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.
http://cmde.tabrizu.ac.ir/article_307_17289efa2e1cc599ee284fe876cc5c65.pdf
2013-10-01T11:23:20
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78
95
Delay differential equation
Bernstein polynomial
Hybrid of block-pulse function
Operational matrix
M.
Behroozifar
m_behroozifar@nit.ac.ir
true
1
Babol University of Technology
Babol University of Technology
Babol University of Technology
AUTHOR
S. A.
Yousefi
s-yousefi@sbu.ac.ir
true
2
Shahid Beheshti University
Shahid Beheshti University
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.
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Int. J. Control 46, (1987), 1621-1631.
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tic, Singapore, 1995.
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Sci. 27, (1996), 623-628.
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lems, Int. J. Syst. Sci. 16, (1985), 855-861.
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by using hybrid Taylor and block-pulse functions, Appl. Math. Comput. 149, (2004),
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hybrid of block-pulse and Lagrange interpolating, IET Control Theory Appl. 3,(2009),
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1363-1369.
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block-pulse and Legendre polynomials, J. Franklin Inst. 34, (2004), 279-293.
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Functions and Taylor Series, J. Vibration and Con. 11, (2005), 1455-1468.
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applications in the calculus of variations, Int. J. Syst. Sci. vol. 33, no. 10,(2002), 805-810.
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J. Control 48, (1988), 887-895.
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block-pulse and Chebyshev functions, Math. Probl. Eng. 6, (2000), 85-97.
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[21] X.T. Wang, Numerical solution of delay systems containing inverse time by hybrid
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functions, Appl. Math. Comput. 173, (2006), 535-546.
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[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),
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[23] S. A. Youse, M. Behroozifar, Operational matrices of Bernstein polynomials and their
46
applications, Int. J. Syst. Sci. 41, (2010), 709-716.
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[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
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of Computational and Applied Mathematics 235, (2011), 5272-5283.
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[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.
http://cmde.tabrizu.ac.ir/article_598_0ef9db406547966aff664044ad1a9c85.pdf
2013-10-01T11:23:20
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96
107
Chebyshev polynomials
orthogonal system
fractional differential equation
fractional-order Chebyshev functions
Operational matrix
Mohammadreza
Ahmadi Darani
ahmadi.darani@sci.sku.ac.ir
true
1
Shahrekord University.
Shahrekord University.
Shahrekord University.
LEAD_AUTHOR
Mitra
Nasiri
mitra65nasiri@yahoo.com
true
2
Shahrekord University.
Shahrekord University.
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.
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[4] Fox, Lslie and Ian Bax Parker, Chebyshev Polynomials in Numerical Analysis, Oxford
8
university press, London, vol. 29, 1968
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Equations, Wiley, New York, 1993.
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15
Equations, Elsevier, San Diego, 2006.
16
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17
collocation method, Commun Nonlinear Sci Numer Simulat., 17 (2012) 3934-3946.
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[11] A. Saadatmandi, M. Dehghan, M. R. Azizi, The Sinc-Legendre collocation method for
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a class of fractional convection-diusion equations with variable coecients, Commun
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Nonlinear Sci Numer Simulat., 17 (2012) 4125-4136.
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[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.
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25
dierential equations, Comput. Math. Appl., 59 (2010) 1326-1336.
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27
fractional-order dierential equations, Applied Mathematical Modelling, In press,
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http://dx.doi.org/10.1016/j.apm.2012.10.026
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[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,
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In press, DOI: 10.1177/1077546312467049.
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[16] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods. Fundamentals
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in Single Domains, Springer-Verlag, Berlin, 2006.
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and Computation 186(2007) 286-293.
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37
with a collocation method based on Muntz polynomials, Comput. Math. Appl., 62
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(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.
http://cmde.tabrizu.ac.ir/article_588_4ab5f973114dd8420b112a7ca9ccee03.pdf
2013-10-01T11:23:20
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108
119
System of integro-differential equations
Volterra equation
Reconstruction of variational iteration method
Homotopy Perturbation method
Esmail
Hesameddini
true
1
Shiraz University of Technology
Shiraz University of Technology
Shiraz University of Technology
AUTHOR
Azam
Rahimi
true
2
Shiraz University of Technology
Shiraz University of Technology
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
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of the rst kind by Adomian decomposition method, Appl. Math. Comput., 139 (2003),
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[3] A. Bratsos, M. Ehrhardt and T. h. Famelis, A discrete Adomian decomposition method
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for discrete nonlinear Schrodinger equations, Appl. Math. Comput. , 197 (2008), 190-
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model, Quart. Appl. Math. 4 (1997) 677686.
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[5] J. A. Cuminato, A. D. Fitt, M. J. S. Mphaka, A. Nagamine, A singular integro-
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dierential equation model for dryout in LMFBR boiler tubes, IMA J. Appl. Math.
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75 (2009) 269290.
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ics, in: Lecture Notes in Biomathematics, vol. 20, Springer, NewYork, 1977.
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ods in heat radiation equations, Int. Commun., Heat and Mass Transfer 33 (3) (2006)
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coupled systems of reaction-diusion equations, Int. J. Nonlinear Sci. Numer. Simul.,
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7 (4) (2006) 411418.
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[9] A. Golbabai and M. Javidi, Application of He's homotopy perturbation method for n-th
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order integro-dierential equations, Appl. Math. Comput., 190 (2007), 1409-1416.
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(1999), 257-262.
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using the Laplace transform, Int. J. of Non. Sci. and Numer. Sim., 10 (2009), 1365-
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[12] A. J. Jerri, Introduction to integral equations with applications, Seconded, Wiley Inter-
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science, 1999.
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Volterra-Fredholm integro-dierential equations, Appl. Math. Comput., 145 (2003), 641-
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equations, Phys. Lett., 367 (2007), 188-191.
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Math. Comput., 102 (1999), 77-86.
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[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.
http://cmde.tabrizu.ac.ir/article_597_aec82c22b058d25675f6cd533c9fac23.pdf
2013-10-01T11:23:20
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120
134
Fractional hybrid differential equations
Distributed order
Extremal solutions
Banach algebra
Hossein
Noroozi
hono1458@yahoo.com
true
1
Shahrekord University
Shahrekord University
Shahrekord University
AUTHOR
Alireza
Ansari
alireza_1038@yahoo.com
true
2
Shahrekord University
Shahrekord University
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.
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3
[3] M. Caputo, Mean fractional-order-derivatives differential equations and filters, Annali
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dell universita di Ferrara. Nuova Serie. Sezione VII. Scienze Mathematiche, 41 (1995)
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[4] M. Caputo, Distributed order differential equations modeling dielectric induction and
6
diffusion, Fractional Calculas and Applied Analysis, 4 (2001) 421-442.
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[5] A. V. Bobyelv and C. Cercignani, The inverse laplace transform of some analytic func-
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tions with an application to the eternal solutions of the Boltzmann equation, Applied
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Mathematics Letters, 15(7) (2002) 807-813.
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[6] B. Davis, Integral Transforms and their applications, 3rd edition, Springer-Verlag, New
11
York, 2001.
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[7] B. C. Dhage, A nonlinear alternative in Banach algebras with applications to functional
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differential equations, Nonlinear Functional Analysis and Applications, 8 (2004) 563-
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Panamerican Mathematical Journal, 9(4) (1999) 93-102.
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periodic boundary conditions, Dynamic Systems and Applications, 18 (2009), 303-322.
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[10] B. C. Dhage, Theorical approximation methods for hybrid differential equations, Dy-
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namic Systems and Applications, 20 (2011) 455-478.
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21
linear Analysis Hybrid, 4 (2010) 414-424.
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[12] B. C. Dhage, V. Lakshmikantham, Quadratic perturbations of boundary value problems
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of second order ordinary differential equations, Differential Equations and Applications,
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2(4) (2010) 465-486.
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26
tinues Differential Equations, Marcel Dekker Inc, New York, 1994.
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[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
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163648, 2012.
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[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.
http://cmde.tabrizu.ac.ir/article_971_26e06445c2a60e5b2c79259ca7107f29.pdf
2013-10-01T11:23:20
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135
145
Lie symmetries
Symmetry analysis
Optimal system
Infinitesimal Generators
Kawahara-KdV equation
Ali
Haji Badali
haji.badali@bonabu.ac.ir
true
1
University of Bonab
University of Bonab
University of Bonab
LEAD_AUTHOR
Mir Sajjad
Hashemi
hashemi@bonabu.ac.ir
true
2
University of Bonab
University of Bonab
University of Bonab
AUTHOR
Maryam
Ghahremani
m_ghahremani90@yahoo.com
true
3
University of Bonab
University of Bonab
University of Bonab
AUTHOR
[1] G. W. Bluman, A. F. Cheviakov and S. C. Anco, Applications of Symmetry Methods to
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Partial Dierential Equations, Springer Science Business Media, LLC (2010).
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World Publishing Corp., (1989).
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[3] B. J. Cantwell, Introduction to Symmetry Analysis, Cambridge University Press, (2002).
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[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.
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[5] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Dierential Equations,
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Wiley, Chichester, (1999).
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[6] N. H. Ibragimov, Invariant Lagrangians and a New Method of Integration of Nonlinear
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Equations, J. Math. Anal. Appl. 304 (2005) 212-235.
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Equations, Commun Nonlinear Sci Numer Simulat 14. (2009) 2551-2558.
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[8] N. H. Ibragimov, Symmetries, Lagrangian and Conservation Laws for the Maxwell Equa-
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tions, Acta. Appl. Math. 105 (2009) 157-187.
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[9] C. M. Khalique and K. R. Adem, Exact Solution of the (2+1)-Dimensional Zakharov-
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Kuznetsov Modied Equal Width Equation Using Lie Group Analysis, Computer Modelling.
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54 (2011) 184-189.
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Solutions to the Generalized Thomas Equations, J. Math. Anal. Appl. 383 (2011) 400-408.
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Equation, Acta Appl Math. 109 (2010) 1107-1119.
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(2010)591-596.
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c-Isochronous Nonlinear Oscillators, J. Math. Anal. Appl. 319 (2006) 357-368.
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Verlag; (1993).
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which Describes HIV Transmission, in: The Geometrical Study of Dierential Equations,
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in: J. A. Leslie, T. P. Robart (Eds.), Contemp. Math., vol. 285 Amer. Math. Soc., Provi-
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dence, RI, (2001) 11-20.
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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.
http://cmde.tabrizu.ac.ir/article_972_0e66618e8fc1bfb24782bd08578d30de.pdf
2013-10-01T11:23:20
2018-08-17T11:23:20
146
157
First integral method
Broer-Kaup equations
Approximate long water wave equations
Ahmad
Neirameh
neirameh.edu@gmail.com
true
1
Department of mathematics,Gonbad University
Department of mathematics,Gonbad University
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:
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transformation and exact analytical solutions for nonlinear reactiondi usion equations,
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Chaos Solitons Fractals 14(3) (2002) 513 522:
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[4] A.M. Wazwaz, Two reliable methods for solving variants of the KdV equation with
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compact and noncompact structures. Chaos Solitons Fractals, 28(2) (2006) 454 462:
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Piskunov equation, Int. J. Non-Linear Mech. 31 (1996) 329-338.
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extended tanh-function method, Chaos Solitons Fractals, 31(4) (2007) 840-852.
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Phys. Lett. A. 277(4 -5) (2000) 212 -218.
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[8] A.M.Wazwaz, The tanh-function method: Solitons and periodic solutions for the Dodd-
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Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations, Chaos Solitons and
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Fractals 25(1) (2005) 55-63.
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Novikov-Vesselov equation, Appl. Math. E-Notes 1, (2001) 139-142.
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[10] A.M. Wazwaz, The sine-cosine method for obtaining solutions with compact and non-
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compact structures, Appl. Math. Comput. 159(2) (2004) 559-576:
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Comput. Modelling, 40(5-6) (2004) 499-508:
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equations by using Sine-Cosine method, Internat. J. Comput. Math. 83(12) (2006) 915-
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