.
LEAD_AUTHOR
Mitra
Nasiri
mitra65nasiri@yahoo.com
2
Shahrekord University.
AUTHOR
[1] A.H. Bhrawy, A.S. Alo, The operational matrix of fractional integration for shifted
1
Chebyshev polynomials, Applied Mathematics Letters, 26 (2013) 25-31.
2
[2] E. H. Doha, A.H. Bhrawy, S. S. Ezz-Eldien , A Chebyshev spectral method based on
3
operational matrix for initial and boundary value problems of fractional order, Comput.
4
Math. Appl., 62 (2011) 2364-2373.
5
[3] A. Nkwanta and E.R. Barnes, Two Catalan-type Riordan arrays and their connections to
6
the Chebyshev polynomials of the rst kind, Journal of Integer Sequences, 15 (2012) 1-19.
7
[4] Fox, Lslie and Ian Bax Parker, Chebyshev Polynomials in Numerical Analysis, Oxford
8
university press, London, vol. 29, 1968
9
[5] K. Diethelm, The analysis of fractional dierential equations, Berlin: Springer-Verlag,
10
[6] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
11
[7] K.S. Miller, B. Ross, An Introduction to The Fractional Calculus and Fractional Dierential
12
Equations, Wiley, New York, 1993.
13
[8] I. Podlubny, Fractional Dierential Equations, Academic Press, San Diego, 1999.
14
[9] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential
15
Equations, Elsevier, San Diego, 2006.
16
[10] X. Li, Numerical solution of fractional dierential equations using cubic B-spline wavelet
17
collocation method, Commun Nonlinear Sci Numer Simulat., 17 (2012) 3934-3946.
18
[11] A. Saadatmandi, M. Dehghan, M. R. Azizi, The Sinc-Legendre collocation method for
19
a class of fractional convection-diusion equations with variable coecients, Commun
20
Nonlinear Sci Numer Simulat., 17 (2012) 4125-4136.
21
[12] M. Lakestani, M. Dehghan, S. Irandoust-pakchin, The construction of operational matrix
22
of fractional derivatives using B-spline functions, Commun Nonlinear Sci Numer Simulat,
23
17 (2012) 1149-1162.
24
[13] A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional-order
25
dierential equations, Comput. Math. Appl., 59 (2010) 1326-1336.
26
[14] S. Kazem, S. Abbasbandy, S. Kumar, Fractional-order Legendre functions for solving
27
fractional-order dierential equations, Applied Mathematical Modelling, In press,
28
http://dx.doi.org/10.1016/j.apm.2012.10.026
29
[15] A. Kayedi-Bardeh, M. R. Eslahchi, M. Dehghan, A method for obtaining the operational
30
matrix of fractional Jacobi functions and applications, Journal of Vibration and Control,
31
In press, DOI: 10.1177/1077546312467049.
32
[16] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods. Fundamentals
33
in Single Domains, Springer-Verlag, Berlin, 2006.
34
[17] Z .M. Odibat, N. T. Shawagfeh, Generalized Taylor's formula, Applied Mathematics
35
and Computation 186(2007) 286-293.
36
[18] S. Esmaeili, M. Shamsi, Y. Luchko, Numerical solution of fractional dierential equations
37
with a collocation method based on Muntz polynomials, Comput. Math. Appl., 62
38
(2011) 918-929.
39
ORIGINAL_ARTICLE
A new numerical scheme for solving systems of integro-differential equations
This paper has been devoted to apply the Reconstruction of Variational Iteration Method (RVIM) to handle the systems of integro-differential equations. RVIM has been induced with Laplace transform from the variational iteration method (VIM) which was developed from the Inokuti method. Actually, RVIM overcome to shortcoming of VIM method to determine the Lagrange multiplier. So that, RVIM method provides rapidly convergent successive approximations to the exact solution. The advantage of the RVIM in comparison with other methods is the simplicity of the computation without any restrictive assumptions. Numerical examples are presented to illustrate the procedure. Comparison with the homotopy perturbation method has also been pointed out.
https://cmde.tabrizu.ac.ir/article_588_4ab5f973114dd8420b112a7ca9ccee03.pdf
2013-10-01
108
119
System of integro-differential equations
Volterra equation
Reconstruction of variational iteration method
Homotopy perturbation method
Esmail
Hesameddini
1
Shiraz University of Technology
AUTHOR
Azam
Rahimi
2
Shiraz University of Technology
AUTHOR
[1] S. Abbasbandy, E. Shivanian, Application of variational iteration method for n-th order
1
integro-dierential equations, Verlag der Zeitschrift furnatur for Schung, 46a (2009),
2
[2] J. Biazar, E. Babolian and R. Islam, Solution of the system of Volterra integral equations
3
of the rst kind by Adomian decomposition method, Appl. Math. Comput., 139 (2003),
4
[3] A. Bratsos, M. Ehrhardt and T. h. Famelis, A discrete Adomian decomposition method
5
for discrete nonlinear Schrodinger equations, Appl. Math. Comput. , 197 (2008), 190-
6
[4] Y. S. Choi, R. Lui, An integro-dierential equation arising from an electrochemistry
7
model, Quart. Appl. Math. 4 (1997) 677686.
8
[5] J. A. Cuminato, A. D. Fitt, M. J. S. Mphaka, A. Nagamine, A singular integro-
9
dierential equation model for dryout in LMFBR boiler tubes, IMA J. Appl. Math.
10
75 (2009) 269290.
11
[6] C. M. Cushing, Integro-dierential Equations and Delay Models in Population Dynam-
12
ics, in: Lecture Notes in Biomathematics, vol. 20, Springer, NewYork, 1977.
13
[7] D. D. Ganji, A. Rajabi, Assessment of homotopy-perturbation and perturbation meth-
14
ods in heat radiation equations, Int. Commun., Heat and Mass Transfer 33 (3) (2006)
15
[8] D. D. Ganji, A. Sadighi, Application of Hes homotopy-perturbation method to nonlinear
16
coupled systems of reaction-diusion equations, Int. J. Nonlinear Sci. Numer. Simul.,
17
7 (4) (2006) 411418.
18
[9] A. Golbabai and M. Javidi, Application of He's homotopy perturbation method for n-th
19
order integro-dierential equations, Appl. Math. Comput., 190 (2007), 1409-1416.
20
[10] J. H. He, Homotopy perturbation technique, Comput. Math. Appl. Mech. Eng., 178
21
(1999), 257-262.
22
[11] E. Hesameddini and H. Latizadeh, Reconstruction of variational iteration algorithm
23
using the Laplace transform, Int. J. of Non. Sci. and Numer. Sim., 10 (2009), 1365-
24
[12] A. J. Jerri, Introduction to integral equations with applications, Seconded, Wiley Inter-
25
science, 1999.
26
[13] K. Maleknejad and Y. Mahmoudi, Taylor polynomial solution of high-order nonlinear
27
Volterra-Fredholm integro-dierential equations, Appl. Math. Comput., 145 (2003), 641-
28
[14] H. Thieme, A model for the spatial spread of an epidemic, J. Math. Biol., 4 (1977)
29
[15] S. Q. Wang and J. H. He, Variational iteration method for solving integro-dierential
30
equations, Phys. Lett., 367 (2007), 188-191.
31
[16] A. M. Wazwaz, A reliable modication of Adomaion's decomposition method, Appl.
32
Math. Comput., 102 (1999), 77-86.
33
[17] X. Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics,
34
vol. 16, Springer, 2003.
35
ORIGINAL_ARTICLE
Extremal Positive Solutions For The Distributed Order Fractional Hybrid Differential Equations
In this article, we prove the existence of extremal positive solution for the distributed order fractional hybrid differential equation$$int_{0}^{1}b(q)D^{q}[frac{x(t)}{f(t,x(t))}]dq=g(t,x(t)),$$using a fixed point theorem in the Banach algebras. This proof is given in two cases of the continuous and discontinuous function $g$, under the generalized Lipschitz and Caratheodory conditions.
https://cmde.tabrizu.ac.ir/article_597_aec82c22b058d25675f6cd533c9fac23.pdf
2013-10-01
120
134
Fractional hybrid differential equations
Distributed order
Extremal solutions
Banach algebra
Hossein
Noroozi
hono1458@yahoo.com
1
Shahrekord University
AUTHOR
Alireza
Ansari
alireza_1038@yahoo.com
2
Shahrekord University
LEAD_AUTHOR
[1] C. Canuto C, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid
1
Dynamic. Englewood Cliffs, NJ: Prentice-Hall, 1998.
2
[2] M. Caputo, Elasticita e Dissipazione, Zanichelli, Bologna, Italy, 1969.
3
[3] M. Caputo, Mean fractional-order-derivatives differential equations and filters, Annali
4
dell universita di Ferrara. Nuova Serie. Sezione VII. Scienze Mathematiche, 41 (1995)
5
[4] M. Caputo, Distributed order differential equations modeling dielectric induction and
6
diffusion, Fractional Calculas and Applied Analysis, 4 (2001) 421-442.
7
[5] A. V. Bobyelv and C. Cercignani, The inverse laplace transform of some analytic func-
8
tions with an application to the eternal solutions of the Boltzmann equation, Applied
9
Mathematics Letters, 15(7) (2002) 807-813.
10
[6] B. Davis, Integral Transforms and their applications, 3rd edition, Springer-Verlag, New
11
York, 2001.
12
[7] B. C. Dhage, A nonlinear alternative in Banach algebras with applications to functional
13
differential equations, Nonlinear Functional Analysis and Applications, 8 (2004) 563-
14
[8] B. C. Dhage, Fixed point theorems in ordered Banach algebras and application,
15
Panamerican Mathematical Journal, 9(4) (1999) 93-102.
16
[9] B. C. Dhage, Nonlinear quadratic first order functional integro-differential equation with
17
periodic boundary conditions, Dynamic Systems and Applications, 18 (2009), 303-322.
18
[10] B. C. Dhage, Theorical approximation methods for hybrid differential equations, Dy-
19
namic Systems and Applications, 20 (2011) 455-478.
20
[11] B. C. Dhage, V. Lakshmikantham, Basic results on hybrid differential equations, Non-
21
linear Analysis Hybrid, 4 (2010) 414-424.
22
[12] B. C. Dhage, V. Lakshmikantham, Quadratic perturbations of boundary value problems
23
of second order ordinary differential equations, Differential Equations and Applications,
24
2(4) (2010) 465-486.
25
[13] S. Heikkila, V. Lakshmikantham, Monotone Iterative Technique For Nonlinear Discon-
26
tinues Differential Equations, Marcel Dekker Inc, New York, 1994.
27
[14] H. Noroozi, A. Ansari, M. Sh. Dahaghin, Existence Results For The Distributed Or-
28
der Fractional Hybrid Differential Equations, Abstract and Applied Analyis, Article ID
29
163648, 2012.
30
[15] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
31
[16] A. Saadatmandi, M. Dehghan, A Legendre collocation method for fractional integro-
32
differential equations, Journal of Vibration and Control, 17(13), (2011) 2050-2058.
33
[17] Y. Zhao, S. Sun, Z. Han, Q. Li, Theory of fractional hybrid differential equations,
34
Computer and Mathematics with Applications, 62 (2011) 1312-1324.
35
ORIGINAL_ARTICLE
Lie symmetry analysis for Kawahara-KdV equations
We introduce a new solution for Kawahara-KdV equations. The Lie group analysis is used to carry out the integration of this equations. The similarity reductions and exact solutions are obtained based on the optimal system method.
https://cmde.tabrizu.ac.ir/article_971_26e06445c2a60e5b2c79259ca7107f29.pdf
2013-10-01
135
145
Lie symmetries
Symmetry analysis
Optimal system
Infinitesimal Generators
Kawahara-KdV equation
Ali
Haji Badali
haji.badali@bonabu.ac.ir
1
University of Bonab
LEAD_AUTHOR
Mir Sajjad
Hashemi
hashemi@bonabu.ac.ir
2
University of Bonab
AUTHOR
Maryam
Ghahremani
m_ghahremani90@yahoo.com
3
University of Bonab
AUTHOR
[1] G. W. Bluman, A. F. Cheviakov and S. C. Anco, Applications of Symmetry Methods to
1
Partial Dierential Equations, Springer Science Business Media, LLC (2010).
2
[2] G. W. Bluman and S. Kumei, Symmetries and Dierential Equations, Springer-Verlag,
3
World Publishing Corp., (1989).
4
[3] B. J. Cantwell, Introduction to Symmetry Analysis, Cambridge University Press, (2002).
5
[4] W. Gang-Wei, L. Xi-Qiang and Z. Ying-Yuan, Lie Symmetry Analysis and Invariant
6
Solutions of the Generalized Fifth-order KdV Equation with Variable Coecients, J. Appl.
7
Math. Informatics Vol. 31 (2013), No. 1-2, pp. 229-239.
8
[5] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Dierential Equations,
9
Wiley, Chichester, (1999).
10
[6] N. H. Ibragimov, Invariant Lagrangians and a New Method of Integration of Nonlinear
11
Equations, J. Math. Anal. Appl. 304 (2005) 212-235.
12
[7] N. H. Ibragimov and S.V. Meleshko, A solution to the Problem of Invariants for Parabolic
13
Equations, Commun Nonlinear Sci Numer Simulat 14. (2009) 2551-2558.
14
[8] N. H. Ibragimov, Symmetries, Lagrangian and Conservation Laws for the Maxwell Equa-
15
tions, Acta. Appl. Math. 105 (2009) 157-187.
16
[9] C. M. Khalique and K. R. Adem, Exact Solution of the (2+1)-Dimensional Zakharov-
17
Kuznetsov Modied Equal Width Equation Using Lie Group Analysis, Computer Modelling.
18
54 (2011) 184-189.
19
[10] H. Liu, J. Li, L. Liu and Y. Wei, Group Classications, Optimal Systems and Exact
20
Solutions to the Generalized Thomas Equations, J. Math. Anal. Appl. 383 (2011) 400-408.
21
[11] H. Liu and J. Li, Lie Symmetry Analysis and Exact Solutions for the Extended mKdV
22
Equation, Acta Appl Math. 109 (2010) 1107-1119.
23
[12] F. Natali, A Note on the Stability for Kawahara-KdV Type Equations, Lett. 23
24
(2010)591-596.
25
[13] M. C. Nucci, P. G. L. Leach and K. Andriopoulos, Lie Symmetries, Quantisation and
26
c-Isochronous Nonlinear Oscillators, J. Math. Anal. Appl. 319 (2006) 357-368.
27
[14] M. C. Nucci, Nonclassical Symmetries and Backlund Transformations, J. Math. Anal.
28
Appl. 178 (1993) 294-300.
29
[15] M. C. Nucci, Iterations of the Nonclassical Symmetries Method and Conditional
30
LieBacklund Symmetries, J. Phys. A: Math. Gen. 29 (1996) 8117-8122.
31
[16] P. J. Olver, Application of Lie Groups to Dierential Equations, New York: Springer-
32
Verlag; (1993).
33
[17] V. Torrisi and M. C. Nucci, Application of Lie Group Analysis to a Mathematical Model
34
which Describes HIV Transmission, in: The Geometrical Study of Dierential Equations,
35
in: J. A. Leslie, T. P. Robart (Eds.), Contemp. Math., vol. 285 Amer. Math. Soc., Provi-
36
dence, RI, (2001) 11-20.
37
[18] J. Zhang, Y. Li, Symmetries and First Integrals of Dierential Equations, Acta. Appl.
38
Math. 103 (2008) 147-159.
39
ORIGINAL_ARTICLE
Solitary Wave solutions of the BK equation and ALWW system by using the first integral method
Solitary wave solutions to the Broer-Kaup equations and approximate long water wave equations are considered challenging by using the rst integral method.The exact solutions obtained during the present investigation are new. This method can be applied to nonintegrable equations as well as to integrable ones.
https://cmde.tabrizu.ac.ir/article_972_0e66618e8fc1bfb24782bd08578d30de.pdf
2013-10-01
146
157
First integral method
Broer-Kaup equations
Approximate long water wave equations
Ahmad
Neirameh
neirameh.edu@gmail.com
1
Department of mathematics,Gonbad University
LEAD_AUTHOR
[1] W.X. Ma, Travelling wave solutions to a seventh order generalized KdV equation, Phys.
1
Lett. A. 180 (1993) 221 224:
2
[2] W. Mal iet, Solitary wave solutions of nonlinear wave equations, Amer. J. Phys.
3
60(7) (1992) 650 654:
4
[3] A.H. Khater, W. Mal iet, D.K. Callebaut and E.S. Kamel, The tanh method, a simple
5
transformation and exact analytical solutions for nonlinear reactiondi usion equations,
6
Chaos Solitons Fractals 14(3) (2002) 513 522:
7
[4] A.M. Wazwaz, Two reliable methods for solving variants of the KdV equation with
8
compact and noncompact structures. Chaos Solitons Fractals, 28(2) (2006) 454 462:
9
[5] W. X. Ma and B. Fuchssteiner, Explicit and exact solutions to a Kolmogorov- Petrovskii-
10
Piskunov equation, Int. J. Non-Linear Mech. 31 (1996) 329-338.
11
[6] S.A. El-Wakil and M.A. Abdou , New exact travelling wave solutions using modi ed
12
extended tanh-function method, Chaos Solitons Fractals, 31(4) (2007) 840-852.
13
[7] E. Fan, Extended tanh-function method and its applications to nonlinear equations,
14
Phys. Lett. A. 277(4 -5) (2000) 212 -218.
15
[8] A.M.Wazwaz, The tanh-function method: Solitons and periodic solutions for the Dodd-
16
Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations, Chaos Solitons and
17
Fractals 25(1) (2005) 55-63.
18
[9] T.C. Xia ,B. Li and H.Q. Zhang, New explicit and exact solutions for the Nizhnik-
19
Novikov-Vesselov equation, Appl. Math. E-Notes 1, (2001) 139-142.
20
[10] A.M. Wazwaz, The sine-cosine method for obtaining solutions with compact and non-
21
compact structures, Appl. Math. Comput. 159(2) (2004) 559-576:
22
[11] A.M. Wazwaz, A sine-cosine method for handling nonlinear wave equations, Math.
23
Comput. Modelling, 40(5-6) (2004) 499-508:
24
[12] E. Yusufoglu and A. Bekir, Solitons and periodic solutions of coupled nonlinear evolution
25
equations by using Sine-Cosine method, Internat. J. Comput. Math. 83(12) (2006) 915-
26
[13] M. Inc and M. Ergut, Periodic wave solutions for the generalized shallow water
27
wave equation by the improved Jacobi elliptic function method, Appl. Math. E-Notes
28
5 (2005) 89 96:
29
[14] Zhang Sheng, The periodic wave solutions for the (2 + 1) dimensional Konopelchenko-
30
Dubrovsky equations, Chaos Solitons Fractals, 30 (2006) 1213-1220.
31
[15] W. X. Ma and J.-H. Lee, A transformed rational function method and exact solutions to
32
the (3+1)-dimensional Jimbo-Miwa equation, Chaos Solitons Fractals, 42 (2009) 1356-
33
[16] Z.S Feng, X.H Wang, The rst integral method to the two-dimensional Burgers-KdV
34
equation, Phys. Lett. A. 308 (2002) 173-178.
35
[17] T.R. Ding and C.Z. Li, Ordinary di erential equations. Peking University Press, Peking,
36
[18] Z.S. Feng, X.H Wang, The rst integral method to the two-dimensional Burgers-KdV
37
equation, Phys. Lett. A. 308 (2002) 173 - 178.
38
[19] K.R. Raslan, The rst integral method for solving some important nonlinear partial
39
di erential equations, Nonlinear Dynam 53 (2008) 281:
40
[20] D.J. Kaup, A higher order water wave equation and method for solving it, Progress of
41
Theoretical physics 54 (1975) 396 - 408.
42
[21] Mingliang Wang, Jinliang Zhang, Xiangzheng Li, Application of the (G0
43
G )-expansion
44
to travelling wave solutions of the Broer-Kaup and the approximate long water wave
45
equations, Appl. Math. Comput. 206 (2008) 321 - 326:
46
[22] G.B. Whitham, Variational methods and application to water waves, Proceedings of the
47
Royal Society of London Series A 299 (1967) 6-25:
48
[23] L.J.F. Broer, Approximate equations for long water waves, Applied Scienti c Research
49
31 (1975) 377-395.
50