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On An Efficient Technique for Solving the Time-Fractional (2+1) - Dimensional Differential Equation
Current Issue
Volume 5, 2018
Issue 4 (July)
Pages: 63-70   |   Vol. 5, No. 4, July 2018   |   Follow on         
Paper in PDF Downloads: 28   Since Aug. 9, 2018 Views: 924   Since Aug. 9, 2018
Authors
[1]
Kamran Ayub, Department of Mathematics, Riphah International University, Islamabad, Pakistan.
[2]
Nawab Khan, Department of Mathematics, Riphah International University, Islamabad, Pakistan.
[3]
Muhammad Yaqub Khan, Department of Mathematics, Riphah International University, Islamabad, Pakistan.
[4]
Attia Rani, Department of Mathematics, University of Wah, Wah Cantt, Pakistan.
[5]
Muhammad Ashraf, Department of Mathematics, University of Wah, Wah Cantt, Pakistan.
[6]
Qazi Mahmood-Ul-Hassan, Department of Mathematics, University of Wah, Wah Cantt, Pakistan.
[7]
Madiha Afzal, Department of Mathematics, Allama Iqbal Open University, Islamabad, Pakistan.
Abstract
Nonlinear mathematical problems and their solutions attain much attention in soliton theory. Under study article is devoted to find soliton wave solutions of nonlinear time fractional (2+1) - dimensional breaking soliton equation via a reliable mathematical technique. By use of proposed technique we attains soliton wave solution of various types. The nonlinear partial differential equation is transformed into an ordinary differential equation by using suitable wave transformation. The regulation of proposed algorithm is demonstrated by consequent numerical results and computational work. It is observed that under discussion technique is user friendly with minimum computational work, also we can extend it for physical problems of different nature.
Keywords
Exp-function Method, Fractional Calculus, Modified Riemann-Liouville Derivative, Breaking Soliton Equation, Maple 18
Reference
[1]
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
[2]
K. S. Miller and B. Ross, an Introduction to the Fractional Calculus and Fractional Differential Equations, a Wiley-Inter science Publication, John Wiley & Sons, New York, NY, USA, 1993.
[3]
Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
[4]
G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. Vol. 135 (1988) No. 2, Pp. 501-544.
[5]
N. T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput. 131 (2-3) (2002) 517-529.
[6]
S. S. Ray and R. K. Bera, An approximate solution of a nonlinear fractional differential equation by Adomian's decomposition method, Appl. Math. Comput. 167, no. 1, (2005) 561--571.
[7]
J. H. He, some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol. 15 (2) (1999) 86--90.
[8]
J. H. He and X. H. Wu, Exp-function method for nonlinear wave equations, Chaos Soliton & Fractals 30 (3) (2006), 700--708.
[9]
J. H. He, an elementary introduction of recently developed asymptotic methods and nanomechanics in textile engineering, Int. J. Mod. Phys. B 22 (21) (2008), 3487-4578.578.
[10]
J. H. He and M. A. Abdou, New periodic solutions for nonlinear evolution equation using Exp-method, Chaos, Solitons & Fractals, 34 (2007), 1421-1429.
[11]
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.
[12]
S. T. Mohyud-Din, M. A. Noor and K. I. Noor, Some relatively new techniques for nonlinear problems, Mathematical Problems in Engineering, Hindawi, 2009 (2009); Article ID 234849, 25 pages, doi: 10.1155/2009/234849.
[13]
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, DOI: 10.1007/s12190-008-0083-y.
[14]
M. A. Noor, S. T. Mohyud-Din and A. Waheed, Exp-function method for generalized traveling solutions of master partial differential equations, Acta Applicandae Mathematicae, 104 (2) (2008), 131--137, DOI: 10.1007/s10440-008-9245-z
[15]
T. Ozis, C. Koroglu, A novel approach for solving the Fisher's equation using Exp-function method, Phys Lett. A 372 (2008) 3836 – 3840.
[16]
X. H. Wu and J. H. He, Exp-function method and its applications to nonlinear equations, Chaos, Solitons & Fractals, (2007), in press.
[17]
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.
[18]
E. Yusufoglu, New solitonary solutions for the MBBN equations using exp-function method, Phys. Lett. A. 372 (2008), 442-446.
[19]
S. Zhang, Application of exp-function method to high-dimensional nonlinear evolution equation, Chaos, Solitons & Fractals, 365 (2007), 448-455.
[20]
S. D. Zhu, Exp-function method for the Hybrid-Lattice system, Inter. J. Nonlin. Sci. Num. Simulation, 8 (2007), 461-464.
[21]
S. D. Zhu, Exp-function method for the discrete m KdV lattice, Inter. J. Nonlin. Sci. Num. Simulation, 8 (2007), 465-468.
[22]
N. A. Kudryashov, Exact soliton solutions of the generalized evolution equation of wave dynamics, J. Appl Math and Mech, 52 (3), (1988), 361.
[23]
S. Momani, An explicit and numerical solutions of the fractional KdV equation, Math. Comput. Simul. 70 (2) (2005) 110-118.
[24]
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.
[25]
X. J. Yang, A new integral transform operator for solving the heat-diffusion problem, Applied Mathematics Letters. 64 (2017) 193-197.
[26]
X. J. Yang, A New integral transform method for solving steady Heat transfer problem, Thermal Science. 20 (2016) S639-S642.
[27]
X. J. Yang, J. A. Tenreiro Machado, D. Baleanu, C. Cattani, On exact traveling wave solutions for local fractional Korteweg-de Vries equation, Chaos. 26 (8) (2016) 084312.
[28]
X. J. Yang, G. Feng, H. M. Srivastava, Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations, Computers & Mathematics with Applications. 73 (2) (2017) 203-210.
[29]
X. J. Yang, J. A. Tenreiro Machado, D. Baleanu, On exact traveling-wave solution for local fractional Boussinesq equation in fractal domain, Fractals. 25 (4) (2017) 1740006.
[30]
X. J. Yang, G. Feng, H. M. Srivastava, Non-Differentiable exact solutions for the nonlinear ODEs defined on fractal sets, Fractals. 25 (4) (2017) 1740002.
[31]
K. M. Saad, E. H. AL-Shareef, M. S. Mohamed, X. J. Yang, Optimal q-homotopy analysis method for time-space fractional gas dynamics equation, The European Physical Journal Plus. 132 (23) (2017) https://doi.org/10.1140/epjp/i2017-11303-6.
[32]
X. J. Yang, G. Feng, A New Technology for Solving Diffusion and Heat Equations, Thermal Science. 21 (1A) (2017) 133-140.
[33]
G. Feng, X. J. Yang, H. M. Srivastava, Exact Travelling wave solutions for linear and nonlinear heat transfer equations, Thermal Science. (2017) doi 10.2298/TSCI161013321G.
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