Approximate solutions of coupled Ramani equation by using RDTM with compared DTM and exact solutions

Year 2016, Volume: 4 Issue: 4, 198 - 212, 31.12.2016

Murat Gubes Galip Oturanc

https://izlik.org/JA64FK57ZG

Abstract

In this paper, we present a new approximate solutions
of famous coupled Ramani Equation. In order to obtain the solution, we use the
semi-analytical methods differential transform method (DTM) and reduced form of
DTM called reduced differential transform method (RDTM). We compare the RDTM
solutions with exact solution and DTM. Numerical results show clearly that DTM
and RDTM are very effective and also provide very accurate solutions. Also, one
can conclude that RDTM is used easier than DTM and converges faster than the
DTM for these kind of problems.

Keywords

Reduced differential transform method (RDTM) , differential transform method (DTM) , coupled ramani equation

References

• Ramani A., Inverse scattering, ordinary differential equations of Painleve-type and Hirota’s bilinear formalism. In: Fourth International Conference Collective Phenomena, New York: Academy of Sciences (1981), p. 54.
• Hirota R., Direct method of finding exact solutions of nonlinear evolution equations, In: Bullough R., Caudrey P., editors. Backlund transformations, Berlin, Springer (1980), p. 1157-75.
• Hu X. B., Wang D. L., Tam H. W., Lax pairs and backlund transformations for a coupled Ramani equation and its related system, Applied Mathematics Letters (2000), 13, 45-48.
• Zhao J. X., Tam H. W., Soliton solutions of a coupled Ramani equation, Appl Math Lett, 16 (2006), 307-313.
• Ablowitz M. J., Clarkson P. A., Solitons, nonlinear evolution equations and inverse scattering transform, Cambridge University press, 1990.
• Ito M., An extension of nonlinear evolution equations K-dV (mK-dV) type to higher order, J. Physc. Soc. Jpn. (1980), 49 (2), 771-778.
• Zhang H., New exact traveling wave solutions for some nonlinear evolution equations, Chaos Solitons and Fractals (2005), 26 (3), 921-5.
• Feng Z., Traveling solitary wave solutions to evolution equations with nonlinear terms any order, Chaos Solitons and Fractals (2003), 17 (5), 861-8.
• Malfliet W., Hereman W., The tanh method: I Exact solutions of nonlinear evolution and wave equations, Physica Sprica (1996), 54, 569-75.
• Li J., Existence of exact families of traveling wave solutions for the sixth-order Ramani equation and a coupled Ramani equation, Int. J. Bifurc. Chaos (2012), 22 (1), 125002.
• Nadjafikhah M., Shirvani-Sh V., Lie symmetries and conservation laws of the Hirota-Ramani equation, Commun. Nonlin. Sci. Numer. Simul. (2012), 17 (11), 4064-4073.
• Yusufoglu E., Bekir A., Exact solutions of coupled nonlinear evolution equations, Chaos Solitons and Fractals (2008), 37, 842-848.
• Wazwaz A. M., Triki H., Multiple soliton solutions for the sixth-order Ramani equation and a coupled Ramani equation, Applied Mathematics and Computation (2010), 216, 332-336.
• Wazwaz A. M., A coupled Ramani equation: multiple soliton solutions, J. Math. Chem. (2014), 52, 2133-2140.
• Jafarian A., Ghaderi P., Golmankaneh A. K., Baleanu D., Homotopy analysis method for solving coupled Ramani Equations, Rom. Journ. Phys. (2014), 59 (1-2), 26-35.
• Wazwaz A. M., Multiple soliton solutions for a new coupled Ramani equation, Physica Scripta (2011), 83, 015002.
• Zhou J. K., Differential Transformation and its Applications for Electrical Circuits. Huarjung University Press, Wuuhahn, China, (1986).
• Jang M. J., Chen C. L., Liu Y. C., On solving the initial value problem using the differential transformation method, Applied Mathematics and Computation (2000), 115, 145-160.
• Chen C. L., Ho S. H., Application of differential transformation to eigenvalue problems, Applied Mathematics and Computation (1996), 79, 173-188.
• Kurnaz A., Oturanç G., Kiris M.E., N-Dimensional differential transformation method for solving PDEs, International Journal of Computer Mathematics (2005), 82(3), 369-380.
• Kurnaz A, Oturanc G., The differential transform approximation for the system of ordinary differential equations, Int J. Comput. Math. (2005), 82,709–719.
• Keskin Y., Kurnaz A., Kiris M. E., Oturanc G., Approximate solutions of generalized pantograph equations by the differential transform method. International Journal of Nonlinear Sciences and Numerical Simulation (2011), 8(2), 159-164.
• Gubes M., Peker H. A., Oturanc G., Application of Differential transform method for El Nino Southern oscillation (ENSO) model with compared Adomian decomposition and variational iteration methods, J. Math. Comput. Sci. (2015), 15, 167-178.
• Abazari R., Borhanifar A., Numerical study of Burgers and coupled Burgers equations by differential transformation method, Comput Math Appl (2010), 59, 2711–2722.
• Srivastava V. K., Awasthi M. K., (1 + n)-Dimensional Burgers’ equation and its analytical solution: A comparative study of HPM, ADM and DTM, Ain Shams Engineering Journal (2014), 5(2), 533-541.
• Abazari R., Abazari M., Numerical simulation of generalized Hirota-Satsuma coupled KdV equation by RDTM and comparison with DTM, Commun. Nonlin. Sci. Numer. Simul. (2012), 17, 619-629.
• Gubes M., Keskin Y., Oturanc G., Numerical solution of time-dependent Foam Drainage Equation (FDE), Computational Methods for Differential Equations (2015), 3(2), 111-122.
• Keskin Y., Oturanc G, Reduced Differential Transform Method for Partial Differential Equations, Int. J. Nonlinear Sci. and Num. Simulat. (2009), 10(6), 741-749.
• Keskin Y., PhD thesis, Selcuk University, 2010 (in Turkish).
• Keskin Y., Oturanc G, Reduced Differential Transform Method for Solving Linear and Nonlinear Wave Equations, Iranian Journal of Science and Technology (2010), Trans A 34, A2.
• Keskin Y., Oturanc G., Reduced Differential Transform Method for fractional partial differential equations, Nonlinear Science Letters A (2010),1(2), 61-72.
• Rawashdeh M. S., An Efficient Approach for Time-Fractional Damped Burger and Time-Sharma-Tasso-Olver Equations Using the FRDTM, Appl. Math. Inf. Sci. (2015), 9(3), 1239-1246.
• Rawashdeh M. S., Obeidat N. A., On Finding Exact and Approximate Solutions to Some PDEs Using the Reduced Differential Transform Method, Appl. Math. Inf. Sci. (2014), 8(5), 2171-2176.
• Srivastava V. K., Awasthi M. K., Chaurasia R. K., Reduced differential transform method to solve two and three dimensional second order hyperbolic telegraph equations, Journal of King Saud University - Engineering Sciences (2016), in press.
• Srivastava V. K., Awasthi M. K., Chaurasia R. K. and Tamsir M., The Telegraph Equation and Its Solution by Reduced Differential Transform Method, Modelling and Simulation in Engineering (2013), Article ID 746351, 6 pages.
• Srivastava V. K., Mishra N., Kumar S., Singh B. K., Awasthi M. K., Reduced differential transform method for solving (1 + n) – Dimensional Burgers’ equation, Egyptian Journal of Basic and Applied Sciences (2014), 1(2), 115-119.
• Srivastava V. K., Awasthi M. K., Tamsir M., RDTM solution of Caputo time fractional-order hyperbolic telegraph equation, AIP Advances 3, 032142 (2013).
• Shukla H. S., Tamsir M., Srivastava V. K. and Kumar J., Approximate Analytical Solution of Time-fractional order Cauchy-Reaction Diffusion equation, Computer Modeling in Engineering and Sciences (2014), 103(1), 1-17.
• Srivastava V. K., Awasthi M. K., Kumar S., Analytical approximations of two and three dimensional time-fractional telegraphic equation by reduced differential transform method, Egyptian Journal of Basic and Applied Sciences (2014), 1(1), 60-66.