Finite Difference Solution to the Space-Time Fractional Partial Differential-Difference Toda Lattice Equation

Year 2018, Volume: 1 Issue: 3, 202 - 205, 30.12.2018

Refet Polat

https://doi.org/10.33187/jmsm.460001

Cited By: 2

Abstract

This paper deals with the numerical solution of space-time fractional partial differential-difference Toda lattice equation $\frac{\partial^{2\alpha} u_n}{\partial x^{\alpha}\partial t^{\alpha}}=(1+\frac{\partial^\alpha u_n}{\partial t^{\alpha}})(u_{n-1}-2u_n+u_{n+1})$, $\alpha \in (0,1)$. The finite differences method (FD-method) is used for numerical solution of this problem. According to the method, we approximate the unknown values $u_n$ of the desired function by finite differences approximation. As an application we demonstrate the capabilities of this method for identification of various values of order of fractional derivative $\alpha$. Numerical results show that the proposed version of FD-method allows to obtain all data from the initial and boundary conditions with enough high accuracy.

Keywords

Finite differences method, Toda lattice equation, Space-time fractional differential-difference equations, Toda lattice equation

References

• [1] M. Toda, Theory of Nonlinear Lattices, Springer-Verlag, New-York, 1989.
• [2] K. Kajiwara,J. Satsuma, The conserved quantities and symmetries of the two-dimensional Toda lattice hierarchy, J. Math. Phys., 32 (1991), 506—514.
• [3] J. J. Mohan, G. V. S. R. Deekshitulu, Fractional order difference equations, Int. J. Differ. Equ., 2012(2012), Article ID 780619, 11 pages, https://doi.org/10.1155/2012/780619.
• [4] M. Cui, Compact finite difference method for the fractional diffusion equation J. Comput. Phys., 228 (2009), 7792–7804.
• [5] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [6] C. Li, F. Zeng, Numerical Methods for Fractional Calculus CRC Press, Boca Raton, 2015.