Abstract:
The aim of this paper is to introduce a more accurate and efficient discrete
fractional order derivative to integrate non-linear time fractional order
reaction diffusion equations. The Fixed Memory method (Short Memory
method (SMM)) and the Full Memory method (FMM) are two established
discrete fractional order derivatives (DFODs). In the fixed memory method,
the tail of the memory at each time step is cut off and hence uncontrollable
error occurs. Also, FMM is not efficient for long time integration of large
systems of fractional differential equations because of higher computational
cost. To overcome these barriers, author propose a new discrete fractional
order derivative. In this method, the number of memory points in the past
are reduced by choosing only a part of the memory points randomly and
decreasing along the tail of the memory (call this the Decreasing Random
Memory Method (DRMM)). Author constructed three semi implicit
numerical schemes, semi implicit scheme with full memory method (SIFMM),
semi implicit scheme with short memory method (SI-SMM) and
semi implicit scheme with decreasing random memory method (SI-DRMM)
and compare accuracies and computational costs (CT) of these three
numerical schemes. To do this comparison, author applied these three
numerical schemes for three fractional reaction diffusion equations whose
exact solutions are known. Numerical experiments show that the error
occurr in proposed SI-DRMM is less than that of SI-SMM. Furthermore,
SI-DRMM is computationally cheaper than the SI-FMM. Therefore, the
proposed DRMM is more accurate than SMM and more efficient than
FMM.
