







Vol.2 , No. 1, Publication Date: Jul. 5, 2017, Page: 1-9
[1] | Qazi Mahmood Ul-Hassan, Department of Mathematics, University of Wah, Wah Cantt, Pakistan. |
[2] | Ayesha Sidiuqa, Department of Mathematics, University of Wah, Wah Cantt, Pakistan. |
[3] | Kamran Ayub, Department of Mathematics, Riphah International University, Islamabad, Pakistan. |
[4] | Madiha Afzal, Department of Mathematics, Allama Iqbal Open University, Islamabad, Pakistan. |
[5] | Muhammad Hashim, Department of Mathematics, NCBA&E, Lahore, Pakistan. |
In this paper, we use the fractional derivatives in Caputo’s sense to construct exact solutions for Burgers equation of fractional order. A generalized fractional complex transform is appropriately used to convert this equation to ordinary differential equation which subsequently resulted into number of exact solutions.
Keywords
Burgers Equation, Fractional Calculus, Exp-function Method
Reference
[01] | A. G. Nikitin, T. A. Barannyk, Solitary waves and other solutions for nonlinear heat equations, Cent. Eur. J. Math. (2)2005 840-858. |
[02] | I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. |
[03] | J. H. He, Some app lications of nonlinear fractional differential equations and their applications, Bull. Sci. Technol., 15(2) (1999) 86-90. |
[04] | K. Diethelm, Y. Luchko, Numerical solution of linear multiterm differential equations of fractional order, J. Comput. Anal. Appl. (6)2004 243-263. |
[05] | Z. B. Li, J. H. He, Application of the fractional complex transform to fractional differential equations, Nonlinear Sci. Lett. A, 2(3) (2011) 121-126. |
[06] | A. Rafiq, M. Ahmed, S. Hussain, A general approach to specific second order ordinary differential equations using homotopy perturbation method, Phys. Lett. A, 372(2008) 372 4973-4976. |
[07] | Z. B. Li, J. H. He, Fractional complex transform for fractional differential equations, Math. And Comput. Appl. 15(2) (2010) 970-973. |
[08] | J. H. He, S. K. Elagan, Z. B. Li, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A, (376)2012 257–259. |
[09] | R. W. Ibrahim, Fractional complex transforms for fractional differential equations, Advan. in Diff. Equat. 2012, vol. 2012:192 doi: 10.1186/1687 -1847-2012-192. |
[10] | J. H. He, An elementary introduction of recently developed asymptotic methods and nanomechanics in textile engineering, Int. J. Mod. Phys. B 22 (21) (2008), 3487-4578 |
[11] | J. H. He and M. A. Abdou, New periodic solutions for nonlinear evolution equation using exp-method, Chaos, Solitons & Fractals, 34 (2007), 1421-1429. |
[12] | S. T. Mohyud-Din, M. A. Noor and A. Waheed, Exp-function method for generalized travelling solutions of good Boussinesq equations, J. Appl. Math. Computg. 30 (2009), 439-445, DOI 10.1007/s12190-008-0183-8. |
[13] | S. T. Mohyud-Din, M. A. Noor and K. I. Noor, Some relatively new techniques for nonlinear problems, Mathematical Problems in Engineering, Hindawi, 25, (2009), doi:10.1155/2009/234849. |
[14] | M. A. Noor, S. T. Mohyud-Din and A. Waheed, Exp-function method for solving Kuramoto-Sivashinsky and Boussinesq equations, J. Appl. Math. Computg. 29 (2008), 1-13. |
[15] | M. A. Noor, S. T. Mohyud-Din and A. Waheed, Exp-function method for generalized travelling solutions of master partial differential equations, Acta Applnda. Mathmtce. (2008), DOI: 10.1007/s10440-008-9245-z. |
[16] | T. Ozis, C. Koroglu, A novel approach for solving the Fisher’s equation using Exp-function method, Phys Lett. A 372 (2008) 3836 - 3840 |
[17] | X. H. Wu, J. H. He, Exp-function method and its application to nonlinear equations, Chaos, Solitons and Fractals 38(3) (2008) 903–910. |
[18] | X. H. Wu and J. H. He, Solitary solutions, periodic solutions and compacton like solutions using the exp-function method, Comput. Math. Appl. 54 (2007), 966-986. |
[19] | E. Yusufoglu, New solitonary solutions for the MBBN equations using exp-function method, Phys. Lett. A. 372 (2008), 442-446. |
[20] | S. Zhang, Application of exp-function method to high-dimensional nonlinear evolution equation, Chaos, Solitons & Fractals, 365 (2007), 448-455. |
[21] | S. D. Zhu, Exp-function method for the Hybrid-Lattice system, Inter. J. Nonlin. Sci. Num. Simulation, 8 (2007), 461-464. |
[22] | S. D. Zhu, Exp-function method for the discrete m KdV lattice, Inter. J. Nonlin. Sci. Num. Simulation, 8 (2007), 465-468. |
[23] | N. A. Kudryashov, Exact soliton solutions of the generalized evolution equation of wave dynamics, J. Appl Math and Mech, 52 (3), (1988), 361 |
[24] | S. Momani, An explicit and numerical solutions of the fractional KdV equation, Math. Comput. Simul. 70 (2) (2005) 110-118. |
[25] | A. Ebaid An improvement on the Exp-function method when balancing the highest order linear and nonlinear terms J. Math. Anal. Appl. 392(2012) 1–5. |
[26] | A. M. Wazwaz, New higher–dimensional fifth–order nonlinear equations with multiple soliton solutions, Phys. Scr. 84 (2011) 025007. |
[27] | W. X. Ma, A. Abdeljabbar, M. G. Asaad, Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation, Appl. Math. Comput. 217 (2011) 10016–10023. |
[28] | M. A. Abdou and A. A. Soliman, Variational iteration method for solving Burger’s and coupled Burger’s equations. J. Comput. and A. Math, 181 (2006), 245-251. |
[29] | H Jafari, N Kadkhoda, CM Khalique,Travelling wave solutions of nonlinear evolution equations using the simplest equation method, Computers & Mathematics with Applications 64 (6), 2084-2088. |
[30] | M. A. Abdou, A. A. Soliman and S. T. Basyony, New application of Exp-function method for improved Boussinesq equation. Phys. Lett. A, 369 (2007), 469-475. |
[31] | J. H. He and M. A. Abdou, New periodic solutions for nonlinear evolution equation using exp-method, Chaos Solitons. Farct.34 (2007), 1421-1429. |
[32] | M. A. Noor, S. T. Mohyud-Din and A. Waheed, Exp-function method for solving Kuramoto-Sivashinsky and Boussinesq equations, J. Appl. Math. Computg. (2008), DOI: 10.1007/s12190-008-0083-y. |