Comparative analysis of three memory selection methods for time integration of Fractional Reaction-Diffusion Equations

By discretising in space, a non-linear time fractional reaction-diffusion equations (TFRDEs) can be converted into a system of time-fractional differential equations (TFDEs). The full memory method (FMM) and short memory method (SMM) are well-established memory selection methods used in the time integration of TFDEs. The main drawbacks of FMM and SMM are higher computational cost and uncontrollable error respectively. The only way to increase the accuracy of SMM is by increasing short memory length which causes an increase in computational cost. Especially when we apply these two methods to integrate TFRDEs, we have to solve a large system of TFDEs. Therefore, the drawbacks of these two methods affect seriously, when these are applied to solve TFRDEs. This paper aims to investigate the accuracy and efficiency of the memory selection method, Exponentially Decreasing Random Memory Method (EDRMM), and compare it with FMM and SMM when these methods apply to integrate TFRDEs. Based on these three memory selection methods, three semi-implicit numerical schemes namely semiimplicit scheme with full memory method (SI-FMM), semi-implicit scheme with short memory method (SI-SMM), and semi-implicit scheme with exponentially decreasing random memory method (SI-EDRMM)) are proposed and the accuracy and CPU time (computational time (CT)) of these three numerical schemes are compared. To do this comparison, these three numerical schemes are applied to four TFRDEs whose exact solutions are known. Numerical experiments confirm that the accuracy and efficiency of the SI-EDRMM are better than that of SI-SMM and the efficiency of SI-EDRMM is higher than that of SI-FMM. Therefore, EDRMM is better than SMM and FMM for the integration of TFRDEs.