An efficient numerical method for fractional ordinary differential equations based on exponentially decreasing random memory on uniform meshes

Abstract

This paper proposes a new numerical method to solve non-linear fractional ordinary diff erential equations (FODEs) of the form , with initial conditions. Here, is a continuous function, is an arbitrary positive real number and the fractional diff erential operator, , is in the sense of Caputo derivative. Fixed (short) memory method (SMM) and full memory method (FMM) are two established numerical methods for fractional diff erential equations. In fi xed memory method, tail of the memory at each time step is cut off and hence an uncontrollable error occurs. Further, full memory method is not suitable for long time integration of fractional diff erential equations because of high computational cost. In the proposed method [hereinafter referred to as decreasing random memory method (DRMM)], number of memory points in the past are chosen randomly and they are decreasing along the tail of the memory. Numerical experiments showed that the error occurs in the proposed DRMM is less than that of SMM. The solutions obtained by FMM and DRMM were also very close to the actual solutions of the considered fractional diff erential equations. The proposed method-DRMM is more accurate than the SMM and the estimated order of convergence (EOC) of DRMM is almost same as that of FMM. The proposed method DRMM is more effi cient than the established methods SMM and FMM.