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

Author:

LW Somathilake

University of Ruhuna, LK
About LW
Senior Lecturer, Department of Mathematics
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Abstract

This paper proposes a new numerical method to solve non-linear fractional ordinary differential equations (FODEs) of the form Dγu(t) = f(t, u(t)), with initial conditions u(0) = u0. Here, f(t,u) is a continuous function, 0 < γ ≤ 1 is an arbitrary positive real number and the fractional differential operator, Dγ , is in the sense of Caputo derivative. Fixed (short) memory method (SMM) and full memory method (FMM) are two established numerical methods for fractional differential equations. In fixed memory method, tail of the memory at each time step is cut off and hence an uncontrollable error occurs. Also, full memory method is not suitable for long time integration of fractional differential equations because of high computational cost. In the proposed method, hereinafter referred to as decreasing random memory method (DRMM), several 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 differential equations. The proposed method-DRMM is more accurate than the SMM and the estimated order of convergence (COC) of DRMM is almost the same as that of FMM. The proposed method DRMM is more efficient than the established methods SMM and FMM.

How to Cite: Somathilake, L., 2020. An efficient numerical method for fractional ordinary differential equations based on exponentially decreasing random memory on uniform meshes. Journal of the National Science Foundation of Sri Lanka, 48(2).
Published on 21 Jul 2020.
Peer Reviewed

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