Authors:
Bhausaheb Sontakke,Rajashri Pandit,DOI NO:
https://doi.org/10.26782/jmcms.2021.02.00005Keywords:
Time Regularized Long Wave equation,Fractional derivative,Adomian Decomposition Method,Convergence,Mathematica,Abstract
In the paper, we develop the Adomian Decomposition Method for the fractional-order nonlinear Time Regularized Long Wave Equation (TRLW) equation. Caputo fractional derivatives are used to define fractional derivatives. We know that nonlinear physical phenomena can be explained with the help of nonlinear evolution equations. Therefore solving TRLW is very helpful to obtain the solution of many physical theories. In this paper, we will solve the time-fractional TRLW equation which may help researchers with their work. We solve some examples numerically, which will show the efficiency and convenience of the Adomian Decomposition Method.Refference:
I. Adomian G., “A Review of Decomposition method in applied mathematics”,J. Math. Anal. Appl., 135, (1988), 501-544.
II. Adomian G., “Solving Frontier Problems of Physics: The Decomposition Method”,Kluwer, Boston, 1994.
III. Adomian G., “The diffusion-Brusselator equation”, Comput. Math. Appl., 29 , (1995), 1-3.
IV. Adomian G., ”Solution of coupled nonlinear partial differential equations by decomposition”, Comput. Math. Appl., 31(6), (1996), 117-120.
V. Biazar J., Babolian E., “Solution of the system of ordinary differential equationsby Adomian Decomposition Method”, Appl.Math.Comput.147 (3), (2004), 713-719.
VI. Bratsos A., Ehrhardt M.,.Famelis I .,“A discrete Adomian Decomposition Method for discrete nonlinear Schrondinger equations”, Appl. Math. Comput., 197,(2008), 190-205.
VII. Daftardar-Gejji V., Jafari H., “Adomian Decomposition: a tool for solving a system of fractional differential equations”, J. Math. Anal. Appl., 301, (2005) , 508-518.
VIII. Daftardar-Gejji V. and Bhalekar S., “Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian Decomposition Method”, Appl. Math. Comput., (202),(2008), 113-120.
IX. Dhaigude D. B., Birajdar G. A,. Nikam V. R., “Adomian decomposition method for fractional Benjamin-Bona-Mahony-Burger’s equation”, int. of appl. math, and mech 8 (12), (2012), 42-51.
X. Jafari H., “Solving a system of nonlinear fractional differential equations using Adomian Decomposition”, Journal of computational and Applied Mathematics 196, (2006).644-651.
XI. Jafari H., Daftardar-Gejji V., “Solving Linear and nonlinear fractional diffusion and wave equations by Adomian Decomposition Method”.Appl. Math. Comput., 180,(2006), 488-497.
XII. Kazi sazzad Hossain., Ali Akbar and Md., “Abul Kalam Azad, Closed form wave solutions of two nonlinear evolution equations”, Cogent Physics,4:1396948, (2017)
XIII. Kulkarni S., Jogdand S., Takale K. C., “Error Analysis of solution of time fractional convection diffusion equation”, JETIR, 6 (3), (2019).
XIV. Li C., And Wang Y., “Numerical algorithm based on Adomian Decomposition Method for fractional differential equations”, Comput. Math. Appl., 57 ,(2009), 1672-1681.
XV. M. Hafiz Uddin, M. Ali Akbar, Md. Ashrafuzzaman Khan, Md. Abdul Haque, : Families of exact traveling wave solutions to the space time fractional modified KdV equation and the fractional Kolmogorov-Petrovskii-Piskunovequation, J. Mech.Cont. & Math. Sci., Vol.-13, No.-1, March – April (2018) Pages 17-33.
XVI. Md. Tarikul Islam, M. Ali Akbar, Md. Abul Kalam Azad, : The exact traveling wave solutions to the nonlinear space-time fractional modified Benjamin-Bona-Mahony equation, J. Mech.Cont. & Math. Sci., Vol.-13, No.-2, May-June (2018) Pages 56-71
XVII. Mittal R. C., and Nigam R., “Solution of fractional integro-differential equations by Adomian Decomposition Method”, Int. J. Appl. Math. Mech., 4 (2), (2008), 8794.
XVIII. Momani S. and Al-Khaled K., “Numerical solutions for system of fractional differential equations by the Decomposition Method”. Appl. Math. Comput. Simul., 70,(2005), 1351-1365.
XIX. Miller K. S., Ross B.,” An Introduction to the Fractional Calculus and Fractional Differential Equations”, New York, (1993).
XX. Momani S., Odibat Z., “ Analytical solutions of a time fractional Navier-Stokes Equation by Adomian decomposition method”, Appl. Math. Comput. 177, 2 , (2006), 488 – 494.
XXI. Oldham K. B., Spanier J., “The fractional calculus, Academic Press”, New York, (1974).
XXII. Podlubny I., “Fractional Differential equations”, Academic Press, San Diago 1999.
XXIII. Rayhanul Islam S. M., Khan K., “Exact solution of unsteady Korteweg-deVries and time regularized long wave equations”, Springer-Plus,4 ; 124 (2015).
XXIV. Sontakke B. R., Sangvikar V. V., “Approximate Solution for Time-Space Fractional Soil Moisture Diffusion Equation and its Applications”, International Journal of Scientific and Technology Research, 5(5)(2016),197-202.
XXV. Sontakke B. R and Shelke A. S., “ Approximate Scheme for Time Fractional Diffusion Equation and Its Applications”, Global Journal of Pure and Applied Mathematics. 13(8) (2017), 4333-4345.
XXVI. Sontakke B. R., and Pandit R.B., “Convergence Analysis and Approximate Solution of fractional differential Equations”, Journal of Mathematics,7(2),(2019),338344.
XXVII. Sontakke B. R, Adomian Decomposition Method for Solving Highly Nonlinear Fractional Partial Differential Equations IOSR Journal of Engineering, 9 (3)( 2019), 39-44.
XXVIII. Saha Ray S., New Approach for General Convergence of the ADM, World Applied Sciences Journal, 32 (11), (2014) 2264-2268.
XXIX. Wazwaz A. M., A reliable modification of Adomian’s Decomposition Method. Appl. Math. Comput., 102,(1999), 77-86.
XXX. Wazwaz A. M., The decomposition method applied to systems of partial differential equations and to the reaction-diffusion Brusselator model. Appl. Math. Comput., 110,(2009), 251-264.
XXXI. Wyss W., and Schneider W.R., Fractional Diffusion and wave equations, J. Math. Phys. 30, (1989).
XXXII. Yang Q., Turner I., Liu F, Analytical and Numerical Solutions for the time and Space-Symmetric Fractional Diffusion Equation, ANZIAM J., 50 (CTA 2008) , (2009) ,pp. C800-C814.
XXXIII. Zahram Emad H. , Khater Mostafa M. A., Exact Travelling Wave Solutions for the system of shallow water wave equations and Modified Liouville equation using extended Jacobian elliptic function expansion method, American Journal of computational Mathematics, 4, (2014), 455-463.