An alternating segment Crank Nicolson parallel difference scheme for the time fractional sub-diffusion equation

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1 Wu et a Advances in Difference Equations (018) 018:87 R E S E A R C H Open Access An aternating segment Crank Nicoson parae difference scheme for the time fractiona sub-diffusion equation Lifei Wu 1, Xiaozhong Yang 1* and Yanhua Cao 1 * Correspondence: yxiaozh@ncepueducn 1 Schoo of Mathematics and Physics, North China Eectric Power University, Beijing, China Abstract In this paper, an aternating segment Crank Nicoson (ASC-N) parae difference scheme is proposed for the time fractiona sub-diffusion equation, which consists of the cassica Crank Nicoson scheme, four kinds of Sau yev asymmetric schemes, and aternating segment technique Theoretica anaysis reveas that the ASC-N scheme is unconditionay stabe and convergent by mathematica induction method Finay, the theoretica anaysis is verified by numerica experiments, which show that the ASC-N scheme is efficient for soving the time fractiona sub-diffusion equation Keywords: Time fractiona sub-diffusion equation; Aternating segment Crank Nicoson scheme; Stabiity; Convergence; Parae computing 1 Introduction At present, the research on the fractiona order differentia and integra equations is attracting more and more attention This kind of fractiona order differentia and integra equations are widey used in various fieds of science and engineering [1, ] Fractiona derivatives have become an important too to describe various compex mechanica behaviors [3 5] Most of fractiona differentia and integra equations cannot be soved anayticay [6, 7] It is necessary and important to deveop numerica methods for soving fractiona differentia and integra equations [8 10] In recent years, many schoars have studied the numerica agorithms of fractiona diffusion equation The finite difference method is sti dominant among the existing agorithms now A cass of unconditionay stabe and convergent impicit difference approximate methods for time fractiona diffusion equation was constructed by Zhuang Pinghui et a [11] Chares Tadjeran et a [1] examined a second order accurate numerica method in time and in space to sove a cass of initia-boundary vaue fractiona diffusion equations with variabe coefficients The cassica Crank Nichoson (C-N) method and spatia extrapoation are used to obtain temporay and spatiay second order accurate numerica estimates [1] Chen Changming et a [13] presented impicit and expicit difference methods for soving the two-dimensiona anomaous subdiffusion equation, and appied a new mutivariate extrapoation to improve the accuracy [13] Zhang, Pu et a [14]constructed a Crank Nicoson-type difference scheme for fractiona subdiffusion equation The Author(s) 018 This artice is distributed under the terms of the Creative Commons Attribution 40 Internationa License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the origina author(s) and the source, provide a ink to the Creative Commons icense, and indicate if changes were made

2 Wu et a Advances in Difference Equations (018) 018:87 Page of 18 with spatiay variabe coefficient [14] Baeanu et a [15] discussed the motion of a bead on a wire by fractiona cacuus [15] They soved the fractiona Euer Lagrange equation numericay by using a discretization technique based on a Grünwad Letnikov approximation for the fractiona derivative Jajarmi et a [16] investigated an efficient iterative scheme with ow computationa effort for the optima contro of noninear fractiona order dynamic systems with externa persistent disturbances [16] Hajipour et a [17] provided a new nonstandard finite difference scheme to study the dynamic treatments of a cass of fractiona chaotic systems and stabiity anaysis of fractiona order systems [17] By combining the impicit difference scheme and the preconditioned conjugate gradient method, Wang Hong et a [18, 19] gave a fast impicit difference method for the two/threedimensiona fractiona diffusion equation The method has a computationa work count of O(N og N) per iteration and a memory requirement of O(N)[18, 19]GaoGuanghuaeta [0] derived a compact finite difference scheme for the sub-diffusion equation, which is fourth order accuracy compact approximate for the second order space derivative [0] Gao Guanghua et a [1] deveoped two higher order difference schemes for numericay soving the one-dimensiona time distributed-order fractiona wave equations, one of which is the second order convergence in time and fourth order convergence in both space and distributed order [1] Yaseen et a [] proposed a finite difference scheme with cubic trigonometric B-spine functions for time fractiona diffusion-wave equation with reaction term [] The scheme is unconditiona stabe and convergent Zaky et a [3] proposed an efficient numerica agorithm for soving the variabe order fractiona Gaiei advection-diffusion equation [3] The method has the advantage of transforming the probem into the soution of a system of agebraic equations, which greaty simpifies it However, the existing seria impicit difference schemes are of reativey high computationa compexity and reativey ow computationa efficiency With the rapid deveopment of muti core and custer technoogy, parae computing is one of the main techniques to improve the efficiency of numerica cacuation [4 7] Zhang Baoin et a proposed the idea which uses the Sau yev asymmetric scheme to construct a segment impicit scheme, and the aternative technique to estabish a variety of expicit-impicit or impicit aternating parae methods The methods have been we appied to numericay sove integer partia differentia equations [8 31] Some progress in parae agorithms of fractiona order partia differentia equations has been made Most of the agorithms are studied on the parae agorithm of agebraic equations from the perspective of numerica agebra Kai Diethem [3] proposed to impement the fractiona version of the second order Adams Bashforth Mouton method on a parae computer and discussed the precise nature of parae method [3] Gong Chunye et a [33, 34] proposed an efficient parae for Riesz and Caputo fractiona reaction-diffusion equation respectivey The parae sover invoves the parae tridiagona matrix vector mutipication, vector vector addition [33 35] Sweiam et a [36] constructed a cass of parae C-N difference schemes for time fractiona paraboic equations The core of the method is to use the precondition conjugate gradient method to sove Ax = b [36] This method is estabished in the view of numerica agebra In this paper, we study the parae agorithm based on the parae of the traditiona difference scheme [37, 38]

3 Wu et a Advances in Difference Equations (018) 018:87 Page 3 of 18 In this paper, we construct an aternating segment C-N (ASC-N) difference scheme for time fractiona sub-diffusion equation The four kinds of Sau yev asymmetric schemes and the cassica C-N scheme are proposed Then we use the four kinds of Sau yev asymmetric scheme and the cassica C-N scheme to construct an ASC-N scheme by the aternating segment technoogy Using the mathematica induction method, we prove that the ASC-N scheme is unconditionay stabe and convergent Finay, the theoretica anaysis is verified by numerica experiments, which show that the ASC-N scheme is effective for soving time fractiona sub-diffusion equation Time fractiona sub-diffusion equation In this paper, we consider the time fractiona sub-diffusion equation of the form [, 3] α u(x, t) t α = u(x, t) x, 0 x L,0 t T (1) Initia boundary conditions u(0, t)=u(l, t)=0, 0 t T, u(x,0)=f (x), 0 x L, where α u(x,t) t α u(x, t): α u(x, t) t α = (0 < α < 1) denotes the Caputo fractiona derivative of order α of the function 1 t u(x, ξ) Ɣ(1 α) 0 ξ dξ (t ξ) α () By taking the finite sine transform and Lapace transform, the anaytica soution for equation (1) with the boundary conditions as above is obtained as [3] u(x, t)= L ( E α a n t α) sin(anx) n=1 t 0 f (r) sin(anr) dr, (3) where E α (z) is a Mittag-Leffer function, E α (z)= z k k=0 Ɣ(αk+1), a = π L Some specia Mittag-Leffer type functions are isted as foows: E 1 ( z)=e z, E 1 (z)= k=1 E ( z ) = cos(z), z k (4) Ɣ( k +1) = ez erfc( z), where erfc(z) is the error function compement defined by erfc(z)= 3 The ASC-N scheme of fractiona sub-diffusion equation π z e t dt In this section, we first define t k = kτ, k =0,1,,,N,andx i = ih, i =0,1,,,M,where τ = T N, h = L M aretimeandspacesteps,respectiveyletuk i be the numerica approximation of u(x i, t k )

4 Wu et a Advances in Difference Equations (018) 018:87 Page 4 of 18 The time fractiona derivative term can be approximated by the foowing scheme: L α h,τ u(x i, t k+1 )= τ α Ɣ( α) k [ b j u(xi, t k+1 j ) u(x i, t k j ) ], (5) where b j =(j +1) α j α, j =0,1,,,N Four kinds of Sau yev asymmetric scheme are constructed for equation (1) L α h,τ u(x i, t k+1 )= 1 [ δ h x u k i + ( u k i+1 uk i uk+1 i + u k+1 )] i 1, (6) L α h,τ u(x i, t k+1 )= 1 h [ δ x u k i + ( u k+1 i+1 uk+1 i u k i + uk i 1)], (7) L α h,τ u(x i, t k+1 )= 1 [ δ h x u k+1 i + ( u k+1 i+1 uk+1 i u k i + i 1)] uk, (8) L α h,τ u(x i, t k+1 )= 1 [ δ h x u k+1 i + ( u k i+1 uk i uk+1 i + u k+1 )] i 1, (9) where δx uk i = u k i+1 uk i + u k τ α i 1, i = 1,,,M 1,k = 0,1,,,N 1Letμ =, h r = μɣ( α) The four kinds of Sau yev asymmetric scheme aso can be rewritten as foows When k =0, (1+ 1 )u r 1i 1 (1 ru1i 1 = ru0i ) r u 0 i + 1 ru0 i 1, 1 ru1 i+1 + ( 1+ 1 r ) u 1 i = 1 ru0 i+1 + ( 1 3 r )u 0 i + ru0 i 1, ru 1 i+1 + ( 1+ 3 r ) u 1 i 1 ru1 i 1 = ( 1 1 r )u 0 i + 1 ru0 i 1, 1 ru1 i+1 + ( 1+ 3 r ) u 1 i ru1 i 1 = 1 ru0 i+1 + ( 1 1 r )u 0 i When k >0, (1+ 1 ) r u k+1 i 1 ruk+1 i 1 = ruk i+1 3 ruk i + 1 k 1 ruk i 1 + c j u k j i + b k u 0 i, 1 ( ruk+1 i ) r u k+1 i = 1 ruk i+1 3 k 1 ruk i + ruk i 1 + c j u k j i + b k u 0 i, ( ru k+1 i ) r u k+1 i 1 ruk+1 i 1 = 1 ruk i + 1 k 1 ruk i 1 + c j u k j i + b k u 0 i, 1 ( ruk+1 i ) r u k+1 i ru k+1 i 1 = 1 ruk i+1 1 k 1 ruk i + c j u k j i + b k u 0 i, where c j = b j b j+1, j =0,,ThecassicaC-Nschemeforequation(1)is L α h,τ u(x i, t k+1 )= 1 ( h δ x u k i + u k+1 ) i (10)

5 Wu et a Advances in Difference Equations (018) 018:87 Page 5 of 18 Figure 1 Point diagram of the ASC-N scheme The C-N scheme aso can be rewritten as foows When k =0, 1 u1 i+1 +(1+r)u1 i 1 ru1 i 1 = 1 ru0 i+1 +(1 r)u0 i + 1 ru0 i 1 When k >0, 1 uk+1 i+1 +(1+r)uk+1 i 1 ruk+1 i 1 = 1 ruk i+1 ruk i + 1 k 1 ruk i 1 + c j u k j i + b k u 0 i The design of the ASC-N scheme for equation (1) is as foows Suppose the numerica vaue of m grid points needs to be cacuated on each time ayer Let m = A (A, are integers, A is odd, and A 3, 3) m grid points wi be divided into A segments, which is named S 1, S,,S A in turn In order to faciitate the description, set m =5, =5Thedesign of ASC-N scheme s segmentation is described by Fig 1Weuse to denote the C-N scheme in Fig 1 S 1, S 5 have one inner boundary point (i =5,i = 1), and four inner points (i = 1,,3,4,i =, 3, 4, 5), respectivey S, S 3, S 4 have two inner boundary points (i = 6, 10, i = 11, 15, i = 16, 0) and three inner points (i =7,8,9,i = 1, 13, 14, i = 17, 18, 19), respectivey We appy the C-N scheme at inner points, appy aternatey the Sau yev asymmetric scheme at inner boundary points in order to contact other inner points Specificay, at inner boundary points (5, k + 1), (15, k + 1), (10, k + ), (0, k +)and(6,k +1), (16, k + 1), (11, k + ), (1, k + ), we respectivey appy Sau yev asymmetric schemes (6), (7) At inner boundary points (10, k + 1), (0, k +1),(5,k + ), (15, k + ) and (11, k +1), (1, k +1),(6,k + ), (16, k + ), we respectivey appy Sau yev asymmetric schemes (8), (9) And we aternatey appy (6)and(8), (7)and(9) at different time steps Therefore, at k +1 time step the space region wi be divided into three segments (S 1, S and S 3, S 4 and S 5 ), which are three impicit segments and can be soved independenty At k +timestep,the space region wi be divided into three impicit segments (S 1 and S, S 3 and S 4, S 5 ) Then

6 Wu et a Advances in Difference Equations (018) 018:87 Page 6 of 18 we get the ASC-N scheme for equation (1) as foows When k =0, (I + rg )U 1 =(I rg 1 )U 0 + H 0 (11) When k >0, (I + rg 1 )U k+1 =(c 0 I rg )U k + c 1 U k c k 1 U 1 + b k U 0 + H k+1, (I + rg )U k+ =(c 0 I rg 1 )U k+1 + c 1 U k + + c k U 1 + b k+1 U 0 + H k+, (1) where I is the unit matrix, k =1,3,5, u k u k 0 u u0 0 H k+1 = r 0, H 0 = r 0, 0 0 u k+1 m+1 + uk m+1 u 1 m 1 m+1 + u0 m+1 m 1 G (1) G 1 = 1 G, G G (1) M M G () G = 1 G, G G () M M G (1) =, G () = G = ( +1)th, ,

7 Wu et a Advances in Difference Equations (018) 018:87 Page 7 of 18 G (1) = G () = ( +1)th ( +1)th 1 1, Using the properties of the function g(x) =x 1 α (x 1), the foowing resuts can be obtained: 1=b 0 > b 1 > b > 0, k j=1 c j =1 b k, j=1 c j =1, 1> 1 α = c 1 > c > c 3 > 0 From the definitions of G 1 and G, we are easy to know that I + G 1 and I + G 1 are stricty diagonay dominant matrices Therefore, we can get that ASC-N scheme (1) fortime fractiona sub-diffusion equation is uniquey sovabe 4 Theoretica anaysis of the ASC-N scheme 41 Stabiity anaysis of the ASC-N scheme In order to discuss the stabiity of the ASC-N scheme, we need the foowing Keogg emma [4] Lemma 1 If ρ >0and P is a nonnegative rea matrix, then (ρi + P) 1 (ρi P)(ρI + P) 1 1 exists and Lemma The matrices G 1 and G of the ASC-N scheme are nonnegative rea matrixes If G 1 and G meet that G 1 + G T 1, G + G T are nonnegative rea matrices, then Lemma wi be estabished Therefore, we ony need to prove G 1 + G T 1, G + G T are nonnegative

8 Wu et a Advances in Difference Equations (018) 018:87 Page 8 of 18 rea matrices Sufficient and necessary conditions for the estabishment of Lemma are G (1) + G (1) T, G () + G () T, G + G T, G (1) + G (1) T, G () + G () T are nonnegative rea matrixes 4 4 G + G T = ( +1)thine 4 4 G + G T is diagonay dominant three diagona matrix, and the main diagona eements are positive rea numbers So G + G T is a nonnegative rea matrix In the same way, we can prove that G (1) + G (1) T, G () + G () T, G (1) + G (1) T, G () + G () T are nonnegative rea matrices Therefore, G 1 + G T 1, G + G T are nonnegative rea matrices In other words, G 1 and G are two nonnegative rea matrices From the definitions of G (1), G (), G (1),andG(), we can know that G(1) and G () have the same eigenvaues, G (1) and G () have the same eigenvaues Therefore, G 1 and G have the same eigenvaues, and rg 1 = rg 0 The growth matrix of the ASC-N scheme for time fractiona sub-diffusion equation is, T =(I + rg ) 1 (I rg 1 )(I + rg 1 ) 1 (I rg ) We can easiy obtain estimates of the growth matrix by Lemmas 1 and Let ˆT =(I + rg )T(I + rg ) 1 =(I rg 1 )(I + rg 1 ) 1 (I rg )(I + rg ) 1, then T = ˆT = (I rg1 )(I + rg 1 ) 1 (I rg )(I + rg ) 1 1 We suppose that ũ k i (i = 0,1,,,M; k = 0,1,,,N) is the approximate soution of ASC-N scheme (1), the error ε i k = ũ k i uk i, Ek =[ε1 k, εk,,εk M 1 ] satisfies (I + rg )E 1 =(I rg 1 )E 0, (I + rg 1 )E k+1 =(c 0 I rg )E k + c 1 E k c k 1 E 1 + b k E 0, (I + rg )E k+ =(c 0 I rg 1 )E k+1 + c 1 E k + + c k E 1 + b k+1 E 0, where k =1,3,5,Thefoowingresutisprovedusingmathematicainduction

9 Wu et a Advances in Difference Equations (018) 018:87 Page 9 of 18 For k =1,λ 1, λ are the eigenvaues of rg 1, rg,respectivey T = { (I rg 1 )(I + rg 1 ) 1 (I rg )(I + rg ) 1 ( ) } 1 λ1 = max 1, 1+λ (I + rg ) 1 (I rg 1 ) } 1 λ = max{ 1+λ 1, 1 E 1 = (I + rg ) 1 (I rg 1 )E 0 (I + rg ) 1 (I rg 1 ) E 0 E 0 For k =, E = (I + rg 1 ) 1[ (c 0 I rg )E 1 + b 1 E 0] (I + rg 1 ) 1 ( (c 0 I rg )E 1 ) + b 1 E 0 { c 0 λ 1 + b 1 max 1+λ } E 0 When c 0 b 1, c 0 λ 1 + b 1 1+λ = c 0 λ 1 + b 1 1+λ = 1 λ 1 1+λ 1 When c 0 b 1,0<b <1 1 < b 1 1<1, c 0 λ 1 + b 1 1+λ = λ 1 c 0 + b 1 1+λ = λ 1 +b λ 1 Therefore, we have } E c 0 λ 1 + b 1 max{ 1+λ E 0 E 0 For k =3, E 3 = (I + rg ) 1[ (c 0 I rg 1 )E + c 1 E 1 + b E 0] = (I + rg ) 1 (c0 I rg 1 )E + c 1 E 1 + b E 0 (I + rg1 ) 1 ( (c0 I rg ) + c 1 + b ) E 0 (I + rg 1 ) 1 ( (c 0 I rg ) ) + b 1 E 0 { c 0 λ 1 + b 1 max 1+λ } E 0 E 0 Suppose that E j E 0, j kwehave E k+1 = (I + rg1 ) 1[ (c 0 I rg )E k + c 1 E k b k E 0] (I + rg1 ) 1 ( c 0 I rg + c b k ) E 0

10 Wu et a Advances in Difference Equations (018) 018:87 Page 10 of 18 = (I + rg1 ) 1 ( c 0 I rg + b 1 ) E 0 E 0, E k+ = (I + rg ) 1[ (c 0 I rg 1 )E k+1 + c 1 E k + + b k+1 E 0] (I + rg ) 1 ( c 0 I rg 1 + c b k+1 ) E 0 = (I + rg ) 1 ( c 0 I rg 1 + b 1 ) E 0 E 0 In summary, we have E k E 0, k =1,,3,Hence,thefoowingtheoremisobtained Theorem 1 ASC-N scheme (1) for time fractiona sub-diffusion equation (1) is unconditionay stabe 4 Accuracy of the ASC-N scheme The ASC-N scheme wi be expanded as the Tayor series at the point (x i, t k+1 ) Let u(x i, t k ) be the exact soution of the time fractiona sub-diffusion equation at the mesh point (x i, t k ) Lemma 3 Suppose that u(x, t) is the exact soution of the time fractiona sub-diffusion equation and u(x, t) C (4,) ([0, L] [0, T]), then L α h,τ u(x i, t k+1 )= α u(x i, t k+1 ) t α θτ α+1 u(x i, t k+1 ) t α+1 + r 1, where r 1 Cτ α, Cisaconstant,0<θ <1 Proof L α h,τ u(x i, t k+1 ) r 1 = = α u(x i, t k+1 ) 1 + t α Ɣ(1 α) k (j+1)τ = α u(x i, t k+1 ) θτ α+1 u(x i, t k+1 ) t α t α Ɣ(1 α) + O ( τ α), 1 Ɣ(1 α) k k (j+1)τ jτ (j+1)τ From equation (1), we have α+1 u(x, t) t α+1 = jτ jτ u(x i, ξ) ξ u(x i, ξ) ξ 1 t u(x, ξ) Ɣ(1 α) 0 ξ u(x i, ξ) (t j+1 ξ)(t j ξ) dξ ξ (t k+1 ξ) α 1 (t k+1 ξ) α 1 (t k+1 ξ) α dξ (t ξ) ( θτ + (t ) j+1 ξ)(t j ξ) dξ ( θτ + (t ) j+1 ξ)(t j ξ) dξ + O ( τ α) α, 0<α 1

11 Wu et a Advances in Difference Equations (018) 018:87 Page 11 of 18 Set c u is a constant depending ony on u(x,t), t 1 Ɣ(1 α) = k c u Ɣ(1 α) (j+1)τ jτ k u(x, ξ) ξ (j+1)τ jτ c u τ 1 α (1 α)ɣ(1 α) = c u(k +1) Ɣ( α) τ 1 α, 1 Ɣ(1 α) Therefore, r 1 = k 1 Ɣ(1 α) (j+1)τ jτ k dξ (t k+1 ξ) α dξ (t k+1 ξ) α k [ ] (k +1 j) (k j) u(x, ξ) (t j+1 ξ)(t j ξ) dξ c ξ (t k+1 ξ) α u τ α (j+1)τ jτ u(x i, ξ) ξ 1 (t k+1 ξ) α ( ) k +1 Ɣ( α) +1 c u τ α + O ( τ α) Cτ α ( θτ + (t ) j+1 ξ)(t j ξ) dξ + O ( τ α) This competes the proof First, the truncation error Tcn i,k+1 for the ASC-N scheme of the C-N scheme wi be anayzed at the inner points T i,k+1 cn = L α h,τ u(x i, t k+1 ) 1 ( h δ x u(xi, t k )+u(x i, t k+1 ) ) = α u(x, t) τ α+1 u(x, t) + r t α t α+1 1 u xx (x i, t k )+ τ u txx(x i, t k+1 ) h 1 u xxxx(x i, t k+1 ) τ 4 u ttxx(x i, t k+1 )+O ( τ α + h ) = O ( τ α + h ) Second, the truncation error T i,k+1 6 of Sau yev asymmetric scheme (6) wibeanayzed at the inner boundary point for the ASC-N scheme for space T i,k+1 6 = L α h,τ u(x i, t k+1 ) 1 h { δ x u(x i, t k )+ [ u(x i+1, t k ) u(x i, t k ) u(x i, t k+1 )+u(x i 1, t k+1 ) ]} = α u(x, t) τ α+1 u(x, t) + r t α 4 t α τ h u tx(x i, t k+1 ) τ 4h u ttx(x i, t k+1 )+ τh 1 u txxx(x i, t k+1 ) + τ 4 u txx(x i, t k+1 ) τ 8 u ttxx(x i, t k+1 ) h 1 u xxxx(x i, t k+1 )+O ( τ α + h )

12 Wu et a Advances in Difference Equations (018) 018:87 Page 1 of 18 The truncation error T i,k+1 8 of Sau yev asymmetric scheme (8) wibeanayzedatthe inner boundary point for the ASC-N scheme for space T i,k+1 8 = L α h,τ u(x i, t k+1 ) 1 h { δ x u(x i, t k )+ [ u(x i+1, t k+1 ) u(x i, t k+1 ) u(x i, t k )+u(x i 1, t k ) ]} = α u(x, t) τ α+1 u(x, t) + r t α 4 t α+1 1 τ h u tx(x i, t k+1 )+ τ 4h u ttx(x i, t k+1 ) τh 1 u txxx(x i, t k+1 ) + τ 4 u txx(x i, t k+1 ) τ 8 u ttxx(x i, t k+1 ) h 1 u xxxx(x i, t k+1 )+O ( τ α + h ) The first three terms τ h u tx(x i, t k+1 ), 4h u τh ttx(x i, t k+1 ), 1 u txxx(x i, t k+1 )oft i,k+1 6 and T i,k+1 8 have the same form but opposite sign So aternating Sau yev asymmetric schemes (6) and (8) can counteract partia error and the three terms wi disappear And the terms τ 4 u txx(x i, t k+1 )of(6)and(8) can disappear with τ α+1 u(x,t) So, aternating Sau yev asymmetric schemes (6)and(8), we aso can have O(τ α + h ) 4 t α+1 In the same way, aternating Sau yev asymmetric schemes (7)and(9)weasocanhave O(τ α + h ) The truncation error of ASC-N at the inner boundary points is O(τ α + h ) Therefore, we have the foowing theorem τ Theorem The truncation error of ASC-N scheme (1) for time fractiona sub-diffusion equation (1) is O(τ α + h ) 43 Convergence anaysis of the ASC-N scheme Define e k i = u(x i, t k ) u k i, ek =(e k 1, ek,,ek M 1 )T,and e k = max 1 i m 1 e k i Usinge0 =0, substitution into ASC-N scheme (1)eadsto (I + rg 1 )e k+1 =(c 0 I rg )e k + c 1 e k c k e 1 + R k+1, (13) (I + rg )e k+ =(c 0 I rg 1 )e k+1 + c 1 e k + + c k+1 e 1 + R k+, where R j = τ α Ɣ( α)(τ α + h ) C 1 (τ + τ α h ), j =1,,3,,C 1 is a constant Lemma 4 Suppose that e k is the soution of (13), then e k b 1 k 1 C(τ + τ α h ), where k =1,,,N and C is a positive constant Proof Lemma 4 can be proved using mathematica induction When k =1, e 1 = (I + rg ) 1 R 1 b 1 0 C 1( τ + τ α h ) When k =, e = (I + rg1 ) 1[ (c 0 I rg )e 1 + R ] b 1 1 (I + rg1 ) 1 ( (c0 I rg )E 1 ) + b 1 R b 1 1 C 1( τ + τ α h )

13 Wu et a Advances in Difference Equations (018) 018:87 Page 13 of 18 Suppose that e j b 1 j C 1 (τ + τ α h ), j =kthenweasohave e k+1 = (I + rg1 ) 1[ (c 0 I rg )e k + c 1 e k c k 1 E 1 + R k+1] b 1 k (I + rg1 ) 1 ( c ) 0 I rg + c c k 1 + b k R k+1 = b 1 k (I + rg1 ) 1 ( c ) 0 I rg + b 1 R k+1 b 1 k C 1( τ + τ α h ), e k+ = (I + rg ) 1[ (c 0 I rg 1 )e k+1 + c 1 e k + + c k e 1 + b k+1 R k+1] b 1 k+1 (I + rg ) 1 ( c ) 0 I rg 1 + c b k+1 R k+ = b 1 k+1 (I + rg ) 1 ( c ) 0 I rg 1 + b 1 R k+ b 1 k+1 C 1( τ + τ α h ) In summary, we have e j b 1 j C 1 (τ +τ α h ), j =1,,,N To sum up, for the ASC-N scheme, we have e k+1 b 1 k C 1(τ + τ α h ), k α τ α T,and b 1 k im k k = im k α k 1 α k (k +1) 1 α = im k1 α k (1 + 1 k )1 α 1 k 1 = im k (1 α)k = 1 1 (1 α) Thus e k Ck α( τ + τ α h ) = Ck α τ α( τ α + h ) CT α( τ α + h ), b where C = C 1 k 1 k α is a constant Hence, the foowing theorem is obtained Theorem 3 Let u k i be the approximate vaue of u(x i, t k ) computed by use of ASC-N scheme (1) Then there is a positive constant C such that u k i u(x i, t k ) C(τ α + h ), i =1,,,M 1,k =1,,,N 5 Numerica exampes In this section, we wi present numerica exampes to demonstrate that the ASC-N scheme is an effective numerica method for time fractiona sub-diffusion equation and verify the theoretica anaysis of the ASC-N scheme We consider the foowing time fractiona sub-diffusion equation [3]: α u(x, t) t α = u(x, t) x, 0 x, t >0 Boundary conditions: u(0, t) =u(, t) =0

14 Wu et a Advances in Difference Equations (018) 018:87 Page 14 of 18 Tabe 1 Comparison of exact soution and numerica soutions CPU times Exact soution s Soution of ID s Soution of ASC-N s Figure Dispacement of ASC-N scheme s numerica soution for various α Initia condition: x, 0 x 05, u(x,0)=f (x)= (4 x)/3, 05 x The function f (x) represents the temperature distribution in a bar generated by a point heat source kept in the point x =05forongenough At t =04,α = 05, we compare the exact soution and the numerica soutions of the impicit difference (ID) scheme and the ASC-N scheme For the exact soution, the series in equation (3) is truncated after 0 terms For numerica soution, we take M =40,N = 1000 The computed resuts and CPU times are isted in Tabe 1 In terms of the computationa accuracy, we can see that the numerica soution of the ASC-N and ID schemes is cose to the exact soution from Tabe 1 In terms of computationa efficiency, the computing times (CPU times) of the ASC-N and ID schemes have big advantage compared with the exact soution Therefore, the difference scheme is effective for soving the time fractiona sub-diffusion equation From Fig, we compare the response of the diffusion wave system with different vaues of α at t = 04 for the numerica soution of the ASC-N scheme The ASC-N scheme can be easiy appied to sove the time fractiona sub-diffusion equation Next we wi anayze the change cure of the reative error (RE) with time steps for the ASC-N scheme We take the exact soution u(x i, t k ) as the contro soution, and the numerica soution u j i of the ASC-N scheme as a perturbation soution The definition of RE is as foows: RE(i)= N u(x i, t j ) u i j u(x i, t j ) j=1

15 Wu et a Advances in Difference Equations (018) 018:87 Page 15 of 18 Figure 3 Change curve of RE with time steps Figure 4 Distribution of DTE at space points Then we wi anayze the distribution of the difference tota energy (DTE) at space grid points The definition of DTE is as foows: DTE(j)= 1 M i=1 ( u(xi, t j ) u i ) j From Fig 3,theREofASC-Nschemeisessthan075TheREisaittebiginthefirstfew steps and decreases rapidy with the time step Therefore, we can know that the ASC-N scheme of the time fractiona sub-diffusion equation is stabe The DTE of ASC-N scheme is between 0 and 005 from Fig 4 It can aso demonstrate that the ASC-N scheme is very cose to the exact soution The DTE appears to fuctuate near the grids 8, 16, 4, 3 And its maximum vaues appear near the grids 8 and 16 The grids 8, 16, 4, 3 are the inter boundary point of the ASC-N scheme The four kinds of Sau yev scheme are aternativey appied at the inter boundary point The cassic C-N scheme is appied at the inter point for the ASC-N scheme The truncation error of the C- N scheme is better than the four kinds of Sau yev scheme So the DTE of inter boundary point is itte bigger than the DTE of inter point The resut of Fig 4 is consistent with theoretica anaysis The next exampe wi be performed to iustrate the computationa efficiency and convergence rate of the ASC-N scheme Denote L h,τ = M ( u(xi, t M ) u M i i=1 ), L τ order1=og L,order=og τ/ Firsty, the numerica accuracy in tempora direction is verified Fixing the spatia step size h (h = 1/80),Tabe gives the computationa resuts with different tempora step sizes L h L h/

16 Wu et a Advances in Difference Equations (018) 018:87 Page 16 of 18 Tabe The numerica errors and convergence orders in tempora direction (h =1/80) α N ID scheme ASC-N scheme L h,τ order 1 CPU (times) L h,τ order 1 CPU (times) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Tabe 3 The numerica errors and convergence orders in spatia direction (α =05,τ = h ) M ID scheme ASC-N scheme L h,τ order CPU (times) L h,τ order CPU (times) e e e e e e e e e e using the same machine with α = 08, 05, 04, respectivey From Tabe, wecanseethat the numerica accuracy in tempora direction is order α for these three cases Secondy, we compute the numerica accuracy in spatia direction Take M = 0, 40, 80, 160, 30 for the ID and ASC-N schemes, and et τ = h (N = M /10)FromTabe3,wecan see that the numerica accuracy in spatia direction is second order for the ID and ASC-N schemes The same convergence orders of spatia and tempora direction are obtained for both of them In terms of computationa efficiency, the CPU times of the ASC-N have big advantage compared with the ID scheme From Tabes and 3, we know that the parae computing advantages of the ASC-N scheme wi be more obvious with the increase of the number of time ayers or space attice points Comparing with the ID scheme the CPU times of the ASC-N scheme can save near 90% Comprehensivey considering the computationa efficiency and computationa accuracy, the ASC-N scheme can be more effective to sove the time fractiona sub-diffusion equation When the ong time course is cacuated, the parae computing advantages of the ASC-N scheme wi be more evident 6 Concusion An aternating segment Crank Nicoson (ASC-N) difference scheme for soving the time fractiona sub-diffusion equation has been constructed The computing accuracy, stabiity, and convergence of the ASC-N scheme have been anayzed The resuts of the numerica experiments are consistent with the theoretica anaysis The computing time of the ASC- N scheme can save near 90% compared with the cassic impicit difference scheme

17 Wu et a Advances in Difference Equations (018) 018:87 Page 17 of 18 The ASC-N scheme has idea computing accuracy and computing efficiency The parae computing advantages of the ASC-N scheme wi be more obvious for the ong time course or the high dimensiona fractiona diffusion equation Funding The work was supported by the Nationa Natura Science Foundation of China (Grant No ) and the Fundamenta Research Funds for the Centra Universities (Grant No 018MS168) Competing interests The authors decare that they have no competing interests Authors contributions They a read and approved the fina version of the manuscript Pubisher s Note Springer Nature remains neutra with regard to jurisdictiona caims in pubished maps and institutiona affiiations Received: 9 Apri 018 Accepted: 8 August 018 References 1 Diethem, K: The Anaysis of Fraction Differentia Equations Springer, Berin (010) Uchaikin, VV: Fractiona Derivatives for Physicists and Engineers, vo II Springer, Berin (013) 3 Agrawa, OP: Soution for a fractiona diffusion-wave equation defined in a boundary domain J Noninear Dyn 9, (00) 4 Bao, JD: Introduction to Abnorma Statistica Dynamics Science Press, Beijing (01) (in Chinese) 5 Yang, XJ, Machado, JAT, Baeanu, D: Anomaous diffusion modes with genera fractiona derivatives within the kernes of the extended Mittag-Leffer type functions Rom Rep Phys 69, 151 (017) 6 Chen, W, Sun, HG, Li, XC: Fractiona Derivative Modeing of Mechanics and Engineering Probems Science Press, Beijing (010) (in Chinese) 7 Yang, XJ, Machado, JAT: A new fractiona operator of variabe order: appication in the description of anomaous diffusion Phys A, Stat Mech App 481, (017) 8 Guo, BL, Pu, XK, Huang, FH: Fractiona Partia Differentia Equations and Their Numerica Soutions Science Press, Beijing (011) (in Chinese) 9 Sun, ZZ, Gao, GH: Finite Difference Method for Fractiona Differentia Equations Science Press, Beijing (015) (in Chinese) 10 Liu, FW, Zhuang, PH, Liu, QX: Numerica Methods and Appications of Fractiona Partia Differentia Equations Science Press, Beijing (015) (in Chinese) 11 Zhuang, P, Liu, F: Impicit difference approximation for the time fractiona diffusion equation J App Math Comput (3), (006) 1 Tadjeran, C, Meerschaert, MM, Scheffer, HP: A second-order accurate numerica approximation for the fraction diffusion equation J Comput Phys 13(1), (006) 13 Chen, CM, Liu, FW, Turner, I, et a: Numerica schemes and mutivariate extrapoation of a two-dimensiona anomaous sub-diffusion equation Numer Agorithms 54(1),1 1 (010) 14 Zhang, P, Pu, H: The error anaysis of Crank Nicoson-type difference scheme for fractiona subdiffusion equation with spatiay variabe coefficient Bound Vaue Prob 017,15 (017) 15 Baeanu, D, Jajarmi, A, Asad, JH, Baszczyk, T: The motion of a bead siding on a wire in fractiona sense Acta Phys Po 131(6), (017) 16 Jajarmi, A, Hajipour, M, Mohammadzadeh, E, Baeanu, D: A new approach for the noninear fractiona optima contro probems with externa persistent disturbances J Frankin Inst 355, (018) 17 Hajipour, M, Jajarmi, A, Baeanu, D: An efficient non-standard finite difference scheme for a cass of fractiona chaotic systems J Comput Noninear Dyn 13, (018) 18 Wang, H, Treena, SB: A fast finite difference method for two-dimensiona space-fractiona diffusion equations SIAM J Sci Comput 34(5), A444 A458 (01) 19 Wang, H, Du, N: A fast finite difference method for three-dimensiona time-dependent space-fractiona diffusion equations and its efficient impementation J Comput Phys 53(45),50 63 (013) 0 Gao, GH, Sun, ZZ: A compact finite difference scheme for the fractiona sub-diffusion equations J Comput Phys 30(3), (011) 1 Gao, GH, Sun, ZZ: Two difference schemes for soving the one-dimensiona time distributed-order fractiona wave equations Numer Agorithms 74, (017) Yaseen, M, Abbas, M, Nazir, T, Baeanu, D: A finite difference scheme based on cubic trigonometric B-spines for a time fractiona diffusion-wave equation Adv Differ Equ 017, 74 (017) 3 Zaky, MA, Baeanu, D, Azaidy, JF, Hashemizadeh, E: Operationa matrix approach for soving the variabe-order noninear Gaiei invariant advection-diffusion equation Adv Differ Equ 018, 10 (018) 4 Zhang, BL, Yuan, GX, Liu, XP, Chen, J: Parae Finite Difference Methods for Partia Differentia Equations Science Press, Beijing (1994) (in Chinese) 5 Zhang, BL, Gu, TX, Mo, ZY: Principes and Methods of Numerica Parae Computation Nationa Defence Industry Press, Beijing (1999) (in Chinese) 6 Zhang, SC: Finite Difference Numerica Cacuation for Paraboic Equation with Boundary Condition Science Press, Beijing (010) (in Chinese)

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