Analytical Solutions of Classical and Fractional KP-Burger Equation and Coupled KdV Equation
Ghosh Uttam 1*, Sarkar Susmita 1**, Das Shantanu 2†
1 Department of Applied Mathematics, University of Calcutta, Kolkata, India
∗E-mail: uttam_math@yahoo.co.in, **susmita62@yahoo.co.in
2 Reactor Control Systems Design Section E & I Group BARC Mumbai India
†E-mail: shantanu@barc.gov.in
Received:
Received: 05 April 2016; accepted: 10 May 2016; published online: 24 August 2016
DOI: 10.12921/cmst.2016.0000016
Abstract:
Development of new analytical and numerical methods and their applications for solving non-linear partial differential equations (both classical and fractional) is a rising field of Applied Mathematical research because of its applications in Physical, Biological and Social Sciences. In this paper we have used a generalized Tanh method to find the exact solution of KP-Burger equation and coupled KdV equation. The fractional Sub-equation method has been used to find the solution of fractional KP-Burger equation and fractional coupled KdV equations. The exact solution obtained by the fractional sub-equation method reduces to classical solution when the order of fractional derivative tends to one. Finally numerical simulation has been done. The numerical simulation justifies that the solutions of two fractional differential equations reduce to shock solution for KP-Burger equation and soliton solution for coupled KdV equations when the order of derivative tends to one.
Key words:
coupled KdV equation, fractional differential equation, fractional sub-equation method, generalized tanh-method, Jumarie fractional derivative, KP-Burger equation
References:
[1] A.M.A. El-Sayed, M. Gaber, The adomian decomposition method for solving partial differential equations of fractal order in finite
domains, Phys. Lett. A 359(20):175-182 (2006).
[2] A.M.A. El-Sayed, S.H. Behiry, W.E. Raslan, Adomian’s decomposition method for solving an intermediate fractional advection-
dispersion equation, Comput. Math. Appl 59(5):1759-1765 (2010).
[3] S. Das. Functional Fractional Calculus 2nd Edition, Springer-Verlag 2011.
[4] J. Cang, Y. Tan, H. Xu and S. J. Liao, Series solutions of non-linear Riccati differential equations with fractional order, Chaos,
Solitons and Fractals 40, 1-9 (2009).
[5] J.H. He. Homotopy perturbation technique, Comput Methods. Appl. Mech. Eng. 178(3-4), 257-262 (1999).
[6] J.H. He, Inter. J. Non-Linear Mech. 35, 37-43 (2000).
[7] H. Jafari, S. Momani, Solving fractional diffusion and wave equations by modified homotopy perturbation method, Phys. Lett. A 370,
388-396 (2007).
[8] Z. Odibat and S. Momani, Appl. Math. Lett. 21, 194-199 (2008).
[9] L. Huibin, W. Kelin, Exact solutions for two nonlinear equations: I, J. Phys. A: Math. Gen. 23, 3923-3928 (1990).
[10] A.M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations, Applied Mathematics and Computation 154,
713-723 (2004).
[11] E. Fan, Extended Tanh-Function Method and Its Applications to Nonlinear Equations, Physics Letters A 277(4-5), 212-218 (2000).
[12] S. Zhang, W. Wang and J. Lin Tong. Electronic Journal of Theoretical Physics Exact Non-traveling Wave and Coefficient Function
Solutions for (2+1)-Dimensional Dispersive Long Wave Equations, 5(19), 177-190 (2008).
[13] T.B. Dinh, V.C. Long and K.W. Wojciechowski, Solitary waves in auxetic rods with quadratic nonlinearity: Exact analytical solutions
and numerical simulations, Phys. Status Solidi B 252(7),1587-1594 (2015).
[14] T. Bui Dinh, V. Cao Long, K. Dinh Xuan, and K.W. Wojciechowski, Computer simulation of solitary waves in a common or auxetic
elastic rod with both quadratic and cubic nonlinearities, Phys. Status Solidi B 249, 1386-1392 (2012).
[15] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA.
1999; 198.
[16] S. Zhang and H. Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A, 375,
1069 (2011).
[17] G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of non differentiable functions further results, Comput.
Math. Appl. 51(9-10), 1367-1376 (2006).
[18] U. Ghosh, S. Sengupta, S. Sarkar and S. Das, Analytic solution of linear fractional differential equation with Jumarie derivative in
term of Mittag-Leffler function, American Journal of Mathematical Analysis 3(2), 32-38 (2015).
[19] E. Fan, Y.C. Hon, Generalized tanh Method Extended to Special types of Non-linear equations, Z. Naturforsh 57, 692-700 (2002)
Development of new analytical and numerical methods and their applications for solving non-linear partial differential equations (both classical and fractional) is a rising field of Applied Mathematical research because of its applications in Physical, Biological and Social Sciences. In this paper we have used a generalized Tanh method to find the exact solution of KP-Burger equation and coupled KdV equation. The fractional Sub-equation method has been used to find the solution of fractional KP-Burger equation and fractional coupled KdV equations. The exact solution obtained by the fractional sub-equation method reduces to classical solution when the order of fractional derivative tends to one. Finally numerical simulation has been done. The numerical simulation justifies that the solutions of two fractional differential equations reduce to shock solution for KP-Burger equation and soliton solution for coupled KdV equations when the order of derivative tends to one.
Key words:
coupled KdV equation, fractional differential equation, fractional sub-equation method, generalized tanh-method, Jumarie fractional derivative, KP-Burger equation
References:
[1] A.M.A. El-Sayed, M. Gaber, The adomian decomposition method for solving partial differential equations of fractal order in finite
domains, Phys. Lett. A 359(20):175-182 (2006).
[2] A.M.A. El-Sayed, S.H. Behiry, W.E. Raslan, Adomian’s decomposition method for solving an intermediate fractional advection-
dispersion equation, Comput. Math. Appl 59(5):1759-1765 (2010).
[3] S. Das. Functional Fractional Calculus 2nd Edition, Springer-Verlag 2011.
[4] J. Cang, Y. Tan, H. Xu and S. J. Liao, Series solutions of non-linear Riccati differential equations with fractional order, Chaos,
Solitons and Fractals 40, 1-9 (2009).
[5] J.H. He. Homotopy perturbation technique, Comput Methods. Appl. Mech. Eng. 178(3-4), 257-262 (1999).
[6] J.H. He, Inter. J. Non-Linear Mech. 35, 37-43 (2000).
[7] H. Jafari, S. Momani, Solving fractional diffusion and wave equations by modified homotopy perturbation method, Phys. Lett. A 370,
388-396 (2007).
[8] Z. Odibat and S. Momani, Appl. Math. Lett. 21, 194-199 (2008).
[9] L. Huibin, W. Kelin, Exact solutions for two nonlinear equations: I, J. Phys. A: Math. Gen. 23, 3923-3928 (1990).
[10] A.M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations, Applied Mathematics and Computation 154,
713-723 (2004).
[11] E. Fan, Extended Tanh-Function Method and Its Applications to Nonlinear Equations, Physics Letters A 277(4-5), 212-218 (2000).
[12] S. Zhang, W. Wang and J. Lin Tong. Electronic Journal of Theoretical Physics Exact Non-traveling Wave and Coefficient Function
Solutions for (2+1)-Dimensional Dispersive Long Wave Equations, 5(19), 177-190 (2008).
[13] T.B. Dinh, V.C. Long and K.W. Wojciechowski, Solitary waves in auxetic rods with quadratic nonlinearity: Exact analytical solutions
and numerical simulations, Phys. Status Solidi B 252(7),1587-1594 (2015).
[14] T. Bui Dinh, V. Cao Long, K. Dinh Xuan, and K.W. Wojciechowski, Computer simulation of solitary waves in a common or auxetic
elastic rod with both quadratic and cubic nonlinearities, Phys. Status Solidi B 249, 1386-1392 (2012).
[15] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA.
1999; 198.
[16] S. Zhang and H. Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A, 375,
1069 (2011).
[17] G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of non differentiable functions further results, Comput.
Math. Appl. 51(9-10), 1367-1376 (2006).
[18] U. Ghosh, S. Sengupta, S. Sarkar and S. Das, Analytic solution of linear fractional differential equation with Jumarie derivative in
term of Mittag-Leffler function, American Journal of Mathematical Analysis 3(2), 32-38 (2015).
[19] E. Fan, Y.C. Hon, Generalized tanh Method Extended to Special types of Non-linear equations, Z. Naturforsh 57, 692-700 (2002)
