Abstract
This study examines the time fractional KdV Burgers equation with the initial conditions by using the extended result on Caputo formula, finite difference method (FDM). For this reason, various fractional differential operators are defined and analyzed. In order to check the stability of the numerical scheme, the Fourier-von Neumann technique is used. By presenting an example of KdV Burgers equation above mentioned issues are discussed and numerical solutions of the error estimates have been found for the FDM. For the errors in $L_2$ and $L_\infty$ the method accuracy has been controlled. Moreover, the obtained results have been compared with the exact solution for different cases of non-integer order and the behavior of the potentials u is presented as a graph. The numerical results have been shown in tables.
Highlights
NfPDE will get the shape of integer-order differential equations
The formulas giving approximate values of partial derivatives according to finite difference method (FDM) are as follows
We present the consistency of Eq (1) with FDM
Summary
Some important notations are needed to define the forward FDM, these are: a) ∆x, which is the spatial step b) ∆t, which is the time step c) xi a + i∆x, i. Uij , where ui,j will is the numerical approximations of u (x, t) at the point (xi, tj). Htui,j = ui,j − ui,j−1, Hxui,j = ui+1,j − ui−1,j , Hxxui,j = ui+1,j − 2ui,j + ui−1,j, Hxxxui,j = ui+2,j − 2ui+1,j + 2ui−1,j − ui−2,j ,. The formulas giving approximate values of partial derivatives according to FDM are as follows
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