An effi cient numerical method for fractional ordinary diff erential equations-based on exponentially decreasing random memory on uniform meshes

This article is published under the Creative Commons CC-BY-ND License (http://creativecommons.org/licenses/by-nd/4.0/). This license permits use, distribution and reproduction, commercial and non-commercial, provided that the original work is properly cited and is not changed in anyway. 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.

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

INTRODUCTION
It is accepted that fractional diff erential equations are more suitable than standard diff erential equations to describe some real world problems. Applications of fractional diff erential equations are found in many areas such as physics and engineering (Schiessel et al., 1995;Diethelm & Freed, 1999;Sabatier et al., 2007), chemistry and biochemistry (Oldham, 2010), fi nance (Scalas et al., 2000) mechanics and system biology, etc. Fractionalorder models are more adequate than the standard order (with integer order derivatives and integrals) models, because fractional derivatives and integrals bring forward the temporal and spatial memory heredity of the materials and processes. This is the main advantage of fractional order models when compared with integer order models. It is very diffi cult if not impossible to obtain analytical solutions for most fractional diff erential equations. However, some analytical (Podlubny, 1999;Diethelm, 2010;Gafi ychuk & Datsko, 2010;Kirane et al., 2014) and numerical (Yuste & Acedo, 2005;Zhuang & Liu, 2006;Yu et al., 2008;Rida et al., 2010;Scherer et al., 2011;Xu & He, 2011;Deng & Li, 2012;Deng et al., 2016) techniques play an important role in identifying the solution behaviour and obtaining approximate solutions of such fractional diff erential equations.

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Journal of the National Science Foundation of Sri Lanka 48 (2) However, numerical integration of fractional ordinary diff erencial equations (FODEs) on fi ner meshes and on long time periods is a computationally expensive process and reducing computational cost by controlling numerical errors is the main challenge. The aim of this paper is to face this challenge by introducing an effi cient numerical scheme to integrate FODEs.

Fractional derivatives
There are diff erent defi nitions for fractional derivatives. Riemann-Liouville defi nition and Caputo defi nition are the popular defi nitions. These defi nitions are reported in previous studies (Podlubny, 1999;Daftardar-Gejji & Jafari, 2005;Diethelm & Kai, 2010;Pindza & Owolabi, 2016;Garrappa, 2018 where denotes the Euler gamma function defi ned by is the fractional order of the derivative, is the smallest integer greater than or equal to q and a is the initial time. Defi nition 1.4 [Caputo fractional derivative (Pindza & Owolabi, 2016)]. Let . The Caputo fractional derivative operator of order is defi ned as: ...(02)

METHODOLOGY
In this paper numerical simulations of fractional diff erential equations are performed using numerical schemes based on two established discrete fractional order derivatives and newly introduced discrete fractional order derivative. These discrete fractional order derivatives are constructed based on fi nite diff erence formula of Caputo derivative proposed in Karatay et al. (2011).

Discrete fractional order derivatives (DFOD)
Fractional-order derivative with full memory Consider the fractional diff erential equation f is continuous function on a suitable set. According to Diethelm and Ford (2002) the above fractional diff erential equation has a unique solution on . Discretising the time interval into N number of partitions with equal step size , one gets a uniform mesh.
Finite diff erence formula of fi rst order on a uniform mesh for Caputo derivative of order is reported in (Karatay et al., 2011)  In full memory method, all the memory points up to the considered time step are taken into account in the calculation of the fractional order derivative at that point ( Figure 1).

Journal of the National Science Foundation of Sri Lanka 48(2)
June 2020 Fractional order derivative with fi xed (short) memory length In full memory method, the unknown values depend on all the past time steps. Applying full memory method in solving fractional diff erential equations is computationally ineffi cient. However, in fi xed (short) memory principle (Podlubny, 1999), it is assumed that the unknown values at a time step depend only on the values at recent past time steps corresponding to time length L ( Figure 2). Based on this assumption, Grünwald-Letnikov fractional-order derivative with fi xed memory length L, is defi ned in Abdelouahab and Hamri (2016). This method is computationally effi cient but computational error is higher due to ignoring the tail of the memory.
Based on the short memory principle, the fractional order derivative with short memory L, , corresponding to can be written in the form: ... (03) where denotes the smallest integer greater than or equal to for . Although, this scheme is computationally cheap, the error propagation due to neglecting the tail of the memory in each time step is uncontrollable without increasing the fi xed memory length. When memory length is increased the computation becomes diffi cult again.
A new discrete fractional order derivative with exponentially decreasing Random memory In this section, a new discrete fractional order derivative on uniform meshes is introduced by choosing memory points randomly and they are decreasing along the tail of the memory. Suppose that the time interval is discretised into N number of partitions with equal step size . Let M and N 1 be integers such that N 1 = N/M. That is the interval is divided into N 1 number of partitions with equal length (Figure 3). Now consider the case of integration up to time Let Generate number of random integers within the partition for Here is a real number which determines the decreasing speed of the number of random points through the partitions as i increases from 1 to . Let be the set of number of random integers chosen on partition sorted in ascending order. Denote the set as Let (Figure 4). Choose memory points on as follows: June 2020 Journal of the National Science Foundation of Sri Lanka 48 (2) Fractional-order derivative with random memory is defi ned as follows.
... (04) Numerical schemes This section introduces three numerical schemes based on the above introduced fractional order derivatives to simulate fractional diff erential equations of the form: ... (05) where is a continuous function. where The weights are introduced to reduce the error, which occurs at the partition due to ignoring number of memory points. Figure 5 shows the variation of the distribution of random memory points over the time interval [0, 1] at diff erent levels of . These distributions are not fi xed as memory points are chosen randomly. It can be observed that as increases the sparsity of the distribution of random memory points increases. Figure 6 depicts how the random memory points distribute over the space upto diff erent time levels. for m = 1, 2, ...N. When integrating a FODE from 0 to T using this scheme, total number of summations (TNOS) have to be done in the process. June 2020 Journal of the National Science Foundation of Sri Lanka 48 (2) A fully explicit scheme with fi xed (short) memory A fully explicit scheme with short memory length L can be obtained in the following form by replacing fractional order derivative of (05) by (03) and rearranging the terms.
A fully explicit scheme with decreasing random memory Replacing fractional order derivative of (05) by (04) and rearranging the terms, the following fully explicit scheme with decreasing random memory can be obtained.
.   Computational cost of the above schemes mainly depend on the corresponding TNOS. According to Table 1 TNOS of DRMM (for = 2, 3, 4, 5) is less than that of SMM (for PML = 20) and FMM. Therefore, the computational cost of DRMM (for = 2, 3, 4, 5) should be less than that of SMM (for PML = 20) and FMM.

Numerical examples
Accuracies of the above three numerical schemes are compared with the exact solutions of the following fractional diff erential equations with and the initial conditions : ... (07) ... (08) ... (09) The exact solutions of the FODEs (07), (08) Table  5 the computational cost of FMM is higher than that of DRMM. Now compare the errors between numerical solutions obtained by the numerical schemes FMM, SMM, DRMM and exact solutions. Table 2 shows the relative mean square error (Re-MSE) between exact solutions and numerical solutions at the end point of the considered time ranges of the FODEs (07), (08) and (09) obtained by the three numerical schemes. In these simulations = 3, M = 100 for DRMM, PML = 20 for SMM and = 0.0001. Re-MSE of DRMM is less than that of SMM. This implies that DRMM is more accurate than SMM.

Estimated (or experimental) order of convergence (EOC)
The estimated (or experimental) order of convergence (Diethelm et al., 2004;Li & Zeng, 2013;Garrappa, 2015;2018;Liu et al., 2018) an estimator for convergence order of a numerical scheme, is measured by Here, is the numerical value of and is the exact solution at EOCs of FMM and DRMM corresponding to the fractional diff erential equations (07), (08) and (09) are shown in Tables 3 and 4, respectively. EOC of SMM is not taken into account as it is very low. Convergence orders of FMM and DRMM are approximately equal as EOCs of DRMM and FMM are approximately equal.

Computational cost
In this section, the computational time (CPU time) of the three numerical schemes FMM, DRMM and SMM are compared. Algorithms for the numerical schemes FMM, SMM and DRMM were developed and solved using Matlab on a 2.3GHz, Intel core i5 laptop computer that had 8GB of RAM and Microsoft Windows 10. Now defi ne the two terms, computational time (CT) reduction percentage (CTRP) and increment percentage ( ) between two methods FMM and DRMM as follows:   (07), (08) and (09) for γ = 0.8

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Journal of the National Science Foundation of Sri Lanka 48 (2) (f) Solutions of (09) for 1.58 ≤ t ≤ 1.78 (a) Solutions of (07) for 0 ≤ t ≤ 20 (b) Solutions of (07) for 19 ≤ t ≤ 20 (c) Solutions of (08) for 0 ≤ t ≤ 1.0 (d) Solutions of (08) for 0.8 ≤ t ≤ 0.9 (e) Solutions of (09) for 0 ≤ t ≤ 2.0 Journal of the National Science Foundation of Sri Lanka 48(2) June 2020 Step Size of FODE EOC of FODE (∆t = 1/2 p )    (07), (08) and (09) (07) up to T = 100 using SMM for γ = 0.8 and mesh size ∆t = 10 −3  Table 5 shows the  and  of  and when FODE (07) is integrated up to diff erent time levels. Computational cost of DRMM has been reduced approximately by 20 % when compared with FMM. It can be seen that CTRP increases and decreases as time increases. Therefore, the DRMM is computationally more effi cient than the FMM in integration of FODEs on more fi ner meshes or on larger time ranges. of DRMM for = 0.8, T = 100, = 0.0001, and = 3 is 0.238 (Table 5). According to Figure 8, it can be seen that the computational cost of SMM corresponding to = 0.238 is approximately 119.4. This computational cost is higher than that of DRMM (91.4200). Therefore, DRMM is computationally effi cient than SMM when computational error is fi xed to a same value of both methods.

CONCLUSION
The relative mean square error between exact solution and numerical solutions of the considered FODEs obtained by DRMM is less than that of SMM. Therefore, the proposed method, DRMM, is more accurate than SMM. The values of EOC of FMM corresponding to the considered three FODs are approximately equal to one. These results agree with the fact that the order of convergence of FMM is one. The values of EOC Journal of the National Science Foundation of Sri Lanka 48 (2) June 2020 of DRMM is approximately equal to that of FMM. Therefore, EOC of the proposed method, DRMM, is good enough for numerical simulations of fractional diff erential equations. Computational cost of DRMM is less than that of FMM and SMM when the accuracy of SMM and DRMM are in the same level. Based on these facts, it can be concluded that the proposed method, DRMM, is more effi cient than the established methods, SMM and FMM, in numerical computations of FODEs. Future research would consider the applicability of newly proposed DRMM to higher order numerical schemes for fractional diff erential equations. Further, it is expected to investigate the suitability of DRMM to solve time fractional reaction diff usion systems.