From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations.

Transcription

1 From tochatic procee to numerical method: A new cheme for olving reaction ubdiffuion fractional partial differential equation. C. N. Angtmann, I. C. Donnelly, B. I. Henry School of Mathematic and Statitic, UNSW Autralia, Sydney NSW 2052 Autralia B. A. Jacob School of Computer Science and Applied Mathematic, Univerity of the Witwaterrand, Johanneburg, Private Bag 3, Wit 2050, South Africa DST-NRF Centre of Excellence in Mathematical and Statitical Science CoE-MaSS) T. A. M. Langland Department of Mathematic and Computing, Univerity of Southern Queenland, Toowoomba QLD 4350 Autralia J. A. Nichol School of Mathematic and Statitic, UNSW Autralia, Sydney NSW 2052 Autralia Abtract We have introduced a new explicit numerical method, baed on a dicrete tochatic proce, for olving a cla of fractional partial differential equation that model reaction ubdiffuion. The cheme i derived from the mater equation for the evolution of the probability denity of a um of dicrete time random walk. We how that the diffuion limit of the mater equation recover the fractional partial differential equation of interet. Thi limiting procedure guarantee the conitency of the numerical cheme. The poitivity of the olution and tability reult are imply obtained, provided that the underlying proce i well poed. We alo how that the method can be applied to tandard reaction-diffuion equation. Thi work highlight the broader applicability of uing dicrete tochatic procee to provide numerical cheme for partial differential equation, including fractional partial differential equation. 1. Introduction Reaction ubdiffuion fractional partial differential equation have been widely ued in recent year a mathematical model of ytem of particle ubject to, trapping, obtacle and reaction [1, 2, 3, 4, 5, 6, 7, 8]. Subdiffuion, characteried by a mean quared diplacement of diffuing particle that grow lower Preprint ubmitted to Journal of Computational Phyic November 9, 2015

2 than linear with time, ha been oberved in hydrogeology [9, 8], phyic [10], biology [11], finance [12] and chemitry [13]. Reaction ubdiffuion fractional partial differential equation may be derived from generalied continuou time random walk CTRW) [14, 15] by incorporating reaction kinetic into the proce [1, 2, 3, 5, 7]. In tandard reaction-diffuion partial differential equation the reaction term and the diffuion term are additive [16, 17, 18, 19], whilt in reaction ubdiffuion equation, derived from CTRW, the reaction kinetic and the diffuion are entwined in the fractional partial differential equation [2, 3, 5]. In general it i not poible to obtain cloed form algebraic olution for non-linear reaction ubdiffuion equation. Thi ha timulated a great deal of interet in the development of numerical method for thee equation. Some of the numerical method developed for ubdiffuion include, explicit and implicit finite difference method [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], pectral method [32, 33, 34, 35], and Galerkin method [36, 37]. One of the implet method i to approximate the continuou fractional derivative uing a Grünwald-Letnikov fractional derivative [38, 39]. There have been numerou article publihed on numerical method for time fractional reaction-diffuion equation. Mot of the article in thi area conider fractional diffuion with the ad-hoc addition of tandard reaction, or addition of reaction including the time fractional derivative on the reaction, or a combination of both [40, 41, 42, 43, 44]. The ad-hoc addition of ubdiffuion and reaction may lead to unphyical negative olution [2]. Thi can be avoided by taking a more phyical approach, deriving the reaction ubdiffuion equation from an underlying tochatic proce, a continuou time random walk [2, 5, 6, 7]. The numerical method that we derive here i alo baed on an underlying tochatic proce, a dicrete time random walk DTRW). In [45], we introduced a novel numerical cheme for olving fractional Fokker- Planck equation that wa baed on a DTRW. Rather then dicretiing a continuum et of equation the numerical cheme i contructed by conidering a dicrete time and pace tochatic proce. The proce wa choen uch that, in the diffuion limit, the evolution equation of the probability denity that decribed the ytem would become a fractional Fokker-Planck equation. A the proce i dicrete the probability denity can be calculated recurively and ued to approximate the diffuion limit denity, and hence the olution of the fractional Fokker-Planck equation. A imilar approach ha alo been ued to olve et of fractional ordinary differential equation ariing from a modified SIR epidemic model [46]. In thi article we have extended the DTRW formalim to provide a numerical cheme for olving reaction ubdiffuion fractional partial differential equation of the form u t = D 2 [ α x 2 e t 0 au,x,t ) dt 0Dt 1 α e )] t 0 au,x,t ) dt u + cu, x, t) au, x, t)u, where u = ux, t), au, x, t) and cu, x, t) are non-negative and 0 D 1 α t 1) i a 2

3 Riemann-Liouville fractional derivative of order 1 α. Equation of thi form can be derived from a phyically conitent theory of continuou time random walk [7], including extenion to multiple pecie. The numerical cheme i obtained by firt deriving a et of dicrete time and pace mater equation for an enemble of particle undergoing a DTRW with reaction. The evolution of thee dicrete equation provide the bai for a numerical olution of the fractional partial differential equation. The reulting numerical cheme i an explicit forward time-tep method, Eq. 66) in thi manucript. Provided that the underlying proce i well poed the olution i guaranteed to be poitive, a i required for reaction diffuion procee, and tability reult follow. The numerical cheme can be implemented without requiring a knowledge of tochatic procee, however additional ymmetrie, if known, can be incorporated into the cheme. The remainder of the paper i organied a follow. In Section 2 we preent the formalim for the DTRW with reaction, and derive the appropriate mater equation. In Section 3 we conider pecific form for the jump and waiting time probability ma function pmf) in the DTRW. In Section 4, we how that in the continuou pace and time limit, depending on the jump and waiting time pmf, the mater equation converge to tandard reaction-diffuion PDE or reaction ubdiffuion fractional PDE. In Section 5, we outline how to implement the dicrete time mater equation a a numerical cheme for the PDE. In Section 6 we derive tability reult from the underlying tochatic proce. We how that the numerical cheme ha poitive olution and i table for phyically conitent reaction kinetic. Detailed example uing the cheme are preented in Section The Mater Equation for Dicrete Time Random Walk With Reaction In thi ection we derive the dicrete time mater equation of DTRW with reaction an ubdiffuion. Thee mater requation underpin the numercial cheme. The derivation of the mater equation follow the framework that we developed in [45] for the derivation of dicrete time mater equation of DTRW ubject to forcing. The additional complication of reaction in the dicrete time mater equation are treated in a manner imilar to our treatment of reaction in continuou time mater equation [47, 7] where the reaction are modelled a a birth annihilation) and death creation) proce. The derivation of the mater equation i broken into two main part. Firtly we conider the evolution of a ingle particle ubject to a pace- and time-dependent probability of annihilation. Next, by conidering an enemble of uch particle ariing from a pace- and time-dependent creation proce, we contruct the dicrete mater equation that govern the evolution of the reaction diffuion ytem A Dicrete Time Random Walk with an Annihilation Rate We begin by conidering a particle travering a one-dimenional lattice V = L 1,..., i 1, i, i + 1,..., L 2 }, with L 1, L 2 Z, in dicrete time n N. At each 3

4 time tep the particle will randomly do one of three thing; either remain at the current ite, undergo a jump, or be annihilated and removed from further conideration. The probability of the action are independent but may depend on the current ite and time tep, and the time tep on which the particle arrived at the ite. For a particle that arrived at the ite i on the m th time tep the probability of jumping to ite j on the n th time tep i given by a tranition probability ma function Ψj, n i, m). Thi function completely decribe the evolution of the proce. We make a further aumption that thi probability ma function pmf) can be decompoed into two independent pmf, Ψj, n i, m) = λj i)ψn m), 2) where ψn m) i the pmf for waiting n m time tep before tranitioning and λj i) i the pmf for jumping to ite j conditional on being on ite i. A uual, the pmf are normalied, and ψn) = 1, 3) n=0 L 2 It i alo important to note that we aume j=l 1 λj i) = 1. 4) ψ0) = 0, 5) a we do not allow multiple jump in the ame time tep. It i poible to incorporate a patial dependence into the waiting time probabilitie, and time dependence into the jump probabilitie but thi i not included here for implicity. The probability that a particle will be annihilated at ite i on the n th time tep i denoted Ai, n). Thi time- and pace-dependence allow for the conideration of arbitrary function provided that 0 Ai, n) 1. The probability of not being annihilated over a number of time tep, between time m and n at ite i, i then given by the urvival function Θi, n, m) = l=m 1 Ai, l)), 6) and we ue the convention, Θi, n, n) = 1. It i ueful to note the emigroup property of Θ; Θi, n, m) = Θi, n, k) Θi, k, m) m k n. 7) 4

5 The probability of a particle arriving at ite i on the n th time tep, given that it i created and begin walking at ite i 0 on the n th 0 time tep, i recurively defined by the probability flux; Qi, n i 0, n 0 ) = δ i,i0 δ n,n0 + L 2 j=l 1 Ψi, n j, m) Θj, n, m) Qj, m i 0, n 0 ) 8) where Qi, m i 0, n 0 ) = 0 for all m < n 0. Thi equation expree that the flux into ite i at time tep n i the um of all the fluxe into ite j at the earlier time tep m that urvived until time tep n when they tranition to ite i. Note that the upper limit of the um could be n, but Ψi, n j, n) = 0 due to Eq. 5). The probability of a particle at a ite not jumping by time tep n, given the particle arrived at the earlier time m i given by the urvival probability n m Φn m) = 1 ψk). 9) The probability of the particle being at ite i on the n th time tep can then be written n Xi, n i 0, n 0 ) = Φn m) Θi, n, m) Qi, m i 0, n 0 ). 10) The right hand ide i the um over all poibilitie of the particle arriving at an earlier time tep, m, and not being annihilated or jumping before the n th time tep. The change in probability ma X between time tep n and i obtained directly from Eq. 10): Xi, n i 0, n 0 ) Xi, n 1 i 0, n 0 ) = Thi can be rewritten a k=0 n Φn m) Θi, n, m) Qi, m i 0, n 0 ) Xi, n i 0, n 0 ) Xi, n 1 i 0, n 0 ) = Qi, n i 0, n 0 ) + Φn 1 m) Θi, n 1, m) Qi, m i 0, n 0 ). Θi, n, m)φn m)qi, m i 0, n 0 ) 11) Θi, n 1, m)φn 1 m)qi, m i 0, n 0 ). 12) 5

6 From Eq. 9) we have that Φn m) = Φn 1 m) ψn m), and from Eq. 7) we ee that Θi, n, m) = Θi, n, n 1)Θi, n 1, m), hence Eq. 12) can be rewritten a Xi, n i 0, n 0 ) Xi, n 1 i 0, n 0 ) = Qi, n i 0, n 0 ) ψn m)θi, n, m)qi, m i 0, n 0 ) Φn 1 m)θi, n 1, m) 1 Θi, n, n 1)) Qi, m i 0, n 0 ). Noting that from Eq. 6), 1 Θi, n, n 1) = Ai, n 1) and we obtain Xi, n i 0, n 0 ) Xi, n 1 i 0, n 0 ) = Qi, n i 0, n 0 ) Ai, n 1) 13) ψn m)θi, n, m)qi, m i 0, n 0 ) Φn 1 m)θi, n 1, m)qi, m i 0, n 0 ). 14) Then by uing the definition in Eq. Xi, n 1 i 0, n 0 ) to arrive at 10), we ubtitute the lat term with Xi, n i 0, n 0 ) Xi, n 1 i 0, n 0 ) = Qi, n i 0, n 0 ) We define the outgoing flux of ite i at time tep n to be ζi, n i 0, n 0 ) = ψn m)θi, n, m)qi, m i 0, n 0 ) Ai, n 1)Xi, n 1 i 0, n 0 ). 15) ψn m) Θi, n, m) Qi, m i 0, n 0 ). 16) Note from the definition of Q, Eq. 8), and recalling Eq. 2), that Qi, n i 0, n 0 ) δ n,n0 δ i,i0 = L 2 j=l 1 λj i) ζj, n i 0, n 0 ). 17) Subtituting thi into Eq. 15) give Xi, n i 0, n 0 ) Xi, n 1 i 0, n 0 ) = L 2 j=l 1 λj i) ζj, n i 0, n 0 ) + δ n,n0 δ i,i0 ζi, n i 0, n 0 ) Ai, n 1)Xi, n 1 i 0, n 0 ). 18) 6

7 To obtain the generalied mater equation GME) governing the evolution of the probability ma for the particle we ue the emigroup property of Θ, given in Eq. 7), to write Eq. 10) a Xi, n i 0, n 0 ) Θi, n, 0) Similarly for Eq. 16) we get ζi, n i 0, n 0 ) Θi, n, 0) = = n n Qi, m i 0, n 0 ) Θi, m, 0) Qi, m i 0, n 0 ) Θi, m, 0) Φn m). 19) ψn m). 20) To proceed further, we make ue of the ingle-ided Z-tranform [48] defined by Z n Y n) z} = Y n)z n. 21) Taking the Z-tranform of Eq. 19) and 20) give } } Xi, n i0, n 0 ) Z n Qi, Θi, n, 0) z n i0, n 0 ) = Z n Θi, n, 0) z Z n Φn) z} 22) and Z n ζi, n i0, n 0 ) Θi, n, 0) } z n=0 = Z n Qi, n i0, n 0 ) Θi, n, 0) } z Z n ψn) z}. 23) Similar to the analyi of CTRW [7], it i convenient to define a dicrete memory kernel Kn) by the Z-tranform relation Z n Kn) z} = Z nψn) z} Z n Φn) z}. 24) Note that from Eq. 5) and Eq. 9), we have K0) = 0. Dividing Eq. 23) by Eq. 22) and inverting the Z-tranform we can expre ζ in term of X, ζi, n i 0, n 0 ) = Θi, n, 0) Kn m) Xi, m i 0, n 0 ). 25) Θi, m, 0) We can ubtitute thi expreion for ζ in to Eq. 18) to obtain the following generalied mater equation for the evolution of a ingle particle probability ma, X, ubject to an annihilation proce; Xi, n i 0, n 0 ) Xi, n 1 i 0, n 0 ) = L 2 j=l 1 λj i) Kn m) Θj, n, m) Xj, m i 0, n 0 ) Kn m) Θi, n, m)xi, m i 0, n 0 ) Ai, n 1) Xi, n 1 i 0, n 0 ) + δ n,n0 δ i,i0. 26) 7

8 2.2. An Enemble of Dicrete Time Random Walk With Creation and Annihilation Under our model of reacting and diffuing particle the ytem comprie of an enemble of particle that are created at ome point, undergo a random walk and are annihilated at ome other point. The evolution of thi enemble can be found by conidering the evolution of the ingle particle in the enemble. The GME for the ingle particle ubject to an annihilation proce, Eq. 26), can be thought of a propagating each ingle particle from ome initial point onward. We aume that the creation proce i Markovian and defined uch that the expected number of particle created at lattice ite i on time tep n i given by the arbitrary function, Ci, n) 0 for all i and n. The expected number of particle from the enemble at poition i at time n can then be found by propagating the creation of all the particle forward in time. The number of particle at ite i on time tep n i then given by, Ui, n) = L 2 i 0=L 1 n 0=0 n Xi, n i 0, n 0 )Ci 0, n 0 ). 27) A the creation proce i Markovian, Ci, n), can depend on the tate of the ytem at the previou time tep, Uj, )} j. To find the evolution of U with time we multiply the ingle particle GME, Eq. 26), by Ci 0, n 0 ) and then um over all poible tarting point; i 0 from L 1 to L 2 and n 0 from 0 to n. Uing the definition in Eq. 27) we obtain Ui, n) Ui, n 1) = L 2 j=l 1 λj i) Kn m) Θj, n, m) Uj, m) Kn m) Θi, n, m)ui, m) Ai, n 1) Ui, n 1) + Ci, n). 28) Thi i the generalied mater equation for a ingle pecie dicrete general reaction diffuion proce. Note that Ai, n 1) may alo be dependent on Uj, n 1)} j. With the appropriate choice of the waiting time pmf, and hence the memory kernel K, Eq. 28) may model the cae of reaction-diffuion or reaction ubdiffuion Interacting Enemble of Dicrete Time Random Walk With Annihilation and Creation Similar to the approach in [7] we can generalie thi mater equation for the cae of a multi-pecie enemble of population that have interaction between them. Thi can repreent, for example, ytem uch a chemical reaction or microbiological population dynamic. To do o we allow the creation and annihilation rate to depend on all population. A we have worked with creation and annihilation probabilitie that are arbitrary in pace and time we may 8

9 incorporate non-linear dependencie on population into thee probabilitie. We calculate Θ k i, n, m) from the annihilation probabilitie through Θ k i, n, m) = 1 A k i, l)), 29) l=m where k i the pecie number. Note that A k i, l) may alo be dependent on the tate of the ytem U p j, l)} j,p at time l. Thu the multi-pecie DTRW mater equation with non-linear reaction i given by U k i, n) U k i, n 1) = L 2 j=l 1 λ k i j) K k n m) Θ k j, n, m) U k j, m) K k n m) Θ k i, n, m) U k i, m) A k i, n 1)U k i, n 1) + C k i, n). 30) In thee equation C k i, n), can depend on the tate of the ytem at the previou time tep, U p j, n 1)} j,p. 3. Jump and Waiting Time Probability Ma Function The dicrete generalied mater equation can be ued to formulate a numerical method for olving continuum reaction-diffuion type equation, including fractional reaction-diffuion equation. To obtain a numerical method for a given continuum reaction-diffuion equation from the dicrete GME appropriate choice need to be made for the jump and waiting time ditribution. Once thee choice have been made, the dicrete GME can be ued a an explicit numerical cheme for approximating reaction-diffuion PDE that are the continuum limit of the dicrete GME. The correponding continuum limit of the dicrete GME will be obtained in ection 4. Thi convergence in the continuum limit etablihe the conitency of the numerical cheme Jump Probability Ma Function We conidering a jump proce compoed of nearet neighbour and elf jump. The jump pmf i given by λj i) = r 2 δ i+1,j + r 2 δ i 1,j + 1 r)δ i,j. 31) Here r [0, 1] i the probability that a jump will not be a elf jump. Thi jump pmf i ymmetric. The incorporation of aymmetric pace and time dependent jump ha previouly been conidered to model a pace and time dependent force [45]. 9

10 Subtituting the jump pmf, Eq. 31), into the dicrete GME, Eq. 28), give, Ui, n) Ui, n 1) = r 2 + r 2 r Kn m) Θi + 1, n, m) Ui + 1, m) Kn m) Θi 1, n, m) Ui 1, m) Kn m) Θi, n, m)ui, m) Ai, n 1) Ui, n 1) + Ci, n). 32) Uing Eq. 7), we can expre the um in the above equation a dicrete convolution, Ui, n) Ui, n 1) = r 2 Θi 1, n, 0) + r 2 Θi + 1, n, 0) rθi, n, 0) Ui + 1, m) Kn m) Θi + 1, m, 0) Ui 1, m) Kn m) Θi 1, m, 0) Ui, m) Kn m) Ai, n 1) Ui, n 1) + Ci, n). Θi, m, 0) 33) 3.2. Waiting Time Probability Ma Function Markovian Auming that the probability of the particle jumping to a new ite on any given time tep i ω and i independent of the time that the particle arrived at the current ite, then it follow that waiting time pmf i given by, The correponding urvival function i, ψn) = ω1 ω) n. 34) Φn) = 1 ω) n. 35) Equation 24) can now be ued to obtain an explicit expreion for the memory kernel, Kn) = ωδ 1,n. 36) 3.3. Sibuya The waiting time pmf i non-markovian when the probability of the particle jumping on any given time tep depend on how many time tep the particle ha waited for. A pecial cae i when the probability of jumping on a given time tep decreae the longer the particle wait without jumping. Sibuya waiting time arie by conidering a particle that ha a probability α n of jumping after 10

11 waiting n time tep for ome 0 < α 1 [49]. The waiting time urvival function i given by n Φn) = 1 α ), 37) k k=1 with Φ0) = 1. The correponding waiting time pmf i, ψn) = α n k=1 1 α ), 38) k with ψ0) = 0 and ψ1) = α. A in [45], we can conider the Z-tranform to obtain an analytic expreion for the memory kernel, K, from Eq. 24), Z n Kn) z} = Z nψn) z} Z n Φn) z} = 1 1 z 1 ) α 1 z 1 ) α 1 = 1 z 1 ) 1 α 1 z 1 ), 39) and hence it can be hown that the kernel, for n 1 can be given by with K0) = 0. coefficient, Kn) = n k=1 1 2 α ) + δ 1,n, 40) k We may alo write the Sibuya kernel in term of binomial ) 1 α Kn) = 1) n δ 0,n + δ 1,n, 41) n and the kernel can be obtained from a recurion relation [45]. It i intereting to note that the term in the memory kernel can be related to the Grünwald- Letnikov fractional derivative via D 1 α fx)) = lim h 0 k=0 n Kn) + δ 0,n δ 1,n ) fx kh) h 1 α. 42) 4. Continuum Limit of the Dicrete Generalied Mater Equation In thi ection we derive continuum limit of the GME, Eq. 33), correponding to Markovian, and Sibuya waiting time. Thee continuum limit yield the tandard reaction-diffuion PDE and the fractional reaction-diffuion PDE repectively. Thi how that the dicrete GME atify the conitency condition for an approximation to the PDE, and can therefore be ued a a conitent explicit numerical method. 11

12 4.1. Markovian Waiting Time Uing the Markovian memory kernel, Eq. 36), the dicrete GME, Eq. 33), implifie to, Ui, n) Ui, n 1) = rω Θi 1, n, n 1)Ui 1, n 1) 2 ) 2Θi, n, n 1)Ui, n 1) + Θi + 1, n, n 1)Ui + 1, n 1) 43) Ai, n 1) Ui, n 1) + Ci, n). Uing Eq. 6) we have Ui, n) Ui, n 1) = rω 1 Ai 1, n 1))Ui 1, n 1) 2 ) 21 Ai, n 1))Ui, n 1) + 1 Ai + 1, n 1))Ui + 1, n 1) Ai, n 1) Ui, n 1) + Ci, n). 44) We now relate our olution Ui, n) of the dicrete GME to a function in continuou pace and time. We conider a uniform grid with pacing x and t, and aociate the point i, n) Z 2 with point i x, n t) R 2. We aociate the dicrete function Ui, n) with a continuou function u x, t) that i dependent on the grid pacing x and t. The function are related by requiring equality on the grid point, u i x, n t) = Ui, n). At the grid point the function u i the olution of a two parameter family of dicrete GME. The continuum limit i obtained when the eparation between grid point, x and t, goe to zero, that i, lim u x, t) = ux, t). 45) x 0, t 0 In order for thi limit to exit we take a diffuion limit, which require the ratio of x 2 and t to remain contant [50]. We alo define a continuou verion of the annihilation probability, a x, t), where a i x, n t) = Ai, n). The dependence in the continuou function define a x, t) a the probability of an annihilation event between t and t + t. We can then define an annihilation rate a the limit, ax, t) = lim t 0, x 0 In a imilar manner, we define a continuou creation rate, cx, t) = where c i x, n t) = Ci, n). lim t 0, x 0 a x, t). 46) t c x, t), 47) t 12

13 With the above definition of u, a, c the dicrete GME, Eq. 44), then become, u i x,n t) u i x, n 1) t) = rω 1 a i 1) x, n 1) t))u i 1) x, n 1) t) 2 21 a i x, n 1) t))u i x, n 1) t) ) + 1 a i + 1) x, n 1) t))u i + 1) x, n 1) t) a i x, n 1) t)u i x, n 1) t) + c i x, n t). 48) We now introduce continuou variable x, t at the point x = i x, t = n t and expand the continuou function, u, a, c, in Taylor erie about x and t, with a x, t) = o t) and c x, t) = o t), to obtain t u t + o t 2 ) = rω 2 ) ) 2 x2 x 2 u x, t) + o t) + o x 2 ) a x, t)u x, t) + c x, t). 49) Finally we divide by t and conider a equence of procee correponding to the above equation in the limit x 0 and t 0, uch that D = r x 2 lim x 0, t 0 2 t, 50) exit. Thu we obtain the diffuion limit of the GME arriving at a reactiondiffuion PDE, ux, t) t = ωd 2 ux, t) x 2 ax, t)ux, t) + cx, t), 51) where ax, t) and cx, t) can be determined from the reaction kinetic for ux, t) and may depend explicitly on ux, t). The parameter ω, which i the probability of the particle jumping to a new ite on any given time tep in the dicrete GME, can be interpreted a a time cale parameter in the continuum equation Sibuya Waiting Time When uing the Sibuya waiting time pmf a different approach need to be taken when finding the continuum limit. The um over the memory kernel i not amenable to direct calculation. However the um can be written a a dicrete convolution, which enable u to exploit propertie of tranform method in finding the continuum limit. The approach from the dicrete to the continuum can be carried out in general by conidering invere Laplace tranform with limit to continuou time, and continuou pace of tar tranform from dicrete time, and dicrete pace. To begin, we conider a general function of dicrete pace and time Y i, n) and 13

14 define the unilateral tar tranform with repect to the dicrete time variable, n, a Zn Y i, n), t} = Y i, n) e n t. 52) We define a bilateral tar tranform with repect to the dicrete pace variable, i, a Ẑi Y i, n) q, x} = Y i, n) e i xq. 53) n=0 i= Thee tranform are related to the unilateral Z-tranform with z = e t and the bilateral Z-tranform with z = e xq. Similar to the Markovian cae, we conider a equence of continuou function, y, uch that y i x, n t) = Y i, n). In thi manner we aociate an interval t between the time tep n and n + 1 and an interval x between the pace grid point i and i + 1. The function y x, t) will be different for different interval ize and the ubcript denote thi functional dependence. The continuum limit, x 0, t 0, can then be obtained from the invere unilateral Laplace tranform with limit to continuou time and the invere bilateral Laplace tranform with limit to continuou pace of the unilateral tar tranform from dicrete time and the bilateral tar tranform from dicrete pace. Explicitly, we have yx, t) = lim x 0, t 0 lim x 0, t 0 t x ˆL 1 q L 1 Ẑ i Z n Y i, n), t} q, x} t} x 54) where L 1 F ) t} denote the invere unilateral Laplace tranform to continuou time t, and ˆL q Gq) x} denote the invere bilateral Laplace tranform 1 to continuou pace x. To ee the reult in Eq. 54) we note that } 1 t x ˆL q Ẑ i Zn Y i, n),, t} q, x} t} x L 1 1 = lim t x ˆL q x 0, t 0 = lim = x 0, t 0 i= n=0 0 = yx, t), L 1 i= n=0 }, } } y i x, n t) e n t e qi x d dq t x, y i x, n t) δt n t)δx i x) t x, yx, t ) δt t )δx x ) dt dx provided that t > 0. The continuum limit of the dicrete generalized mater equation with a Sibuya waiting time ditribution can be found by firt taking the unlilateral tar tranform with repect to the dicrete time variable and the bilateral tar tranform with repect to the dicrete pace variable on each ide of the GME, Eq. 33). The exponential function ariing in the tar tranform 14

15 are expanded in Taylor erie ee Appendix A) and then we take the invere unilateral Laplace tranform to continuou time, and the invere bilateral Laplace tranform to continuou pace. The limit x 0, t 0, i then evaluated with the requirement that D α = r x 2 lim x 0, t 0 2 t α 55) exit. The analyi, which i hown in Appendix A, reult in the continuum diffuion limit of the dicrete generalized mater equation, ux, t) t = D α 2 x 2 [ θx, t, 0) 0 D 1 α t ux, t) θx, t, 0) )] ax, t)ux, t)+cx, t). 56) Thi recover the fractional reaction-diffuion equation derived from the diffuion limit of continuou time random walk in [7] where 0 Dt 1 α i the Riemann- Liouville fractional derivative, formally equivalent to the Grünwald-Letnikov fractional derivative in thi etting. We note that Eq. 56) i potentially a non-linear equation, a the annihilation and creation rate, ax, t) and cx, t) repectively, are defined by the reaction kinetic, which may be dependent on the concentration ux, t). 5. Numerical Implementation In thi ection we preent a implified method of matching parameter and implementing the explicit numerical cheme to olve the reaction-ubdiffuion equation, Eq. 56). The tandard reaction-diffuion equation can be recovered with α = 1. Firt we et the tep ize t by inverting the relation for the diffuion coefficient given by D α = r x2 2 t α, 57) and treat x and r a free parameter, thu defining ) 1 r x 2 α t =. 58) 2D α We note that r may be a ueful parameter for decoupling x, t and D α. In the cae of multiple pecie where each pecie ha a different diffuion coefficient the lattice for each of the pecie need to be the ame. Thi can only be achieved by letting r be different for each pecie. If we have two pecie, A and B, with diffuion coefficient D αa > D αb, then we need to et, r b = r ad αb D αa. 59) Thi i done to enure that 0 r b 1, given 0 r a 1. Thi choice will then give the ame t, for a given x, for each pecie. 15

16 In modelling a phyical ytem the annihilation and creation rate are precribed by the reaction kinetic. In a tochatic reaction-diffuion ytem the annihilation and creation procee may be modelled by inhomogeneou Poion procee. In general, for an inhomogeneou Poion proce with Nt) event up to time t, and rate function λx, t), the expected number of event in the time interval n 1) t, n t) i given by E [Nn t) Nn 1) t)] = n t ) t λx, t ) dt. 60) For the ame proce, the probability that there i no event in the time interval n 1) t, n t) i given by ) n t P [Nn t) Nn 1) t) = 0] = exp λx, t ) dt. 61) ) t If we aume that the annihilation proce i modelled by an inhomogeneou Poion proce with rate parameter ax, t), then the probability of an annihilation event between n 1) t and n t i one minu the probability that there i no annihilation event in thi time. Thu ) n t Ai, n 1) = 1 exp ai x, t ) dt. 62) ) t The urvival probability function for no annihilation event between time m t and n t i given by ) n t Θi, n, m) = exp ai x, t ) dt. 63) If the creation proce i independent of the annihilation proce and may alo be modelled a an inhomogeneou Poion proce with rate cx, t) then the expected number of particle created between n 1) t and n t i given by Ci, n) = n t ) t m t ci x, t ) dt. 64) If the creation proce i dependent on an annihilation proce, uch a in the cae of multiple interacting chemical pecie, then a different approach may be ued to find the expected number of particle created in the time tep. The dependence of the procee may mean that a flux balance argument can be ued to model the creation event. Conidering a reaction uch that the decay of a particle of pecie A, become a particle of pecie B. Then the number of particle of pecie B that are created would imply be the number of particle of pecie A lot. If we model the lo of pecie A by a Poion proce, i.e. 16

17 with Ai, n) given by Eq. 62), then the expected number of pecie B particle created would be given by, C b i, n) = 1 exp n t ) t a a i x, t ) dt )) u a i x, n t) 65) where the ubcript a and b denote pecie A and B repectively. Whilt the dependent and independent expected number of particle created in a time tep look very different, they hare the ame limiting behaviour a t 0. Both the cae of dependent and independent creation procee are dealt with in the example in Section 7. In reaction dynamic obtained from the law of ma action the reaction term are multinomial with poitive coefficient identifying creation term cx, t) and negative coefficient identifying annihilation term ax, t)ux, t). In general a numerical quadrature rule may be needed to evaluate the integral in Eq. 62), 63), 64), and 65). Taking the independent creation proce, the explicit finite difference cheme for Eq. 56) can be written a, Ui, n) = ) 1 α n m + r 2 exp + exp n t m t n t ) t where t i given by Eq. 58) Boundary Condition 1) n m δ 0,n m + δ 1,n m ) r 2 exp ai + 1) x, t ) dt ) Ui + 1, m) r exp n t m t n t ai 1) x, t ) dt ) Ui 1, m) m t ) n t ai x, t ) dt Ui, n 1) + ci x, t ) dt, ) t 66) On the domain, x [l 1, l 2 ], where l 1 = L 1 x, and l 2 = L 2 x with L 1, L 2 Z, boundary condition can be et in the following manner: Dirichlet: For ul 1, t) = b 1 t) and ul 2, t) = b 2 t), 67) et UL 1, n) = b 1 n t) and UL 2, n) = b 2 n t). 68) Zero-Flux: The condition for a zero flux boundary can be found by integrating Eq. 56) over it domain. Thi give, [ ] l2 [ )] ux, t) ux, t)dx = D α θx, t, 0) 0 Dt 1 α t l 1 x θx, t, 0) x=l 2 l2 + l x=l 1 1 ai x, t ) dt ) Ui, m) cx, t) ax, t)ux, t)dx. 69) ) 17

18 Thu the zero flux boundary condition are given by [ )] x=l ux, t) 2 D α θx, t, 0) 0 Dt 1 α = 0. 70) x θx, t, 0) x=l 1 Thee condition i guaranteed to hold if we take ux, t) ux, t) x = θx, t, 0) x=l1 x = x=l2 x = x=l1 θx, t, 0) x = 0. 71) x=l2 For the numerical cheme thee boundary condition are implemented by etting ghot point for all n a follow, UL 1 1, n) = UL 1, n) and UL 2 + 1, n) = UL 2, n). 72) Thi i equivalent to redirecting outgoing flux that i detined to jump out of the domain back in to thoe end-point. Alternatively the domain, x [l 1, l 2 ], can be dicretied uch that, l 1 = L ) x, and l 2 = L ) x with L 1, L 2 Z. Thi reult in the dicrete point each being at the centre of an interval of width x. The Dirichlet and zero flux boundary condition are implemented in the ame manner a before Initial Condition For the initial condition ux, 0) = u 0 x), 73) which we aume to be bounded, we imply ample u 0 x) via Ui, 0) = u 0 i x). 74) However we may alo treat unbounded initial condition, provided that the integral of the initial condition over the domain i bounded. For example, if the initial condition i a Dirac delta function, we take ux, 0) = δx x 0 ), 75) Ui, 0) = δ i,i 0 76) x where δ i,i0 i a Kronecker delta, i 0 = x0 x. The jutification for Eq. 76) follow from the identification l2 L2 x ux, 0) dx = lim u x, 0) dx l x 0, t 0 1 L 1 x = lim x 0, t 0 = lim x 0, t 0 L 2 i=l 1 u i x, 0) x L 2 i=l 1 Ui, 0) x. with the left hand ide and right hand ide both equating to one. 18

19 6. Stability Analyi For the numerical method to be table we require the ditance between the olution and it approximation from the numerical cheme to be bounded for all n, i.e. Ui, n) ui x, n t) M 77) i where M R +. To how thi bound exit it i ufficient to how that both the olution i bound, and the approximation i bound, i.e. Ui, n) M 1, 78) i ui x, n t) M 2, 79) i with M 1, M 2 R +. If ux, t) i non-negative, and Riemann integrable then the condition in Eq. 79) i atified if, l2 l 1 ux, n t)dx M 3, 80) with M 3 R +. The temporal evolution of the left hand ide i given by Eq. 69). Taking zero-flux boundary condition thi further implifie to, [ ] l2 l2 ux, t)dx = cx, t) ax, t)ux, t)dx. 81) t l 1 l 1 Hence provided that the initial condition i bounded and l2 for all t then, l 1 cx, t) ax, t)ux, t)dx 0, 82) l2 l 1 ux, t)dx l2 l 1 ux, 0)dx, 83) and the condition in Eq. 80) will be atified. Equation 82) put a ufficient condition on the reaction in order for the olution of the fractional PDE to remain bounded and imply tate that in total at leat a many particle are annihilated a created in the domain. We alo need to enure that the condition in Eq. 78) i met. By contruction, a a um of pmf in Eq. 27), Ui, n) i non-negative, provided that the tochatic proce i well poed. Thi i guaranteed if r, Ai, n), ψn) and λi j) are all probabilitie, and thu retricted to [0, 1], for all i and n, and Ci, n) and Ui, 0), are non-negative. Thi permit u to replace the condition in Eq. 78) with Ui, n) M 1. 84) i 19

20 To obtain further condition for the tability of the numerical cheme, we rearrange the GME, Eq. 28), and um over i. Again taking zero-flux boundarie thi give, Ui, n) = Ui, n 1) + i i n λj i) δ i,j ) Kn m) Θj, n, m) Uj, m) i i j Ai, n 1)Ui, n 1) + Ci, n)) = i Ui, n 1) Ai, n 1)Ui, n 1) + Ci, n)). 85) where we have ued the fact that by Eq. 4), λi j) δ i,j ) = 0. 86) Provided that then i Ci, n) Ai, n 1)Ui, n 1)) 0, 87) i Ui, n) i i Ui, n 1) 88) and hence if i Ui, 0) i bounded then the condition in Eq. 78) will be atified. Equation 87) i an analogou condition on the reaction to the continuou condition Eq. 82), and they are equivalent in the continuum limit. The olution of the numerical cheme i therefore table, a defined by Eq. 77), when the underlying tochatic proce i well poed and Eq. 82) and 87) hold. It hould be emphaied that thi tability reult i valid for non-linear reaction. The implicity of the provided tability analyi i due to deriving the numerical cheme from a tochatic proce. An alternative derivation of the tability of the numerical cheme, in the abence of reaction, via a von Neumann type analyi i provided in Appendix B. 7. Numerical Example 7.1. Example 1: Non-linear morphogen death rate on emi-infinite domain In [51], the author conidered ubdiffuion with elf enhanced degradation a a model for morphogen concentration in developmental biology. The fractional reaction ubdiffuion equation in thi cae i ux, t) t = 2 x 2 [ D α e t 0 kux,)d 0Dt 1 α [e ]] t 0 kux,)d ux, t) kux, t) 2, 89) 20

21 on the emi-infinite domain x [0, ). The total ma i conerved over the domain by injecting a flux, which i equal to the integral of the reaction term, at the origin. Thi correpond to the boundary condition D α [e t x 0 kux,)d 0D 1 α t e )] t 0 kux,)d ux, t) x=0 = 0 kux, t) 2 dx. 90) Stationary Ditribution The tationary ditribution i equivalent to the tationary ditribution derived in [52]. In the long time limit we can replace the Riemann-Lioville fractional derivative with Weyl fractional derivative [46]. To find the tationary ditribution, u t x) we ue the aymptotic relation e kutx)t 0D 1 α t Thu u t x) atifie the ordinary differential equation with boundary condition e kutx)t) ku t x)) 1 α. 91) 1 α d2 D α k ut dx 2 x) 2 α) ku t x) 2 = 0 92) D α k 1 α d dx ut x) 2 α) 0 = ku t x)dx. 93) 0 The olution i given by [52, 51] u t x) = u t 0) 1 + x ) 2 α µ 94) where and u t 0) = ) 2 4 α g 4 α, 95) Dα k 2 α 4 2α g = 0 ku t x) 2 dx, 96) µ = 4 2α ) 2 α 4 α α g 4 α Dα 4 2α k α. 97) 21

22 Numerical Solution The numerical cheme for thi problem i found by equating the proce parameter in the fractional PDE, Eq.89), with the dicrete proce parameter in the GME, Eq. 32), with the Sibuya waiting time kernel. The numerical cheme require four parameter, x, t, r, and α, a well a two function, Ai, n), and Ci, n). Comparing Eq. 89) with the general form given in Eq. 56) we ee that we have a continuou annihilation proce with a rate of ax, t) = kux, t), and no creation proce, i.e. cx, t) = 0. A there are no particle created in each time tep we et Ci, n) = 0. Uing Eq. 62), we et ) n t Ai, n 1) = 1 exp kui x, t )dt ) t 1 exp kui x, n 1) t) t) = 1 exp kui, n 1) t). Here we have ued a imple one point approximation for the integral, and the relation between the continuou and dicrete olution on the lattice point. We obtain Θi, n, m) from Ai, n) through Eq. 6). The anomalou exponent in the fractional PDE, α, i equivalent to the fractional exponent parameter in the Sibuya memory kernel. The choice of the parameter x, t, and r, i dependent on the diffuion coefficient D α and the fractional exponent from the fractional PDE. Treating x and r a free parameter we can et t uing Eq. 58), ) 1 r x 2 α t =. 99) 2D α Note that r i a probability and hence 0 r 1 and, for a given x, the maximum t correpond to r = 1. The fractional PDE i defined over a emi-inifite domain, x [0, ). Thi i approximated by taking an aborbing boundary a long ditance from the origin, i.e. ul, t) = 0, with l 0. In the numerical cheme, we take l = L x and thi boundary condition correpond to UL, n) = 0. The conervation of ma boundary condition at the origin, Eq. 90), i implemented through a flux of particle inerted at the origin, equal to particle lot from the annihilation proce and the flux at the x = l boundary. Subtituting thee function and parameter in to the GME Eq. 32), we obtain Ui, n) Ui, n 1) = [ 2 exp Kn m) r 2 p=m exp t kui, p) p=m ) t kui 1, p) Ui, m) + exp [1 exp t kui, n 1))] Ui, n 1), ) Ui 1, m) p=m t kui + 1, p) 98) ) 100) Ui + 1, m) ] 22

23 for 0 < i < L, where Kn m) i the Sibuya kernel from Eq. 40). The boundary point are given by, U0, n) U0, n 1) = and L [1 exp k t Ui, n 1))] Ui, n 1) i=0 + exp Kn m) r 2 p=m [ exp t ku0, p) ) p=m U0, m) t kul 1, p) ] ) UL 1, m) 101) UL, n) = ) The dicrete olution i related to the continuou olution at the lattice point, hence the numerical olution will be given uing the relation, ui x, n t) = Ui, n). 103) To obtain a olution we take k = 10, r = 1, D α = 1, and α = 0.9. We alo chooe an initial condition with g = 1 in Eq. 96), and tart near the aymptotic teady tate given by Eq.94), Ui, 0) = u t i x). 104) A the teady tate reult i valid in the limit l, we examine the convergence of the numerical method to thi olution a we increae the length of the domain. We alo examine a range of different x to demontrate convergence to the teady tate. The numerical olution were run to t = 2. At thi time the numerical olution wa found to have converged in the ene that where L Ui, n) Ui, n 1)) 2 x 10 3 U u t 2 105) i=0 U u t 2 = L Ui, n) u t i x)) 2 x. 106) i=0 Figure 1 how the analytic olution and the convergence of the numerical olution. The method i hown to converge with decreaing x and increaing domain ize l = L x. 23

24 Figure 1: Left: Analytic teady tate olution for example 1 at t = 2. Right: Convergence between the DTRW olution and the analytic teady tate at t = 2 a a function of x for l = 5 cro), l = 10 plu), l = 20 quare) and l = 50 circle) Example 2: Multi-pecie chemical reaction model In the econd example, we conider a imple two-pecie chemical reactionubdiffuion model with a imple tranition from pecie A to pecie B and vice-vera with a contant rate k, A k B. We denote the population denitie in time and pace of A and B by u a and u b repectively. A a et of fractional partial differential equation, the model can then be expreed a [4], u a x, t) t u b x, t) t 2 [ = D α e kt x 2 0 Dt 1 α e kt u a x, t) )] ku a x, t) + ku b x, t) e kt u b x, t) )] + ku a x, t) ku b x, t). 2 [ = D α e kt x 2 0 Dt 1 α 107) We conider the cae of zero flux boundarie located at x = l, and x = l. Thi give the condition, D α x D α x [ e kt 0 Dt 1 α e kt u a x, t) )] [ e kt 0 Dt 1 α e kt u a x, t) )] x= l x=l = 0, = 0, 108) with equivalent condition on u b x, t). Thee condition will be atified if, u a x, t) = 0, x x= l 109) u a x, t) = 0, x x=l 24

25 and again equivalent condition for u b x, t). To facilitate a imple analytic olution we take the initial condition to be a Dirac delta function located at the origin for pecie A and zero every where for pecie B, i.e. u a x, 0) = δ0 x) u b x, 0) = 0 110) Analytic Solution By umming both equation in Eq. 107) and ubtituting u z x, t) = e kt u a x, t)+ u b x, t)), we obtain the equation, u z x, t) t = D α 0 D 1 α t [ 2 u z x, t) x 2 ] + ku z x, t). 111) Thi equation i of the ame form a the fractional cable equation [53], where it wa olved for the cae k < 0. The boundary condition are then, u z x, t) = 0, x x= l 112) u z x, t) = 0, x x=l and the initial condition i, u z x, 0) = δ0 x). 113) Thi equation can then be olved analytically via the method of image yielding [ ] 1 kt) j u z x, t) = H 2,0 x + 4nl) 2 1 α 4πDα t α 1,2 j! 4D j=0 n= α t α 2 + j, α) 0, 1) j, 1) [ ]) +H 2,0 2l x + 4nl) 2 1 α 1,2 4D α t α 2 + j, α) 0, 1) 1, 2 + j, 1) 114) where H are Fox H function [54]. Alternatively the olution can be found by eparation of variable [53] giving, u z x, t) = expkt) 2L + m=1 1 mπx ) L co L j=0 kt) j j! E j) α,1+1 α)j where E m) α,β z) i the mth derivative of a Mittag-Leffler function. m2 π 2 D α t α L 2 ), 115) 25

26 Numerical Solution To contruct the numerical olution to the et of fractional PDE, Eq 107), we mut parameterie an equivalent et of GME. The coupled GME will be of the ame form a Eq. 32), uing a Sibuya memory kernel and with the additional complication of a dependence in the annihilation and creation procee. The numerical cheme will require the identification of five parameter, x, t, r a, r b, and α, a well a four function A a i, n), A b i, n), C a i, n), and C b i, n). Similar to the firt example we compare the fractional PDE with the general form preented in Eq. 56). From thi we ee that that the continuou annihilation prece for both pecie A and B ha a contant rate, i.e., a a x, t) = a b x, t) = k. The creation proce arie out of a flux balance with the annihilation proce, i.e. c a x, t) = ku b x, t), and c b x, t) = ku a x, t). The diffuion coefficient for pecie A and B are both equal to D α. The dicrete annihilation probabilitie are found from Eq. 62), ) n t A a i, n 1) = A b i, n 1) = 1 exp kdt 116) = 1 exp k t). ) t Again note that etting Ai, n) alo et Θi, n, m) through Eq. 6). A we have a dependent creation proce we will ue the expected number of created particle of the form of Eq. 65), thi will give, C a i, n) = 1 exp n t ) t = 1 exp k t))) U b i, n), kdt )) u b i x, n t) 117) where the correpondence between the dicrete and continuou olution ha been ued. Similarly we take, C b i, n) = 1 exp k t))) U a i, n) 118) The fractional exponent ued in the Sibuya memory kernel mut match the fractional exponent in Eq. 107). The choice of the parameter x, t, r a, and r b, i dependent on the diffuion coefficient, D α, and the fractional exponent, α, of the fractional PDE. A the diffuion coefficient of the two pecie are equal then the two r parameter mut alo be equal, i.e. r a = r b = r. Treating x and r a free parameter t will be given by Eq. 58). The continuou boundary condition are taken to be zero flux at x = l and x = l. We let l = L x, and define the dicrete boundary condition are found by creating ghot point at L 1 and L + 1 according to Eq. 72). Putting thee parameter and function into the GME baed on Eq.32) 26

27 yield the numerical cheme, U a i, n) U a i, n 1) = Kn m) exp k tn m)) r 2 Ua i 1, m) 2U a i, m) + U a i + 1, m) ) [1 exp k t)] U a i, n 1) + [1 exp k t)] U b i, n 1), 119) for L < i < L, where the Kn) i the Sibuya memory kernel given by Eq. 40). The boundary point evolve according to U a L, n) U a L, n 1) = Kn m) exp k tn m)) r 2 Ua L, m) + U a L + 1, m) ) [1 exp k t)] U a L, n 1) + [1 exp k t)] U b L, n 1), 120) and U a L, n) U a L, n 1) = Kn m) exp k tn m)) r 2 Ua L 1, m) U a i, m) ) [1 exp k t)] U a L, n 1) + [1 exp k t)] U b L, n 1). 121) Equivalent equation are alo found for U b. A the continuum equation initial condition involve a Dirac delta function we ue Eq. 76) to obtain dicrete initial condition, U a i, 0) = δ i,0 x, U b i, 0) = ) The numerical olution i obtained by noting the correpondence between the dicrete and continuou olution, i.e. u a i x, n t) = U a i, n) u b i x, n t) = U b i, n) 123) To obtain a numerical olution we have taken α = 0.5, D α = 1 and k = 1, with a patial domain of [ 1, 1]. The olution wa run up to time t = 0.1. A range of value for x were conidered. Two eparate dicretiation of the domain were conidered to examine the effect of the boundary point on the numerical olution. In the firt the domain wa dicretied o that the dicrete point aligned with the boundary point, i.e. l = L x. The econd dicretiation placed each point in the centre of an interval of width x, i.e. l = L ) x. 27

28 For comparion with the analytical olution we conider the um of the two pecie olution, U Z i, n) = U a i, n) + U b i, n)) expkn t). Thi i compared with the olution obtained from Eq. 114) in Fig. 2 for the aligned boundary grid, and in Fig. 3 for the centred grid. We define the norm of the difference U Z u z 2 in a imilar fahion to Eq. 106). In each cae the numerical olution converge to the analytical olution. For the cae of the aligned boundary the numerical olution converge with order x 2, and for the cae of the centred grid the convergence i of the order x 4. Thi ugget, for zero-flux boundary condition, the centred grid dicretiation of the domain i preferable to the aligned grid dicretiation. Figure 2: Solution of example 2 uing aligned boundary grid dicretiation. Left): The olid line denote the analytical olution u zx, t), at t = 0.1. The point how the numerical olution, U Z x, t), at t = 0.1 with x = 1 5 Blue Circle), x = 1 Purple Square), and 10 x = 1 15 Orange Cro). Right): Convergence of the numerical olution, U Zx, t), to the analytic olution u zx, t). For each x the olution i compared at the time t = 0.1 t t. Figure 3: Solution of example 2 uing centred grid dicretiation. Left): The olid line denote the analytical olution u zx, t) at t = 0.1. The point how the numerical olution, U Z x, t), with x = Blue Circle), x = Purple Square), and x = Orange Cro) Each numerical olution i evaluated at t = 0.1 t t. Right): Convergence of the numerical olution, U Z x, t), to the analytic olution u zx, t). For each x the olution i compared at the time t = 0.1 t t. 28

29 8. Summary Numerou numerical cheme have been propoed for olving nonlinear reaction diffuion procee. The implet cheme to implement i an explicit finite difference cheme in which the time and pace derivative are replaced by local difference operator. However uch cheme are known to be numerically untable for general parameter and a formal tability analyi i required to invetigate thi. Here we have preented a new approach to obtaining numerical cheme for reaction diffuion equation and fractional reaction ubdiffuion equation. The numerical cheme are implemented a explicit finite difference cheme but they are guaranteed to be table, in the ene that the difference between the numerical olution and the exact olution remain bounded for all time, by contruction, provided the tochatic proce i well poed. The contruction involve the derivation of dicrete time mater equation for a tochatic proce with the conitency that the dicrete time mater equation converge to the reaction diffuion equation of interet in the continuum pace and time limit. The utility of thi contruction a the bai of a table numerical cheme i perhap unurpriing. Reaction diffuion equation, including fractional reaction ubdiffuion equation, are themelve formally derived from a continuou time tochatic proce [7]. In implementing the numerical cheme, further information on the tochatic proce, uch a ymmetrie in the reaction kinetic, can be incorporated in a natural way. More generally it may be poible to formulate dicrete time tochatic procee that converge to partial differential equation, and fractional partial differential equation, even in cae where the partial differential equation have not previouly been obtained from a continuou time tochatic proce. Thi open up the poibility of new clae of explicit finite difference cheme that are inherently table. Appendix A In thi appendix we evaluate the continuum limit of the dicrete generalized mater equation, Eq. 33), baed on tranform method with } 1 yx, t) = t x ˆL q L 1 Ẑ i Zn Y i, n), t} q, x} t} x, where lim x 0, t 0 Z n Y i, n), t} = 124) Y i, n) e n t, 125) i the unilateral tar tranform with repect to n, Ẑi Y i, n) q, x} = Y i, n) e i xq, 126) n=0 i= i the bilateral tar tranform with repect to i, L 1 F ) t} i the invere 1 unilateral Laplace tranform to continuou time and ˆL q Gq) x} i the invere bilateral Laplace tranform to continuou pace. 29

30 In taking the continuum limit we make ue of a product rule 1 t x ˆL q L 1 Ẑ i Zn Y 1 i, n)y 2 i, n), t} q, x} t} lim x 0, t 0 1 = lim t x ˆL q x 0, t 0 = lim x 0, t 0 i= n=0 L 1 i= n=0 } x } } y 1, i x, n t)y 2, i x, n t)e n t e i xq t x y 1, i x, n t)y 2, i x, n t)δt n t)δx i x) t x = y 1 x, t)y 2 x, t), ) = lim y 1, i x, n t)δt n t) t x 0, t 0 n=0 lim x 0, t 0 n=0 ) y 2, i x, n t)δt n t) t. 127) which i valid for t > 0 and provided that both limit exit. We alo make ue of the hift propertie Z n Y i, n k), t} = e k t Z n Y i, n), t}, 128) Ẑ i Y i k, n) q, x} = e k xq Ẑ i Y i, n) q, x}, 129) and, for notational convenience, we write Ŷ, q) = Ẑ i Z n Y i, n), t} q, x}. 130) We now take the unilateral tar tranform with repect to the dicrete time variable and the bilateral tar tranform with repect to the dicrete pace variable on each ide of the GME, Eq. 33), to obtain Û, q) e t Û, q) = r 2 e xq +e xq 2) ˆX, q) e t ˆB, q)+ĉ, q), where and ˆX, q) = Ẑ i Z n Θi, n, 0) 131) } } n Ui, m) Kn m), t q, x, Θi, m, 0) 132) ˆB, q) = Ẑ i Z n Ai, n)ui, n), t} q, x}. 133) We can expand the exponential function in Eq. 131) to write tû, q) t2 2 Û, q) + O t 3 ) = r 2 2 x2 q 2 ˆX, q) ˆB, q) + t ˆB, q) + t2 2 ˆB, q) + 2 Ĉ, q) + O x 4 ) + O t 3 ). 134) 30