A spatially second-order accurate implicit numerical method for the space and time fractional Bloch-Torrey equation

Transcription

1 Noname manuscript No. (will be inserted b the editor) A spatiall second-order accurate implicit numerical method for the space and time fractional Bloch-Torre equation J. Song Q. Yu F. Liu I. Turner Received: date / Accepted: date Abstract In recent ears, it has been found that man phenomena in engineering, phsics, chemistr and other sciences can be described ver successfull b models using mathematical tools from Fractional Calculus. Recentl, a new space and time fractional Bloch-Torre equation (ST-FBTE) has been proposed [3], and successfull applied to analse diffusion images of human brain tissues to provide new insights for further investigations of tissue structures. In this paper, we consider the ST-FBTE with a nonlinear source term on a finite domain in three-dimensions. The time and space derivatives in the ST-FBTE are replaced b the Caputo and the sequential Riesz fractional derivatives, respectivel. Firstl, we propose a spatiall second-order accurate implicit numerical method (INM) for the ST-FBTE whereb we discretize the Riesz fractional derivative using a fractional centered difference. Secondl, we prove that the implicit numerical method for the ST-FBTE is uniquel solvable, unconditionall stable and convergent, and the order of convergence J. Song Sunshine College, Fuzhou Universit, Fuzhou, Fujian 355, China geigeisong@26.com Q. Yu School of Mathematical Sciences, Queensland Universit of Technolog, GPO Bo 2434, Brisbane, QLD 4, Australia F. Liu Corresponding author: School of Mathematical Sciences, Queensland Universit of Technolog, GPO Bo 2434, Brisbane, QLD 4, Australia f.liu@qut.edu.au I. Turner School of Mathematical Sciences, Queensland Universit of Technolog, GPO Bo 2434, Brisbane, QLD 4, Australia

2 2 J. Song, Q. Yu, F. Liu, I. Turner of the implicit numerical method is O(τ 2 α + τ + h 2 + h 2 + h 2 z). Finall, some numerical results are presented to support our theoretical analsis. Kewords Fractional Bloch-Torre equation Fractional Calculus implicit numerical method fractional centered difference solvabilit stabilit convergence Introduction It is now well accepted that man phenomena in engineering, phsics, chemistr and other sciences can be described ver successfull b models that emplo the theor of derivatives and integrals of fractional order [4, 5,,, 8, 2, 22, 28, 29]. At present, fractional order equations have been applied to model dnamical sstems in science and engineering [2, 26]. These new fractional models are more adequate than the previousl used integer order models [7], because fractional order derivatives and integrals enable the description of the memor and hereditar properties of different substances. In phsics, particularl when applied to diffusion, fractional order dnamics lead to an etension of Brownian motion to what is called anomalous diffusion [3]. Anomalous diffusion concerns the theor of diffusing particles in environments that are not locall homogeneous, including disorder that is not well-approimated b assuming a unified change in the diffusion constant. Such sstems include diffusion in complicated structures such as brain tissue. Hall and Barrick [7] show that the usual manner to stud the diffusive dnamics is to investigate the mean square displacement r 2 (t) of the particles, namel r 2 (t) t α, t, () where α is the anomalous diffusion eponent. A ver interesting and particular class of comple phenomena arises in the nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) fields, and Fractional Calculus ma help to epress a phsical meaning b utilizing the fractional derivative operator [4, 5]. If the comple heterogeneous structure, such as spatial connectivit, can facilitate the movement of particles at a certain scale, fast motions ma no longer obe the classical Fick s law and ma indeed have a probabilit densit function that follows a power-law. For eample, if C(, t) represents the concentration of the diffusing species in one-dimension, then a space-time Riesz-Caputo fractional diffusion equation of the form C Dt α β C(, t) C(, t) = K β, (2) emerges from Fick s first law in the continuit equation [6], where K is the generalized diffusion coefficint, C Dt α is the Caputo time fractional derivative of order α ( < α < ) with respect to t with starting point at t = defined as [7]: C D α t C(, t) = t Γ ( α) C (, τ) dτ, (3) (t τ) α

3 A spatiall second-order accurate implicit numerical method for ST-FBTE 3 and R β = β is the Riesz fractional derivative of order β ( < β 2) with β respect to, which is defined in equation (5) below. Recentl, some authors have used Fractional Calculus to investigate the connection between fractional order dnamics and diffusion b solving the Bloch-Torre equation [3, 23, 24]. The have demonstrated that a Fractional Calculus based diffusion model can be successfull applied to analzing d- iffusion images of human brain tissues and provide new insights for further investigations of tissue structures. The following new diffusion model was proposed for solving the Bloch-Torre equation using Fractional Calculus with respect to time and space [3]: τ α C D α t M (r, t) = λm (r, t) + Dµ 2(β ) R β M (r, t), (4) where λ = iγ(r G(t)), r = (,, z), G(t) is the magnetic field gradient, γ and D are the gromagnetic ratio and the diffusion coefficient, respectivel. In addition, R β = (R β + R β + R β z ) is a sequential Riesz fractional order operator in space [8]. M (r, t) = M (r, t) + im (r, t), where i =, comprises the transverse components of the magnetization; τ α and µ 2(β ) are the fractional order time and space constants needed to preserve units, respectivel ( < α, and < β 2). The fractional order dnamics derived from the space fractional Bloch-Torre equation can be used to fit the signal attenuation in diffusion-weighted images obtained from Sephade gels, human articular cartilage and a human brain [3], and can also be used to analse diffusion images of health human brain tissues in vivo at high b values up to 47 sec/mm 2 (b is the degree of diffusion sensitization defined b the amplitude and the time course of the magnetic field gradient pulses used to encode molecular diffusion displacements [9]) [27]. Compared with the considerable work carried out on the theoretical analsis, relativel little work has been done on the numerical solution of (4). Magin et al. [3] derived the analtical solutions with fractional order dnamics in space (i.e., α =, β an arbitrar real number, < β 2) and time (i.e., < α <, and β = 2), respectivel. Yu et al. [24] derived an analtical solution for solving (4) using a fractional Laplacian based model and an effective implicit numerical method for solving (4) using a Riesz fractional based model. The also considered the stabilit and convergence of the implicit numerical method. However, due to the computational overheads necessar to perform the simulations for solving (4) in three dimensions, Yu et al. [24] presented a preliminar stud based on a two-dimensional eample to confirm their theoretical analsis. Yu et al. [23] proposed a fractional alternating direction implicit scheme to overcome the computational bottlenecks described in [24], the also proved the stabilit and convergence of the proposed method. However, the order of convergence in [23,24] is O(τ 2 α + h + h + h z ) first order in space. The Grünwald-Letnikov derivative approimation scheme of order O(h) is generall used to approimate the Riesz fractional derivative [7, 8, 2, 23, 24, 29]. In order to obtain a better approimation, Ortigueira [6] defined the

4 4 J. Song, Q. Yu, F. Liu, I. Turner fractional centered derivative and proved that the Riesz fractional derivative of an analtic function can be represented b the fractional centered derivative. Celik and Duman [3] used the fractional centered derivative to approimate the Riesz fractional derivative and applied the Crank-Nicolson method to a fractional diffusion equation that utilises the Riesz fractional derivative, and showed that the method is unconditionall stable and convergent with O(h 2 ) accurac. Yu et al. [25] derived an effective implicit numerical method for solving (4) in two-dimensions with a linear source term using the fractional centered derivative to approimate the Riesz fractional derivative. The also considered the stabilit and convergence of the implicit numerical method, however, the did not consider the method s solvabilit, and the order of convergence in [25] is O(τ 2 α + τ + h 2 + h 2 ). In this paper, we build upon the work in [25] and use a fractional centered derivative to approimate the Riesz fractional derivative, and propose a new effective implicit numerical method for the space and time fractional Bloch- Torre equation (ST-FBTE) with a nonlinear source term with initial and boundar conditions on a finite domain in three-dimensions, and prove that the implicit numerical method for the ST-FBTE is uniquel solvable, unconditionall stable and convergent. The convergence order is O(τ 2 α +τ +h 2 +h 2 +h 2 z). The remainder of this article is arranged as follows. Some mathematical preliminaries related to Fractional Calculus are introduced in section 2. In section 3, we propose a new effective implicit numerical method for the ST-FBTE. The solvabilit, stabilit and convergence of the implicit numerical method are investigated in sections 4, 5 and 6, respectivel. Finall, some numerical results are presented to show that our new implicit numerical method can obtain second order space accurac, which is O(τ 2 α + τ + h 2 + h 2 + h 2 z). 2 Preliminar knowledge In this section, we give some preliminar information that is assumed throughout this paper. Definition. [23] The Riesz fractional operator R β for n < β n on a finite domain [, L ] [, L 2 ] [, L 3 ] is defined as R β C(,, z, t) = β C(,, z, t) β = c β ( D β + D β L )C(,, z, t), (5) where c β = 2 cos( πβ 2 ), β, D β C(,, z, t) = D β L C(,, z, t) = Γ (n β) n n ( )n Γ (n β) n n L C(ξ,, z, t)d ξ ( ξ) β+ n, C(ξ,, z, t)d ξ (ξ ) β+ n.

5 A spatiall second-order accurate implicit numerical method for ST-FBTE 5 Similarl, we can define the Riesz fractional derivatives R β C(,, z, t) = β C(,,z,t) and R β β z C(,, z, t) = β C(,,z,t) of order β ( < β 2) with z β respect to and z. Now, we present our solution techniques for the ST-FBTE for a finite domain. Firstl, the ST-FBTE (4) can be rewritten as: K α C D α t M (r, t) = λm (r, t) + K β R β M (r, t), (6) where K α = τ α and K β = Dµ 2(β ). For the numerical solutions of the ST-FBTE, we equate real and imaginar components to epress (6) as a coupled sstem of partial differential equations for the components M and M, namel K α C D α t M (r, t) = K β ( β β + K α C D α t M (r, t) = K β ( β β + β β + β β + β z β )M (r, t) + λ G M (r, t), (7) β z β )M (r, t) λ G M (r, t), (8) where λ G = γ(r G(t)). For convenience, (7) and (8) are decoupled, which is equivalent to solving K α C D α t M(r, t) = K β ( β β + β β + β )M(r, t) + f(m, r, t), (9) z β where M can be either M or M, and f(m, r, t) = λ G M (r, t) if M = M, and f(m, r, t) = λ G M (r, t) if M = M. 3 Implicit numerical method for the ST-FBTE In this section, we propose a new implicit numerical method for the following space and time fractional Bloch-Torre equation with initial and boundar conditions on a finite domain: K α C D α t M(r, t) = K β ( β β + β β + β )M(r, t) + f(m, r, t), () z β M(r, ) = M (r), () M(r, t) Ω =, (2) where < α, < β 2, < t T, r = (,, z) Ω, Ω is the finite rectangular region [, L ] [, L 2 ] [, L 3 ], and Ω is R 3 Ω, the nonlinear source term f(m, r, t) is assumed locall Lipschitz continuous. Remark : [,2] We sa that f : X X is globall Lipschitz continuous if for some L >, we have f(u) f(v) L u v for all u, v X, and is locall Lipschitz continuous, if the latter holds for u, v M with L = L(M) for an M >.

6 6 J. Song, Q. Yu, F. Liu, I. Turner Baeumer et al. [, 2] showed how to solve nonlinear reaction-diffusion equations of tpe () b an operator splitting method when the abstract function f is onl locall Lipschitz (see [,2,2]). Thus, we assume that for all k =, 2,, N, u(,, z, t k ), v(,, z, t k ) M k with a constant M k > for an r = (,, z) Ω, we have f(u(,, z, t k )) f(v(,, z, t k )) L(M k ) u(,, z, t k ) v(,, z, t k ) = L k u(,, z, t k ) v(,, z, t k ), (3) where we have defined L k = L(M k ) and L ma = ma L k. k N Let h = L /N, h = L 2 /N 2, h z = L 3 /N 3, and τ = T/N be the spatial and time steps, respectivel. For i, j, k N and n N, we denote the eact and numerical solutions M(r, t) at a point ( i, j, z k ) at the moment of time t n as m( i, j, z k, t n ) and m n i,j,k, respectivel. Similar notations for f(m( i, j, z k, t n ), i, j, z k, t n ) and fi,j,k n. Firstl, utilizing the discrete scheme in [8], we can discretize the Caputo time fractional derivative of m( i, j, z k, t n+ ) as C Dt α m( i, j, z k, t n+ ) = τ α Γ (2 α) n b l [m( i, j, z k, t n+ l ) l= m ( i, j, z k, t n l )] + O(τ 2 α ), (4) where b l = (l + ) α l α, l =,,, N. Secondl, adopting the fractional centered difference scheme in [3], we can discretize the Riesz fractional derivative as β β m( i, j, z k, t n+ ) = h β i p= N +i where the coefficients ω p are defined b Similarl, ω p m( i p, j, z k, t n+ ) + O(h 2 ), (5) ( ) p Γ (β + ) ω p = Γ ( β 2 p + )Γ ( β p =,, 2,. (6) 2 + p + ), β β m( i, j, z k, t n+ ) = h β j q= N 2+j ω q m( i, j q, z k, t n+ )+O(h 2 ), (7) β z β m( i, j, z k, t n+ ) = k h β ω r m( i, j, z k r, t n+ ) + O(h 2 z). z r= N 3+k (8) The nonlinear source term can be treated either eplicitl or implicitl. In this paper, we use an eplicit method and evaluate the nonlinear source term at the previous time step: f n+ i,j,k = f n i,j,k + O(τ). (9)

7 A spatiall second-order accurate implicit numerical method for ST-FBTE 7 In this wa, we avoid solving a nonlinear sstem at each time step and obtain an unconditionall stable and convergent numerical scheme, as shown in sections 5 and 6. However, the shortcoming of the eplicit method is that it generates additional temporal error, as shown in (9). Then we can obtain the implicit numerical scheme: with K α τ α Γ (2 α) + h β j n l= q= N 2 +j b l (m n+ l i,j,k ω q m n+ i,j q,k + h β z m n l i,j,k ) = K β( h β k r= N 3 +k i p= N +i ω p m n+ i p,j,k ω r m n+ i,j,k r ) + f n i,j,k. (2) Thus, we have the following implicit difference approimation: m n+ i,j,k + µ +µ 3 k r= N 3 +k i p= N +i ω r m n+ ω p m n+ i p,j,k + µ 2 i,j,k r = n j q= N 2+j ω q m n+ i,j q,k (b l b l+ )m n l i,j,k + b nm i,j,k + µ fi,j,k, n (2) l= i =, 2,, N, j =, 2,, N 2, k =, 2,, N 3, m i,j,k = g i,j,k = g( i, j, z k ), m n+,j,k = mn+ N,j,k = mn+ i,,k = mn+ i,n 2,k = mn+ i,j, = mn+ i,j,n 3 =, (i =,,, N, j =,,, N 2, k =,,, N 3 ) where µ = τ α Γ (2 α) K α, µ = K βτ α Γ (2 α) K α h β, µ 2 = K βτ α Γ (2 α) K α h β, µ 3 = K βτ α Γ (2 α), K α h β z and noting that the coefficients µ, µ, µ 2, µ 3 > for < α and < β 2. Remark 2: If we use the implicit method to approimate the nonlinear source term, the numerical method of the ST-FBTE can be written as: m n+ i,j,k + µ +µ 3 k r= N 3 +k i p= N +i ω r m n+ ω p m n+ i p,j,k + µ 2 i,j,k r = n j q= N 2 +j ω q m n+ i,j q,k (b l b l+ )m n l i,j,k + b nm i,j,k + µ f n+ l= i,j,k,(22) namel, replace fi,j,k n n+ in (2) with fi,j,k. This numerical method is stable and convergent when the nonlinear source term f(m, r, t) satisfies the local Lipschitz condition (3) (see [2]). Lemma. The coefficients b l (l =,, 2, ) satisf: () b =, b l > for l =, 2, ; (2) b l > b l+ for l =,, 2,. Proof: See [].

8 8 J. Song, Q. Yu, F. Liu, I. Turner Lemma 2. The coefficients ω p (p N) satisf: () ω, ω k = ω k for all k ; (2) ω p = ; p= (3) For an positive integer n, m with n < m, we have Proof: See [3, 6]. n p= m+n ω p >. 4 Solvabilit of the implicit numerical method We let M n+ = [m n+,,, mn+ 2,,,, mn+ N,N 2,N 3 ]T, N = (N )(N 2 )(N 3 ), and M = [g,,, g 2,,,, g N,N 2,N 3 ] T, respectivel, then the implicit difference approimation (2) can be written in matri form as n (I + A)M n+ = (b l b l+ )M n l + b n M + µ F n, (23) l= where F n = [f,,, n f2,,, n, fn n,n 2,N 3 ]T, I R N N is the identit matri. A = [a sv ] R N N is a coefficient matri. If s = v for s =, 2,, N, then we obtain a ss = ω (µ + µ 2 + µ 3 ). (24) In addition, let s = (k )(N 3 ) + (j )(N 2 ) + i, then we have N 3 r= N 2 q= N p= = µ i p= N +i,p a s,(r )(N 3 )+(q )(N 2 )+p a s,s ω p + µ 2 j q= N 2+j,q ω q + µ 3 k r= N 3+k,r ω r, (25) for i =, 2,, N, j =, 2,, N 2, k =, 2,, N 3. Theorem. The difference equation defined b (23) is uniquel solvable. Proof: Let λ be the eigenvalue of the matri A. Then b the Gerschgorin s circle theorem [9] and Lemma 2, we have λ ω (µ + µ 2 + µ 3 ) r i = µ i p= N +i,p ω p + µ 2 j q= N 2 +j,q ω q + µ 3 k r= N 3 +k,r < ω (µ + µ 2 + µ 3 ), (26) where ω p = ω, ω q = ω and ω r = ω, that p=,p q=,q r=,r is, we have < λ < 2ω (µ + µ 2 + µ 3 ). (27) Hence the spectral radius of the matri (I + A) is greater than one. Therefore, the difference equation defined b (23) is uniquel solvable. ω r

9 A spatiall second-order accurate implicit numerical method for ST-FBTE 9 5 Stabilit of the implicit numerical method In this section, we prove the stabilit of the implicit numerical method for the ST-FBTE. Let m n i,j,k be the approimate solution of the implicit numerical method (2), and set E n = [ψ,,, n ψ2,,, n, ψn n,n 2,N 3 ]T, where ψi,j,k n = mn i,j,k m n i,j,k, and let f i,j,k n be the approimation of f i,j,k n. Assuming that E n = ma i N, j N 2, k N 3 ψn i,j,k, then we can obtain the following theorem b mathematical induction. Theorem 2. The implicit numerical method defined b (2) is unconditionall stable, and there is a positive constant C, such that E n+ C E, n =,, 2,. Proof: From (2), the error ψ n i,j,k satisfies ψ n+ i,j,k + µ i p= N +i ω p ψ n+ i p,j,k + µ 2 j q= N 2 +j ω q ψ n+ i,j q,k + µ 3 k r= N 3+k ω r ψ n+ i,j,k r n = (b m b m+ )ψ n m i,j,k + b nψi,j,k + µ (fi,j,k n f i,j,k), n (28) m= for i =, 2,, N, j =, 2,, N 2, k =, 2,, N 3. Since f satisfies the local Lipschitz condition (3), we have f n i,j,k f n i,j,k L n ψ n i,j,k. (29) When n =, assume that E = ma i N ψ i,j,k =, j N 2, k N 3 ψi,,k. Using Lemma 2, and noting that µ, µ 2, µ 3 > we have E = ψ ψi,,k + µ ω p ψi,,k + µ 2 ω q ψi,,k p= N + q= N 2 + k +µ 3 r= N 3 +k ω r ψ = [ + ω (µ + µ 2 + µ 3 )]ψ + µ +µ 2 q= N 2 +,q ω q ψ + µ 3 [ + ω (µ + µ 2 + µ 3 )]ψ + µ p= N +,p k r= N 3 +k,r p= N +,p ω p ψ ω r ψ ω p ψ p,,k

10 J. Song, Q. Yu, F. Liu, I. Turner +µ 2 q= N 2+,q ω q ψ, q,k + µ 3 k r= N 3+k,r ω r ψ,,k r. With the well known inequalit Z Z 2 Z Z 2, using Lemma we have E = ψi,,k [ + ω (µ + µ 2 + µ 3 )]ψi,,k + µ ω p ψi p,,k p= N +,p k +µ 2 ω q ψi, q,k + µ 3 ω r ψ,,k r q= N 2+,q r= N 3+k,r = ψ,,k + µ ω p ψi p,,k + µ 2 ω q ψi, q,k p= N + q= N 2 + k +µ 3 ω r ψ,,k r r= N 3+k = b ψ + µ (f i,j,k f i,j,k). Using local Lipschitz condition (29), we have E ψ + µ L ψ ψ + µ L ma ψ = ( + µ L ma )ψ,,k = ( + µ L ma ) E. Let ξ = + µ L ma, thus, E ξ E. Now, we suppose that E m ξ E, m =, 2,, n. Assuming E n+ = ma i N, j N 2, k N 3 ψn+ i,j,k = ψn+, and using Lemma 2 again, we can obtain E n+ = ψ n+ ψ n+,,k + µ k +µ 3 r= N 3 +k ω r ψ n+ p= N + ω p ψ n+,,k + µ 2 = [ + ω (µ + µ 2 + µ 3 )]ψ n+,,k + µ +µ 2 q= N 2 +,q ω q ψ n+ + µ 3 q= N 2+ ω q ψ n+,,k p= N +,p k r= N 3 +k,r ω p ψ n+,,k ω r ψ n+

11 A spatiall second-order accurate implicit numerical method for ST-FBTE [ + ω (µ + µ 2 + µ 3 )]ψ n+ + µ +µ 2 q= N 2 +,q ω q ψ n+, q,k + µ 3 p= N +,p k r= N 3 +k,r ω p ψ n+ p,,k ω r ψ n+ r. Using inequalit Z Z 2 Z Z 2 and Lemma again, we have E n+ [ + ω (µ + µ 2 + µ 3 )]ψ n+ + µ +µ 2 ψn+ q= N 2 +,q = + µ ω q ψ n+, q,k + µ 3 ω p ψ n+ p,,k + µ 2 p= N + p= N +,p k r= N 3 +k,r ω p ψ n+ p,,k ω r ψ n+ r q= N 2 + ω q ψ n+, q,k k +µ 3 ω r ψ n+ r r= N 3 +k n = (b m b m+ )ψ n m,,k + b n ψi,,k + µ (fi,j,k n f i,j,k) n. m= Using local Lipschitz condition (29) again, we have n E n+ (b m b m+ ) E n m + b n E + µ L n ψi n,,k m= n ξ (b m b m+ ) E + b n E + µ L ma E n m= (b n + b ξ b n ξ) E + ξµ L ma E = (ξ 2 b n µ L ma ) E. From Lemma, we know that b = and b n as n. Also, note that ξ = + µ L ma, then it is eas to know that ξ 2 b n µ L ma >. Hence, let C = ξ 2 b n µ L ma, we have E n+ C E. (3) Hence the implicit numerical method defined b (2) is unconditionall stable. Remark 3: If we use an implicit method to approimate the nonlinear source term, as shown in Remark 2, we can prove that the numerical method defined in (22) is stable when µ L ma >, which is independent of the spatial

12 2 J. Song, Q. Yu, F. Liu, I. Turner step. In fact, when the time step is small, the condition µ L ma > is generall satisfied. 6 Convergence of the implicit numerical method In this section, we prove the convergence of the implicit numerical method for the ST-FBTE. Setting θi,j,k n = m( i, j, z k, t n ) m n i,j,k, and denoting R n = [θ n,,, θ n 2,,,, θ n N,N 2,N 3 ] T, where R =. Note that R n and are ((N ) (N 2 ) (N 3 )) vectors, respectivel. From ()-(2), the error θ n i,j,k satisfies θ n+ i,j,k + µ i p= N +i ω p θ n+ i p,j,k + µ 2 j q= N 2+j ω q θ n+ i,j q,k + µ 3 k r= N 3 +k n = (b m b m+ )θ n m + µ (f(m( i, j, z k, t n ), i, j, z k, t n ) fi,j,k) n m= i,j,k ω r θ n+ i,j,k r + C τ α (τ 2 α + τ + h 2 + h 2 + h 2 z), (3) for i =, 2,, N, j =, 2,, N 2, k =, 2,, N 3. Since f satisfies the local Lipschitz condition (3), we have Assuming R n+ = f(m( i, j, z k, t n ), i, j, z k, t n ) f n i,j,k L n θ n i,j,k. (32) ma i N, j N 2, k N 3 θn+ i,j,k, then we can obtain the following theorem b mathematical induction. Theorem 3. The implicit difference approimation defined b (2) is convergent, and there is a positive constant C, such that R n+ C (τ 2 α + τ + h 2 + h 2 + h 2 z), n =,, 2,. (33) Proof: When n =, assume that R = ma i N θ i,j,k =, j N 2, k N 3 θi,,k. Similarl, using Lemma 2, and noting that µ, µ 2, µ 3 >, we have R = θ θi,,k + µ ω p θi,,k + µ 2 ω q θi,,k p= N + q= N 2+ k +µ 3 r= N 3 +k ω r θ

13 A spatiall second-order accurate implicit numerical method for ST-FBTE 3 = [ + ω (µ + µ 2 + µ 3 )]θ,,k + µ +µ 2 q= N 2 +,q ω q θ + µ 3 [ + ω (µ + µ 2 + µ 3 )]θ,,k + µ +µ 2 q= N 2 +,q ω q θ, q,k + µ 3 p= N +,p k r= N 3 +k,r p= N +,p k r= N 3 +k,r ω p θ,,k ω r θ ω p θ p,,k ω r θ r. Let V = C τ α (τ 2 α + τ + h 2 + h 2 + h 2 z), using inequalit Z Z 2 Z Z 2, Lemma and local Lipschitz condition (32), we have R [ + ω (µ + µ 2 + µ 3 )]θi,,k + µ ω p θi p,,k p= N +,p k +µ 2 ω q θi, q,k + µ 3 ω r θ r q= N 2 +,q r= N 3 +k,r = θ + µ ω p θi p,,k + µ 2 ω q θi, q,k p= N + q= N 2 + k +µ 3 ω r θ r r= N 3 +k = µ (f(m( i, j, z k, t ), i, j, z k, t ) f ) + V µ L θ + V = b V. Now, we suppose that R m R n+ = b m V, m =, 2,, n. Assuming,,k, and using Lemma 2 again we have ma i N, j N 2, k N 3 θn+ i,j,k = θn+ R n+ = θ n+ θ n+ + µ k +µ 3 r= N 3 +k ω r θ n+ p= N + ω p θ n+ + µ 2,,k = [ + ω (µ + µ 2 + µ 3 )]θ n+ + µ q= N 2 + ω q θ n+ p= N +,p ω p θ n+

14 4 J. Song, Q. Yu, F. Liu, I. Turner +µ 2 q= N 2+,q ω q θ n+ + µ 3 [ + ω (µ + µ 2 + µ 3 )]θ n+ + µ +µ 2 q= N 2+,q ω q θ n+, q,k + µ 3 k r= N 3+k,r p= N +,p k r= N 3+k,r ω r θ n+ ω p θ n+ p,,k ω r θ n+ r. Using inequalit Z Z 2 Z Z 2 and Lemma again, we have R n+ [ + ω (µ + µ 2 + µ 3 )]θ n+ + µ +µ 2 θn+ q= N 2 +,q = + µ ω q θ n+, q,k + µ 3 ω p θ n+ p,,k + µ 2 p= N + p= N +,p k r= N 3 +k,r ω p θ n+ p,,k ω r θ n+ r q= N 2 + ω q θ n+, q,k k +µ 3 ω r θ n+,,k r r= N 3 +k n = (b m b m+ )θ n m + µ (f(m( i, j, z k, t n ), i, j, z k, t n ) m= f n ) + V. Using local Lipschitz condition (32) again, we have n R n+ (b m b m+ )θ n m,,k + µ L n θi n,,k + V m= n (b m b m+ )b n m V + µ L ma b n V + V m= n (b m b m+ )b n V + µ L ma b n V + V m= = b n (b b n + µ L ma + b n )V = b n (b + µ L ma )V = ξb n V.

15 A spatiall second-order accurate implicit numerical method for ST-FBTE 5 Noting that V = C τ α (τ 2 α + τ + h 2 + h 2 + h 2 z), and lim n b n n α = lim n α n (n + ) α n α = α, therefore there eists a positive constant C 2, such that R n+ ξc C 2 n α τ α (τ 2 α + τ + h 2 + h 2 + h 2 z). Finall, noting that nτ T is finite, so there eists a positive constant C, such that R n+ C (τ 2 α + τ + h 2 + h 2 + h 2 z) for n =,, 2,. Hence, the implicit numerical method defined b (2) is convergent. Remark 4: If we use an implicit method to approimate the nonlinear source term, as shown in Remark 2, we can prove that the numerical method defined in (22) is convergent when µ L ma >, which is independent of the spatial step. In fact, when the time step is small, the condition µ L ma > is generall satisfied. 7 Numerical results Due to the computational overheads necessar to perform the simulations for the space and time fractional Bloch-Torre equation in three dimensions, we present here a preliminar stud based on a two-dimensional eample to confirm our theoretical analsis. In eample, we use the same eample in [24], where the source term depends onl on space and time, for comparison to show that our new implicit numerical method can obtain second order space accurac, which is O(τ 2 α + τ + h 2 + h 2 + h 2 z). Eample. The following time and space Riesz fractional diffusion equation with initial and zero Dirichlet boundar conditions on a finite domain is considered (See [24]): where f(r, t) = C K α Dt α M(r, t) = K β ( β β + β )M(r, t) + f(r, t), β (34) M(r, ) =, (35) M(r, t) Ω =, (36) K βt α+β 2cos(βπ/2) (( 2 Γ (3 β) [2 β + ( ) 2 β 2 ] Γ (4 β) [3 β +( ) 3 β 24 ] + Γ (5 β) [4 β + ( ) 4 β ]) 2 ( ) 2 2 +( Γ (3 β) [2 β + ( ) 2 β 2 ] Γ (4 β) [3 β + ( ) 3 β ]

16 6 J. Song, Q. Yu, F. Liu, I. Turner + 24 Γ (5 β) [4 β + ( ) 4 β ]) 2 ( ) 2 ) + K αγ (α + β + ) t β 2 ( ) 2 ) 2 ( ) 2, (37) Γ (β + ) and < α, < β 2, t >, r = (, ) Ω, Ω is the finite rectangular region [, ] [, ], and Ω is R 2 Ω. The eact solution of this problem is M(r, t) = t α+β 2 ( ) 2 2 ( ) 2, which can be verified b substituting directl into (34). When K α =., K β =.5, α =.8, and β =.8, Table lists the maimum absolute error between the eact solution and the numerical solutions obtained b the implicit numerical method, with spatial and temporal steps τ 2 α 2 h = h = /4, /8, /6, /32 at time t =. Table Comparison of maimum error for the implicit numerical method at time t =. τ 2 α 2 h = h Maimum error Error rate From Table, it can be seen that the Error rate = error(h)2 error( 4. 2h)2 This is in good agreement with our theoretical analsis, namel the convergence order of the implicit numerical method for this problem is O(τ 2 α + τ + h 2 + h 2 ). We now ehibit in eample 2 the solution profiles of the time and space Riesz fractional diffusion equation with a nonlinear source term. Eample 2. Nonlinear time and space Riesz fractional diffusion equation with initial and zero Dirichlet boundar conditions on a finite domain: C K α Dt α M(r, t) = K β ( β β + β )M(r, t) + f(m, r, t), (38) β M(r, ) = min{., e 2 2 }, (39) M(r, t) Ω =, (4) where the nonlinear source term is Fisher s growth equation f(m, r, t) =.25M(r, t)[ M(r, t)], and < α, < β 2, t >, r = (, ) Ω, Ω is the finite rectangular region [, ] [, ], and Ω is R 2 Ω. The solution profiles of (38) b the implicit numerical method, with spatial and temporal steps h = h = /32, τ = /322 at time t = 32/322 with

17 A spatiall second-order accurate implicit numerical method for ST-FBTE M(,,t=32/322) M(,,t=32/322) (a) (b) M(,,t=32/322) M(,,t=32/322) (c) (d) Fig. A plot of numerical solutions of ST-FBTE using the implicit numerical method (INM) with spatial and temporal steps h = h = /32, τ = /322 at time t = 32/322 with K α =., t final =. for different α, β and K β. (a) α =., β = 2., K β =.. (b) α =., β = 2., K β = 2.. (c) α =.8, β =.8, K β =.. (d) α =.8, β =.8, K β = 2.. K α =., t final =. for different α, β and K β are listed in Figure. From Figure, it can be seen that the coefficient K β impacts on the solution profiles of (38), whereb a larger value of K β produces more diffuse profiles. In Figure 2, we illustrate the effect of the fractional order in space for this problem, with spatial and temporal steps h = h = /32, τ = /322 at time t = 32/322 with K α =., K β =., t final =. for β fied at 2 and α varing. From Figure 2, it can be seen that as α is decreased the diffusion profiles becomes more pronounced. In Figure 3, we illustrate the effect of the fractional order in time for this problem, with spatial and temporal steps h = h = /32, τ = /322 at time t = 32/322 with K α =., K β =., t final =. for α fied at and β varing. From Figure 3, it can be seen that as β is reduced the diffusion becomes more pronounced. 8 Conclusions In this paper, a new effective implicit numerical method for solving the fractional Bloch-Torre equation in three-dimensions with a nonlinear source term

18 8 J. Song, Q. Yu, F. Liu, I. Turner M(,,t=32/322) M(,,t=32/322) (a) (b) M(,,t=32/322) M(,,t=32/322) (c) (d) Fig. 2 A plot of numerical solutions of ST-FBTE using the implicit numerical method (INM) with spatial and temporal steps h = h = /32, τ = /322 at time t = 32/322 with K α =., K β =., t final =. for β fied at 2. (a) α =.. (b) α =.9. (c) α =.5. (d) α =.2. has been derived. We prove that the implicit numerical method is uniquel solvable, unconditionall stable and convergent. In addition, compared with first order spatial accurac of convergence in [24], and the two-dimensional model with a linear source term in [25], our new implicit numerical method can obtain second order space accurac, which is O(τ 2 α + τ + h 2 + h 2 + h 2 z). Acknowledgements The authors gratefull acknowledge the help and interest in our work b Professor Kerrie Mengersen from QUT. Mr Yu also acknowledges the Centre for Comple Dnamic Sstems Control at QUT for offering financial support for his PhD scholarship. This research was partiall supported b the Australian Research Council grant DP2377. The authors wish to thank the referees for their constructive comments and suggestions. References. B. Baeumer, M. Kovál and M.M. Meerschaert, Fractional reproduction-dispersal equations and heav tail dispersal kernels, Bulletin of Mathematical Biolog, 69, , (27). 2. B. Baeumer, M. Kovál and M.M. Meerschaert, Numerical solutions for fractional reaction-diffusion equations, Computers and Mathematics with Applications, 55, , (28).

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