Formulation and Solution of Space-Time Fractional KdV-Burgers Equation
Abulwafa Essam M., Elgarayhi Ahmed M., Mahmoud Abeer A., Tawfik Ashraf M.
Theoretical Physics Research Group, Physics Department
Faculty of Science, Mansoura University, Mansoura 35516, Egypt
E-mail: abulwafa@mans.edu.eg
Received:
Received: 6 August 2013; revised: 18 November 2013; accepted: 6 December 2013; published online: 17 December 2013
DOI: 10.12921/cmst.2013.19.04.235-243
OAI: oai:lib.psnc.pl:460
Abstract:
The space-time fractional KdV-Burgers equation has been derived using the semi-inverse method and Agrawal’s variational method. The modified Riemann-Liouville definition is used for the fractional differential operators. The derived fractional equation is solved using the fractional sub-equation method.
Key words:
Fractional Euler-Lagrange equation, fractional sub-equation method, modified Riemann-Liouville fractional definition, space-time fractional KdV-Burgers equation
References:
[1] J.A. Tenreiro Machado, Fractional generalization of memris-
tor and higher order elements, Communications in Nonlinear
Science & Numerical Simulation 18(2), 264-275 (2013).
[2] D. Baleanu, J.A. Tenreiro Machado, A.C.J. Luo, Editors,
Fractional Dynamics and Control, Springer, New York, 2012.
[3] M.-A. Polo-Labarrios, G. Espinosa-Paredes, Application of
the fractional neutron point kinetic equation: Start-up of
a nuclear reactor, Annals of Nuclear Energy 46(1), 37-42
(2012).
[4] A. Kadem, D. Baleanu, Analytical method based on Walsh
function combined with orthogonal polynomial for fractional
transport equation, Communications in Nonlinear Science &
Numerical Simulation 15(3), 491-501 (2010).
[5] I.S. Jesus, J.A. Tenreiro Machado, R.S. Barbosa, Control of
a heat diffusion system through a fractional order nonlin-
ear algorithm, Computers & Mathematics with Applications
59(5), 1687-1694 (2010).
[6] D. Baleanu, Z.B. Güvenç, J.A. Tenreiro Machado, Editors,
New Trends in Nanotechnology and Fractional Calculus Ap-
plications, Springer, Dordrecht, 2010.
[7] F. Mainardi, Fractional Calculus and Waves in Linear Vis-
coelasticity: An Introduction to Mathematical Models, Impe-
rial College Press, London, UK, 2010.
[8] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional
Calculus: Models and Numerical Methods, vol. 3 of series
on Complexity, Nonlinearity and Chaos, World Scientific
Publishing, Hackensack, NJ, 2012.
[9] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Ap-
plications of Fractional Differential Equations, vol. 204
of North-Holland Mathematics Studies, Elsevier Science B.
V., Amsterdam, 2006.
[10] I. Podlubny, Fractional Differential Equations, Academic
Press, San Diego, 1999.
[11] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals
and derivatives: theory and applications, Gordon and Breach
Science Publishers, Amsterdam, 1993.
[12] G. Jumarie, On the fractional solution of the equation
f(x+y)=f(x)f(y) and its application to fractional Laplace’s
transform, Applied Mathematics and Computation 219(4),
1625-1643 (2012).
[13] G. Jumarie, An Approach to Differential Geometry of Frac-
tional Order via Modified Riemann-Liouville Derivative,
Acta Mathematica Sinica, English Series 28(9), 1741-1768
(2012).
[14] G. Jumarie, An approach via fractional analysis to non-
linearity induced by coarse-graining in space, Nonlinear
Analysis: Real World Applications 11(1), 535-546 (2010).
[15] G. Jumarie, Table of some basic fractional calculus formulae
derived from a modified Riemann-Liouville derivative for
non-differentiable functions, Applied Mathematics Letters
22(3), 378-385 (2009).
[16] G.-w. Wang, X.-q. Liu, Y.-y. Zhang, Lie symmetry analysis
to the time fractional generalized fifth-order KdV equation,
Communications in Nonlinear Science & Numerical Simula-
tion 18(9), 2321-2326 (2013).
[17] H. Wang, T.-C. Xia, The fractional supertrace identity and its
application to the super Jaulent-Miodek hierarchy, Commu-
nications in Nonlinear Science Numerical Simulation 18(10),
2859-2867 (2013).
[18] Z. Hammouch, T. Mekkaoui, Travelling-wave solutions for
some fractional partial differential equation by means of gen-
eralized trigonometry functions, Int. J. Applied Mathematical
Research 1(2), 206-212 (2012).
[19] R. Almeida, D.F.M. Torres, Fractional variational calculus
for nondifferentiable functions, Computers & Mathematics
with Applications 61(10), 3097-3104 (2011).
[20] G.-c. Wu, A fractional variational iteration method for solv-
ing fractional nonlinear differential equations, Computers &
Mathematics with Applications 61(8), 2186-2190 (2011).
[21] F. Riewe, Nonconservative Lagrangian and Hamiltonian Me-
chanics, Physical Review E 53(2), 1890-1899 (1996).
[22] F. Riewe, Mechanics with Fractional Derivatives, Physical
Review E 55(3), 3581-3592 (1997).
[23] O.P. Agrawal, Formulation of Euler-Lagrange Equations for
Fractional Variational Problems, J. Mathematical Analysis
& Applications 272(1), 368-379 (2002).
[24] O.P. Agrawal, Fractional Variational Calculus in Terms of
Riesz Fractional Derivatives, J. Physics A: Mathematical &
Theoretical 40, 6287-6303 (2007).
[25] O.P. Agrawal, S.I. Muslih, D. Baleanu, Generalized varia-
tional calculus in terms of multi-parameters fractional deriva-
tives, Communications in Nonlinear Science & Numerical
Simulation 16(12), 4756-4767 (2011).
[26] M.A.E. Herzallah, S.I. Muslih, D. Baleanu, E.M. Rabei,
Hamilton-Jacobi and fractional like action with time scaling,
Nonlinear Dynamics 66(4), 549-555 (2011).
[27] M.A.E. Herzallah, D. Baleanu, Fractional Euler-Lagrange
equations revisited, Nonlinear Dynamics 69(3), 977-982
(2012).
[28] S.A. El-Wakil, E.M. Abulwafa, M.A. Zahran, A.A. Mah-
moud, Formulation of Some Fractional Evolution Equations
used in Mathematical Physics, Nonlinear Science Letters A
2(1), 37-46 (2011).
[29] S.A. El-Wakil, E.M. Abulwafa, E.K. El-Shewy, A.A. Mah-
moud, Time-fractional study of electron acoustic solitary
waves in plasma of cold electron and two isothermal ions,
J. Plasma Physics 78(6), 641-649 (2012).
[30] A.M.A. El-Sayed, S.H. Behiry, W.E. Raslan, A numerical
algorithm for the solution of an intermediate fractional ad-
vection dispersion equation, Communications in Nonlinear
Science & Numerical Simulation 15(5): 1253-1258 (2010).
[31] S. Momani, Z. Odibat, Numerical comparison of methods
for solving linear differential equation of fractional order,
Chaos, Solitons & Fractals 31(5), 1248-1255 (2007).
[32] J.H. He, Homotopy perturbation technique, Computer Meth-
ods in Applied Mechanics & Engineering 178(3-4), 257-262
(1999).
[33] E.M. Abulwafa, M.A. Abdou, A.A. Mahmoud,
The Variational-Iteration Method to Solve the Nonlinear
Boltzmann Equation, Zeitschrift für Naturforschung A 63a(3-
4), 131-139 (2008).
[34] S. Zhang, H.-Q. Zhang, Fractional sub-equation method and
its applications to nonlinear fractional PDEs, Physics Letters
A 375(7), 1069-1073 (2011).
[35] S. Guo, L. Mei, Y. Li, Y. Sun, The improved fractional sub-
equation method and its applications to the space-time frac-
tional differential equations in fluid mechanics, Physics Let-
ters A 376(4), 407-411 (2012).
[36] J.-H. He, Semi-inverse Method of Establishing Generalized
Variational Principles for Fluid Mechanics with Emphasis
on Turbo-Machinery Aerodynamics, Int. J. Turbo Jet-Engines
14(1), 23-28 (1997).
[37] J.-H. He, Variational Principles for Some Nonlinear Par-
tial Differential Equations with Variable Coefficients, Chaos,
Solitons & Fractals 19(4), 847-851 (2004).
[38] S. Zhang, Q.A. Zong, D. Liu, Q. Gao, A Generalized Exp-
Function Method for Fractional Riccati Differential Equa-
tions, Communications in Fractional Calculus 1(1), 48-51
(2010).
The space-time fractional KdV-Burgers equation has been derived using the semi-inverse method and Agrawal’s variational method. The modified Riemann-Liouville definition is used for the fractional differential operators. The derived fractional equation is solved using the fractional sub-equation method.
Key words:
Fractional Euler-Lagrange equation, fractional sub-equation method, modified Riemann-Liouville fractional definition, space-time fractional KdV-Burgers equation
References:
[1] J.A. Tenreiro Machado, Fractional generalization of memris-
tor and higher order elements, Communications in Nonlinear
Science & Numerical Simulation 18(2), 264-275 (2013).
[2] D. Baleanu, J.A. Tenreiro Machado, A.C.J. Luo, Editors,
Fractional Dynamics and Control, Springer, New York, 2012.
[3] M.-A. Polo-Labarrios, G. Espinosa-Paredes, Application of
the fractional neutron point kinetic equation: Start-up of
a nuclear reactor, Annals of Nuclear Energy 46(1), 37-42
(2012).
[4] A. Kadem, D. Baleanu, Analytical method based on Walsh
function combined with orthogonal polynomial for fractional
transport equation, Communications in Nonlinear Science &
Numerical Simulation 15(3), 491-501 (2010).
[5] I.S. Jesus, J.A. Tenreiro Machado, R.S. Barbosa, Control of
a heat diffusion system through a fractional order nonlin-
ear algorithm, Computers & Mathematics with Applications
59(5), 1687-1694 (2010).
[6] D. Baleanu, Z.B. Güvenç, J.A. Tenreiro Machado, Editors,
New Trends in Nanotechnology and Fractional Calculus Ap-
plications, Springer, Dordrecht, 2010.
[7] F. Mainardi, Fractional Calculus and Waves in Linear Vis-
coelasticity: An Introduction to Mathematical Models, Impe-
rial College Press, London, UK, 2010.
[8] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional
Calculus: Models and Numerical Methods, vol. 3 of series
on Complexity, Nonlinearity and Chaos, World Scientific
Publishing, Hackensack, NJ, 2012.
[9] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Ap-
plications of Fractional Differential Equations, vol. 204
of North-Holland Mathematics Studies, Elsevier Science B.
V., Amsterdam, 2006.
[10] I. Podlubny, Fractional Differential Equations, Academic
Press, San Diego, 1999.
[11] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals
and derivatives: theory and applications, Gordon and Breach
Science Publishers, Amsterdam, 1993.
[12] G. Jumarie, On the fractional solution of the equation
f(x+y)=f(x)f(y) and its application to fractional Laplace’s
transform, Applied Mathematics and Computation 219(4),
1625-1643 (2012).
[13] G. Jumarie, An Approach to Differential Geometry of Frac-
tional Order via Modified Riemann-Liouville Derivative,
Acta Mathematica Sinica, English Series 28(9), 1741-1768
(2012).
[14] G. Jumarie, An approach via fractional analysis to non-
linearity induced by coarse-graining in space, Nonlinear
Analysis: Real World Applications 11(1), 535-546 (2010).
[15] G. Jumarie, Table of some basic fractional calculus formulae
derived from a modified Riemann-Liouville derivative for
non-differentiable functions, Applied Mathematics Letters
22(3), 378-385 (2009).
[16] G.-w. Wang, X.-q. Liu, Y.-y. Zhang, Lie symmetry analysis
to the time fractional generalized fifth-order KdV equation,
Communications in Nonlinear Science & Numerical Simula-
tion 18(9), 2321-2326 (2013).
[17] H. Wang, T.-C. Xia, The fractional supertrace identity and its
application to the super Jaulent-Miodek hierarchy, Commu-
nications in Nonlinear Science Numerical Simulation 18(10),
2859-2867 (2013).
[18] Z. Hammouch, T. Mekkaoui, Travelling-wave solutions for
some fractional partial differential equation by means of gen-
eralized trigonometry functions, Int. J. Applied Mathematical
Research 1(2), 206-212 (2012).
[19] R. Almeida, D.F.M. Torres, Fractional variational calculus
for nondifferentiable functions, Computers & Mathematics
with Applications 61(10), 3097-3104 (2011).
[20] G.-c. Wu, A fractional variational iteration method for solv-
ing fractional nonlinear differential equations, Computers &
Mathematics with Applications 61(8), 2186-2190 (2011).
[21] F. Riewe, Nonconservative Lagrangian and Hamiltonian Me-
chanics, Physical Review E 53(2), 1890-1899 (1996).
[22] F. Riewe, Mechanics with Fractional Derivatives, Physical
Review E 55(3), 3581-3592 (1997).
[23] O.P. Agrawal, Formulation of Euler-Lagrange Equations for
Fractional Variational Problems, J. Mathematical Analysis
& Applications 272(1), 368-379 (2002).
[24] O.P. Agrawal, Fractional Variational Calculus in Terms of
Riesz Fractional Derivatives, J. Physics A: Mathematical &
Theoretical 40, 6287-6303 (2007).
[25] O.P. Agrawal, S.I. Muslih, D. Baleanu, Generalized varia-
tional calculus in terms of multi-parameters fractional deriva-
tives, Communications in Nonlinear Science & Numerical
Simulation 16(12), 4756-4767 (2011).
[26] M.A.E. Herzallah, S.I. Muslih, D. Baleanu, E.M. Rabei,
Hamilton-Jacobi and fractional like action with time scaling,
Nonlinear Dynamics 66(4), 549-555 (2011).
[27] M.A.E. Herzallah, D. Baleanu, Fractional Euler-Lagrange
equations revisited, Nonlinear Dynamics 69(3), 977-982
(2012).
[28] S.A. El-Wakil, E.M. Abulwafa, M.A. Zahran, A.A. Mah-
moud, Formulation of Some Fractional Evolution Equations
used in Mathematical Physics, Nonlinear Science Letters A
2(1), 37-46 (2011).
[29] S.A. El-Wakil, E.M. Abulwafa, E.K. El-Shewy, A.A. Mah-
moud, Time-fractional study of electron acoustic solitary
waves in plasma of cold electron and two isothermal ions,
J. Plasma Physics 78(6), 641-649 (2012).
[30] A.M.A. El-Sayed, S.H. Behiry, W.E. Raslan, A numerical
algorithm for the solution of an intermediate fractional ad-
vection dispersion equation, Communications in Nonlinear
Science & Numerical Simulation 15(5): 1253-1258 (2010).
[31] S. Momani, Z. Odibat, Numerical comparison of methods
for solving linear differential equation of fractional order,
Chaos, Solitons & Fractals 31(5), 1248-1255 (2007).
[32] J.H. He, Homotopy perturbation technique, Computer Meth-
ods in Applied Mechanics & Engineering 178(3-4), 257-262
(1999).
[33] E.M. Abulwafa, M.A. Abdou, A.A. Mahmoud,
The Variational-Iteration Method to Solve the Nonlinear
Boltzmann Equation, Zeitschrift für Naturforschung A 63a(3-
4), 131-139 (2008).
[34] S. Zhang, H.-Q. Zhang, Fractional sub-equation method and
its applications to nonlinear fractional PDEs, Physics Letters
A 375(7), 1069-1073 (2011).
[35] S. Guo, L. Mei, Y. Li, Y. Sun, The improved fractional sub-
equation method and its applications to the space-time frac-
tional differential equations in fluid mechanics, Physics Let-
ters A 376(4), 407-411 (2012).
[36] J.-H. He, Semi-inverse Method of Establishing Generalized
Variational Principles for Fluid Mechanics with Emphasis
on Turbo-Machinery Aerodynamics, Int. J. Turbo Jet-Engines
14(1), 23-28 (1997).
[37] J.-H. He, Variational Principles for Some Nonlinear Par-
tial Differential Equations with Variable Coefficients, Chaos,
Solitons & Fractals 19(4), 847-851 (2004).
[38] S. Zhang, Q.A. Zong, D. Liu, Q. Gao, A Generalized Exp-
Function Method for Fractional Riccati Differential Equa-
tions, Communications in Fractional Calculus 1(1), 48-51
(2010).
