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Abstrakty
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.
Rocznik
Tom
Strony
235--243
Opis fizyczny
Bibliogr. 38 poz., rys.
Twórcy
autor
- Theoretical Physics Research Group, Physics Department Faculty of Science, Mansoura University, Mansoura 35516, Egypt
autor
- Theoretical Physics Research Group, Physics Department Faculty of Science, Mansoura University, Mansoura 35516, Egypt
autor
- Theoretical Physics Research Group, Physics Department Faculty of Science, Mansoura University, Mansoura 35516, Egypt
autor
- Theoretical Physics Research Group, Physics Department Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Bibliografia
- [1] J.A. Tenreiro Machado, Fractional generalization of memristor 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 nonlinear 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 Applications, Springer, Dordrecht, 2010.
- [7] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial 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 Applications 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 Fractional 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 nonlinearity induced by coarse-graining in space, NonlinearAnalysis: 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 Simulation 18(9), 2321-2326 (2013).
- [17] H.Wang, T.-C. Xia, The fractional supertrace identity and its application to the super Jaulent-Miodek hierarchy, Communications 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 generalized 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 solving fractional nonlinear differential equations, Computers & Mathematics with Applications 61(8), 2186-2190 (2011).
- [21] F. Riewe, Nonconservative Lagrangian and Hamiltonian Mechanics, 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 Theoretical 40, 6287-6303 (2007).
- [25] O.P. Agrawal, S.I. Muslih, D. Baleanu, Generalized variational calculus in terms of multi-parameters fractional derivatives, 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. Mahmoud, 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. Mahmoud, 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 advection 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 Methods 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 subequation method and its applications to the space-time fractional differential equations in fluid mechanics, Physics Letters 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 Partial 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 Equations, Communications in Fractional Calculus 1(1), 48-51 (2010).
Typ dokumentu
Bibliografia
Identyfikator YADDA
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