Higher order numerical methods for solving fractional differential equations

Transcription

1 Noname manuscript No. will be inserted by the editor Higher order numerical methods for solving fractional differential equations Yubin Yan Kamal Pal Neville J Ford Received: date / Accepted: date Abstract In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order < α <. The order of convergence of the numerical method is Oh 3 α. Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adams-type method for a nonlinear fractional differential equation of any order α >. The order of convergence of the numerical method is Oh 3 for α and Oh 2α for < α for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results. Keywords Fractional differential equation finite difference method Caputo fractional derivative error estimates Mathematics Subject Classification 2 26A33 65L7 65L5 Y.Yan Department of Mathematics, University of Chester, CH 4BJ, UK Tel.: Fax: y.yan@chester.ac.uk K. Pal Department of Mathematics, University of Chester, CH 4BJ, UK Tel.: Fax: kamal-pal8@yahoo.co.uk N.J. Ford Department of Mathematics, University of Chester, CH 4BJ, UK Tel.: Fax: njford@chester.ac.uk

2 2 Yubin Yan et al. Introduction We consider numerical methods for solving the fractional differential equation C D α t yt f t,yt, < t < T,. y k y k, k,,2,..., α,.2 where the y k may be arbitrary real numbers and α >. Here C Dα t denotes the differential operator in the sense of Caputo denoted by C D α t yt Γ n α t t u n α y n udu, where n α is the smallest integer α. Existence and uniqueness of solutions for. -.2 have been studied, for example, in Podlubny [2, Diethelm [8, Diethelm and Ford [. Numerical methods for solving fractional differential equations have been considered by many authors and we mention here a few key contributions. Lubich [8 wrote the fractional differential equation in the form of an Abel-Volterra integral equation and used the convolution quadrature method to approximate the fractional integral and obtained approximate solutions of the fractional differential equation. Diethelm [7 wrote the fractional Riemann-Liouville derivative by using the Hadamard finite-part integral and approximated the integral by using a quadrature formula and obtained an implicit numerical algorithm for solving a linear fractional differential equation. Diethelm and Luchko [3 used the observation that a fractional differential equation has an exact solution, which can be expressed as a Mittag-Leffler type function. Then they used convolution quadrature and discretised operational calculus to produce an approximation to this Mittag-Leffler function. Blank [ applied a collocation method to approximate the fractional differential equation. Podlubny [2 used the Grünwald and Letnikov method to approximate the fractional derivative and defined an implicit finite difference method for solving.-.2 and proved that the order of convergence is Oh, where h is the step size. Gorenflo [7 introduced a second order Oh 2 difference method for solving.-.2, but the conditions to achieve the desired accuracy are restrictive. In [, the authors approximated the integral in.5 by using a piecewise linear interpolation polynomial and introduced a fractional Adams-type predictor-corrector method for solving.-.2, proving that the order of convergence of the numerical method is min{2,α} for < α 2 if C Dα t y C 2 [,T. Deng [3 modified the method in [ and introduced a new predictor-corrector method for solving.-.2 and the convergence order is proved to be min{2, 2α} for α,. In [22, the authors introduced a so-called Jacobi-predictor-corrector approach to solve.-.2 which is based on the polynomial interpolation and the Gauss-Lobatto quadrature with respect to some Jacobi-weight function and the computational cost is ON, N /h and any desired convergence order can be obtained. In [2, a high order numerical method for solving.-.2 is obtained where a quadratic interpolation polynomial was used to approximate the integral. Ford, Morgado and Rebelo recently see [6 used a nonpolynomial collocation method to achieve good convergence properties without

3 Higher order numerical methods for solving fractional differential equations 3 assuming any smoothness of the solution. There are also several works that are related to the fixed memory principle and the nested memory concept for solving.-.2, see, e.g., [5, [2, [3, [4, [5, etc. Two approaches are used in this paper to develop new numerical methods for solving.-.2. The first approach is based on a direct discretisation of the fractional differential operator in the form proposed by Diethelm [6. In [7, Diethelm considered the following linear fractional differential equation, with < α <, C D α t yt βyt f t, t,.3 y y,.4 where β <, f is a given function on the interval [,. Diethelm [7 used a firstdegree compound quadrature formula to approximate the Hadamard finite-part integral in the equivalent form of.3-.4 and defined a numerical method for solving.3-.4 and proved that the order of convergence of the numerical method is Oh 2 α, < α <. Here we approximate the Hadamard finite-part integral by using the second-degree compound quadrature formula and obtain an asymptotic expansion of the error for solving.3-.4, which implies that the order of convergence of the numerical method is Oh 3 α, < α <. Moreover, a high order finite difference method Oh 3 α, < α < 2 for approximating the Riemann-Liouville fractional derivative is given, which may be applied to construct high order numerical methods for solving time-space-fractional partial differential equations. Our second approach for solving the fractional differential equation.-.2 is based on the discretisation of the integral in the equivalent form of.-.2, see [. It is well-known that.-.2 is equivalent to the Volterra integral equation α yt ν y ν t ν ν! t t u α f u,yudu..5 Γ α We use piecewise quadratic interpolation polynomials to approximate the integral in.5 and introduce a high order fractional Adams method for solving.5 and prove that the order of convergence of the numerical method is min{3, 2α} for α,2 if C Dα t yt C 3 [,T. This method has higher convergence order than the method in [3. It is easier to implement our numerical algorithm compared with the method in [22 where the Jacobi-Gauss-Lobatto nodes must be calculated at each time level. Our method is simpler than the method in [2 in the sense that we are using a predictor-corrector method and therefore we do not need to solve the nonlinear system at each time level. The paper is organised as follows: in Section 2, we consider how we can extend the Diethelm method for solving a linear fractional differential equation. In Section 3, we consider the fractional Adams method for solving a nonlinear fractional differential equation. Finally in Section 4, we consider some numerical examples.

4 4 Yubin Yan et al. 2 Extending Diethelm s method In this section we will consider a higher order numerical method for solving It is well-known that.3-.4 is equivalent, with < α <, to the following problem: R D α t [yt y βyt f t, t, 2. where R Dα t yt denotes the Riemann-Liouville fractional derivative defined by, with < α <, R Dt α d t yt t u α yudτ. 2.2 Γ α dt The Riemann-Liouville fractional derivative R Dα t yt can be written as [7 R Dt α t yt t u α yudu, 2.3 Γ α where the integral denotes the Hadamard finite-part integral. In [7, Diethelm approximated the Hadamard finite-part integral in 2.3 by piecewise linear interpolation polynomials and defined a numerical method for solving 2.. In this section, we will approximate the Hadamard finite-part integral by using piecewise quadratic interpolation polynomials. Let M be a fixed positive integer and let t < t < t 2 < < t < t < < t 2M be a partition of [, and h the step size. At node t 2M, the equation 2. satisfies R D α t [yt y βyt f t, j,2,...,m, 2.4 and at node t 2M, the equation 2. satisfies R D α t [yt y βyt f t, j,,2,...,m. 2.5 Let us first consider the discretisation of 2.4. Note that R Dt α t yt t τ α yτdτ t α w α yt t wdw. Γ α Γ α 2.6 For every j, we replace gw yt t w in the integral in 2.6 by a piecewise quadratic interpolation polynomial with equispaced nodes, 2 j, 2,...,. We then have w α gwdw w α g 2 wdw R g, 2.7 where g 2 w is the piecewise quadratic interpolation polynomial of gw with equispaced nodes,, 2,..., and R g is the remainder term.

5 Higher order numerical methods for solving fractional differential equations 5 Lemma 2. Let < α <. We have where w α g 2 wdw α α α 2 α α l, k k α k, g, α α 2, for l, α2 2 α, for l, α 2 α α 2 F 2, for l 2, F k, for l 2k, k 2,3,..., j, 2 F 2 k F k, for l 2k, k 2,3,..., j, 2 F 2 j, for l, F k 2k 2k 2k α 2k 2 α α α 2 2k 2k 2k α 2k 2 α α α 2 2k α2 2k 2 α2 α α, F k 2k 22k 2k α 2k 2 α α α 2 2k 2 2k 2k α 2k 2 α α α 2 2k α2 2k 2 α2 α α, and F 2 k 2k 22k 2k α 2k 2 α α α 2 2k 2 2k 2k α 2k 2 α α α 2 2k α2 2k 2 α2 α α. Proof For fixed, let < 2 j < 2 < < be a partition of [,. Denote w l 2 l j, l,,2,...,. We then have, for k,2,..., j, g 2 w w w 2k w w 2k w 2k 2 w 2k w 2k 2 w 2k gw 2k 2 w w 2k 2 w w 2k w 2k w 2k 2 w 2k w 2k gw 2k w w 2k 2w w 2k w 2k w 2k 2 w 2k w 2k gw 2k, for w [w 2k 2,w 2k. 2.9

6 6 Yubin Yan et al. Let us now consider w α g 2 wdw [ w2 w4 w w α g 2 wdw. w 2 w 2 By the definition of the Hadamard finite-part integral [6, we obtain w2 w α g 2 wdw g 2w 2 α w2 α 2 α α α g 2 By using 2.9, we have w α[ w w2 g 2ydy dw w α g 2 w g 2 dw. 2. w2 g 2 ww α dw 2 α w2 α α g w α[ j2 w 2 2 w 2 w gw 2 w 2 w w 2 w g w 2 w w gw 2 dw 2 α α α g α α α 2 α α α 2 α gw 2 α α α α 2 α gw 2. Similarly we have, after a simple calculation, w2k2 α α α 2 α g 2 ww α dw w 2k 2 F kgw 2k 2 F kgw 2k 2 F 2kgw 2k, where F i k,i,,2 and k 2,3,..., j are defined as above. Together these estimates lead to 2.8 and the proof of Lemma 2. is complete. Next we consider the discretisation of 2.5. At the node t 2M, j,2,...,m we have R D α t yt t Γ α t t τ α yτdτ Γ α t τ α yτdτ t α Γ α w α yt t wdw. 2.

7 Higher order numerical methods for solving fractional differential equations Line for α /4 Line. for α /2 Line for α 3/4 α k, k Fig. 2. The weights α k, of the approximation of the fractional derivative with the different fractional order α For j,2,...,m, we replace gw yt t w in the integral in 2. by a piecewise quadratic interpolation polynomial with equispaced nodes,, 2,...,. We then have, for any smooth function gw, w α gwdw w α g 2 wdw R g, 2.2 where g 2 w is the piecewise quadratic interpolation polynomial of gw with the nodes,, 2,..., and R g is the remainder term. Similarly we can prove the following lemma. Lemma 2.2 Let < α <. We have w α g 2 wdw k k α k, g, 2.3 where α k, α k,, k,,2,..., and α k, are given in Lemma 2.. Remark 2. By the direct calculations, we can show that, with < α <, α, 2 α α 2 <, 2.4 α α α 2 α and α k, > for k >, k 2. For k 2, there exists α, such that α 2, for < α < α and α 2, α < α <. To see the weights α k,, k,,2,...,, in Figure 2., we plot α l, for l,, 2,..., with the different fractional order α /4, /2, 3/4. Now solutions of 2. satisfy, with j, 2,..., M, yt [ α, t2 α j Γ αβ t2 α j Γ α f t k α k, yt k y k α k, R g, 2.5

8 8 Yubin Yan et al. and, with j,2,...,m, yt [ α, t2 α j Γ αβ t2 α j Γ α f t y α k, R g t2 α j k t k t τ α yτdτ α k, yt k. 2.6 Here α,l t α l Γ αβ <, l,, which follow from 2.4 and Γ α <, β < and α, α,. Let y yt and y yt denote the approximations of the exact solutions yt and yt, respectively. Assume that the starting values y and y are given. We define the following numerical methods for solving 2., with j,2,...,m, y [ α, t2 α j Γ αβ t2 α j Γ α f t and, with j,2,...,m, y k [ α, t2 α j Γ αβ t2 α j Γ α f t y α k, t2 α j k t α k, y k y α k,, 2.7 t τ α yτdτ k k α k, y k. 2.8 Remark 2.2 In practice, we need to approximate t t τ α yτdτ. One way is to divide the integral [,t into small intervals t t2 tn t with stepsize h h. We first obtain y p yt p, p,2,...,n by using some numerical methods for solving fractional differential equation. Then we apply a quadrature formula to approximate the integral. We have the following asymptotic expansion theorem. Theorem 2. Let < α < and M be a positive integer. Let t < t < t 2 < < t < t < < t 2M be a partition of [, and h the step size. Let yt,yt,y and y be the exact solutions and the approximate solutions of , respectively. Assume that the function y C m2 [,, m 3. Further assume that we obtain the exact starting values y y and y yt. Then there exist coefficients c µ c µ α and c µ c µα such that the sequence {y l },l,,2,...,2m possesses an asymptotic expansion of the form m yt 2M y 2M that is, c µ 2M α µ m yt 2M y 2M c µ h µ α c µ2m 2µ o2m α m, for M, c µh 2µ oh m α, for h,

9 Higher order numerical methods for solving fractional differential equations 9 where is the integer satisfying 2 < m α < 2, and c µ and c µ are certain coefficients that depend on y. To prove Theorem 2., we need the following lemma for the asymptotic expansions for the remainder terms R g and R g in 2.7 and 2.2. Lemma 2.3 Let < α < and g C m2 [,, m 3. Let R g and R g be the remainder terms in 2.7 and 2.2, respectively. Then we have, with l 2,3,...,,,...,2M, m R l g d µ l α µ d µl 2µ ol α m, 2.9 where is the integer satisfying 2 < m α < 2, and d µ and d µ are certain coefficients that depend on g. Proof We follow the proof of Theorem.3 in [4 where the piecewise linear Lagrange interpolation polynomials are used. We first consider the case l for j,2,...,m. Let w < w < w 2 < < w,w k 2 k j,k,,2,..., be a partition of [,. Let h 2 j be the step size. Let g 2 w denote the piecewise quadratic Lagrange interpolation polynomial defined by 2.9 on [w 2l,w 2l2,l,,2,..., j. Then we have R g w α gwdw w α g 2 wdw j w2l2 l w 2l w α gw g 2 w dw j l w 2l 2h s α[ gw 2l 2h s 2 2s 2s 2gw 2l 2s2s 2gw 2l 2 2s2s gw 2l2 2h ds. By using the Taylor formula, we have gw 2l gw 2l 2h s g w 2l 2h s! g w 2l 2h s 3! gw 2l gw 2l 2h s g w 2l 2h s! g w 2l 2h s 3! 2h s g w 2l 2h s 2h s 2 2! 2h s 3 gm w 2l 2h s 2h s m R m m!, h 2h s g w 2l 2h s h 2h s 2 2! h 2h s 3 gm w 2l 2h s h 2h s m R 2 m m!, gw 2l2 gw 2l 2h s g w 2l 2h s! g w 2l 2h s 3! 2h 2h s g w 2l 2h s 2h 2h s 2 2! 2h 2h s 3 gm w 2l 2h s m! 2h 2h s m R 3 m, 2.2

10 Yubin Yan et al. where R i m,i,2,3 denote the remainder terms. Thus we obtain R g 2h j 2h l j l w 2l 2h s α[m 3 r h r3 g r3 w 2l 2h sπ r s ds w 2l 2h s α ε m sds I II, where ε m s depends on the remainder terms R i m,i,2,3 and π rs are some functions of s. For I, we have m 3 I r h r3 j [2h l w 2l 2h s α g r3 w 2l 2h s π r sds. Applying Theorem 3.2 in [2, we have, with w l w 2l, h 2h, 2h j l w 2l 2h s α g r3 w 2l 2h s j h w l h s α g r3 w l h s l m r 3 j a j sh j m r 2 a, j sh j α oh m r 2, j with some suitable functions a j s, j,,...,m r 3 and a, j s, j,,...,m r 2, with r,,2,...,m 3, m 3. Hence we have, noting that h, m 3 I r m 3 r m 3 r m 3 r m h 3r h 3r m r 3 j m r 2 j [ [ m r 3 a j sh j π rsds j m r 2 a, j sh j α π r sds j [ a j sπ r sds h 3r j oh m [ 3r j α a, j sπ r sds h oh m d µ α µ d µ 2µ o m, 2.2 where is the integer satisfying 2 < m α < 2, and d µ and d µ are certain coefficients that depend on g. We remark that the expansion does not contain

11 Higher order numerical methods for solving fractional differential equations any odd integer of powers of which follows from the argument in the proof of Theorem.3 in [4. For II, we have, following the argument of the proof for Theorem.3 in [4, II 2h j l w 2l 2h s α ε m sds o α m. Thus 2.9 holds for l. Next we consider the case l. Denote w 2l, we have R g j w2l2 l w 2l w α gwdw w α gw g 2 w dw w α g 2 wdw j l 2l, w 2l2 2 2l2 j and h w 2l 2h s α[ gw 2l 2h s 2 2s 2s 2gw 2l 2s2s 2gw 2l 2 2s2s gw 2l2 2h ds. Following the same argument as for the case l, we show that 2.9 also holds for l. Together these estimates complete the proof of Lemma 2.3. Proof Proof of Theorem 2. We follow the proof of Theorem 2. in [4 where the piecewise linear Lagrange interpolation polynomials are used to approximate the Hadamard finite-part integral. Let us fix t l c to be a constant for l,2,...2m. We will investigate the difference e l yt l y l, for l, with t l lh l 2M c, where h /2M is the step size. In other words, there is a constant c, independent of M, such that l c 2M, or M l/2c, and consequently, we see that if e l possesses an asymptotic expansion w. r. t. l, then e 2M possesses at the same time one w. r. t. M, and vice versa. We shall prove m e l yt l y l c µ 2M α µ c µ2m 2µ o2m α m, for l, 2.22 for some suitable constants c µ,c µ which we will determine later. Let us first consider the case l. Subtracting 2.7 from 2.5, we have, noting t h 2M c, e α, 2M α Γ αβ c α Γ αβ α, k [ k α k, yt k y k R g α k, e k R g. 2.23

12 2 Yubin Yan et al. Note that g yt t C m2 [,, m 3, we have, by Lemma 2.3, m R g d µ α µ d µ 2µ o α m, for j, 2.24 where is the integer satisfying 2 < m α < 2, and d µ and d µ are certain coefficients that depend on g. Note that /2M c, we can write 2.24 into m R g Choose d µ 2M α µ d µ2m 2µ o2m α m, for j c µ c α d Γ αβ /α µ, µ 3,4,...,m, 2.26 c µ c α d Γ αβ /α µ, µ,2,...,, 2.27 we will prove below that 2.22 holds for the coefficents c µ,c µ defined in 2.26 and We shall use mathematical induction to prove By assumption e,e, hence 2.22 holds for l, with the coefficients given by 2.26 and Let us now consider the case for l 2. We have, noting that α,l applying Lemma 2.3, 2 e 2 yt 2 y 2 c α Γ αβ α,2 k [ m c α Γ αβ α,2 c µ 2M α µ α k,2 e 2 k R 2 g 2 α α22mc α α α α2 and c µ2m 2µ o2m α m 2 k,2 α,2 R 2 g kα Thus we get, noting that 2 k α k,2 /α and α,2 [ 2 α α 22Mc α α α α 2 cα Γ αβ [ m α c µ 2M α µ m d µ 2M α µ 2 α α 22Mc α [ m α α α 2 e 2 c µ2m 2µ o2m α m d µ2m 2µ o2m α m c µ 2M α µ 2 α α22mc α α α α2, c µ2m 2µ o2m α m. 2.29

13 Higher order numerical methods for solving fractional differential equations 3 This shows that the sequence e 2 possesses an asymptotic expansion with respect to the powers of 2M, and it is easy to check that, by comparing with the coefficients of powers of 2M, see [4, m e 2 c µ 2M α µ c µ2m 2µ o2m α m. Assume that 2.22 holds for l,,...,. Then we have, following the same argument for 2.29, noting k α k, /α and applying Lemma 2.3, [ 2 α α 22Mc α α α α 2 cα Γ αβ [ m α c µ 2M α µ m d µ 2M α µ 2 α α 22Mc α [ m α α α 2 e c µ2m 2µ o2m α m d µ2m 2µ o2m α m c µ 2M α µ c µ2m 2µ o2m α m. 2.3 This shows that the sequence e possesses an asymptotic expansion with respect to the powers of 2M, and it is easy to check that, by comparing with the coefficients of powers of 2M, see [4, m e c µ 2M α µ c µ2m 2µ o2m α m. Hence 2.22 holds for l. Finally we assume that 2.22 holds for l,,...,. Then we have, following the same argument for 2.3, noting k α k, k α k, /α, α, α, and applying Lemma 2.3, [ 2 α α 22Mc α α α α 2 cα Γ αβ [ m α c µ 2M α µ m d µ 2M α µ 2 α α 22Mc α [ m α α α 2 e c µ2m 2µ o2m α m d µ2m 2µ o2m α m c µ 2M α µ c µ2m 2µ o2m α m. 2.3

14 4 Yubin Yan et al. This again shows that the sequence e possesses an asymptotic expansion with respect to the powers of 2M, and it is easy to check that, by comparing with the coefficients of powers of 2M, see [4, m e c µ 2M α µ c µ2m 2µ o2m α m. Hence 2.22 holds also for l. Together these estimates complete the proof of Applying l 2M in 2.22, we get The proof of Theorem 2. is now complete. Remark 2.3 In Theorem 2., we assume that y yt exactly. In practice y can be approximated by using the ideas described in Remark A higher order fractional Adams-type method In this section we will consider a higher order numerical method for solving.-.2. For simplicity we only consider the case where < α 2 since the case α > 2 does not seem to be of major practical interest [. To make sure that. -.2 has a unique solution, we assume that f u, satisfies a Lipschitz condition, i.e., there exists a constant L such that f u,x f u,y L x y, x,y R. 3. Let m be a positive integer and let t < t < t 2 < < t < t < < t 2m T be a partition of [,T and h the stepsize. Note that the system.-.2 is equivalent to.5. Let us now consider the discretisation of.5. At node t t, j,2,...,m, we have yt y y t! t t u α f u,yudu. 3.2 Γ α The second of the initial conditions only for < α < 2 of course. At node t t, j,2,...,m, we have t yt y y t t u α f u,yudu! Γ α y y t t t u α f u,yudu! Γ α t t u α f u,yudu Γ α t y y t t t u α f u,yudu! Γ α t t u α f u h,yu hdu 3.3 Γ α

15 Higher order numerical methods for solving fractional differential equations 5 We will replace f u,yu in the integral t t u α f u,yudu in 3.2 by the following piecewise quadratic polynomial, for t 2l u t 2l2, l,,2,... j with j,2,...,m, f u,yu P 2 u u t 2lu t 2l2 t 2l t 2l t 2l t 2l2 f t 2l,yt 2l u t 2l u t 2l2 t 2l t 2l t 2l t 2l2 f t 2l,yt 2l u t 2l u t 2l t 2l2 t 2l t 2l2 t 2l f t 2l2,yt 2l Similarly we will replace f u h, f u h in the integral t t u α f u h,yu hdu in 3.3 by the following piecewise quadratic polynomial, for t 2l u t 2l2, l,,2,... j, j,2,...,m, f u h,yu h Q 2 u u t 2lu t 2l2 t 2l t 2l t 2l t 2l2 f t 2l,yt 2l u t 2l u t 2l2 t 2l t 2l t 2l t 2l2 f t 2l2,yt 2l2 u t 2l u t 2l t 2l2 t 2l t 2l2 t 2l f t 2l3,yt 2l We then have the following lemma: Lemma 3. Let < α 2. We have t t u α P 2 udu k c k, f t k,yt k, 3.6 and t t u α Q 2 udu k c k, f t k,yt k, 3.7 where h α c k, αα α 2 2 F, if k, 2 F l 2 F 2l, if k 2l, l,2,..., j, F l, if k 2l, l,,2,..., j, 2 F 2 j, if k,

16 6 Yubin Yan et al. and F l αα 2l α2 2l 2 α2 αα 2 2 2l 2l 2 2l 2 α 2l α α α 2 2l 2l 2 2l α 2l 2 α, F l αα 2l α2 2l 2 α2 αα 2 2 2l 2l 2 2l 2 α 2l α α α 2 2l 2l 2 2l α 2l 2 α, F 2 l αα 2l α2 2l 2 α2 αα 2 2 2l 2l 2l 2 α 2l α α α 2 2l 2l 2l α 2l 2 α, Proof This follows from a simple calculation. We omit the proof here. We now define a fractional Adams numerical method for solving.5. Let y l yt l denote the approximation of yt l, l,,2,...,2m. The corrector formula is defined by y y y t! Γ α c k, f t k,y k c, f t,y P, j,2,...,m, k 3.8 and y y y t! t t u α f u,yudu Γ α Γ α c k, f t k,y k c, f t,y P k, j,2,...,m. 3.9 The remaining problem is the determination of the predictor formula required to calculate y P k. The idea is the same as the one described above: we replace f u,yu and f u h,yu h of the integrals on the right-hand sides of equations 3.2 and 3.3, respectively, by the piecewise linear interpolation polynomials and obtain y P y y t! Γ α a k, f t k,y k a, f t,y PP, j,2,...,m, k 3. and, with j,2,...,m, y P y y t! Γ α k a k, f t k,y k a, f t,y PP, 3.

17 Higher order numerical methods for solving fractional differential equations 7 where the weights are [ h α n α n αn α, if k, a k,n n k 2 αα α n k α 2n k α if k n,, if k n. Similarly, to calculate y PP k, we replace f u,yu and f u h,yu h in the integrals on the right-hand sides of equations 3.2 and 3.3, respectively, by the piecewise constants and obtain and y PP y y t! Γ α k b k, f t k,y k, j,2,...,m, 3.2 y PP y y t! where the weights [ Γ α k b k, f t k,y k, j,2,...,m. 3.3 b k,n hα n k α n k α. 3.4 α Our basic fractional Adams method, is completely described now by equations Remark 3. In practice, we need to approximate the integral in 3.9. We shall use the same ideas as in Remark 2.2. We have thus completed the description of our numerical algorithm. The remainder of the paper will be devoted to the error analysis of the scheme. We have the following theorem. Theorem 3. Let < α 2 and assume that C Dα t y C 3 [,T for some suitable chosen T. Let yt k and y k,k,,2,...,2m, t 2m T be the solutions of 3.2, 3.3, 3.8, 3.9, respectively. Assume that y y and y yt exactly. Then there exists a positive constant C > such that max yt k y k k 2m To prove this theorem, we need some lemmas. { C h 2α, if < α, C h 3, if < α 2. Lemma 3.2 Theorem 2.4 [ Let < α 2. If z C [,T, then there is a constant C α depending only on α such that t t u α zudu k b k, zt k C α tα h.

18 8 Yubin Yan et al. Lemma 3.3 Theorem 2.5 [ Let < α 2. If z C 2 [,T, then there is a constant C2 α depending only on α such that t t u α zudu k a k, zt k C2 α tα h2. Lemma 3.4 Let < α 2. If z C 3 [,T, then there is a constant C3 α depending only on α such that and t t Proof We have t t u α zudu t u α zudu k t I t u α zudu t t u α zudu k c k, zt k C3 α tα h c k, zt k C3 α tα h k t c k, zt k t u α P 2 udu, 3.7 where P 2 u is the piecewise quadratic interpolation polynomial of zu, defined by 3.4. Thus we have j t2k2 I t u α zu P 2 u du k t 2k j t2k2 k f 3! t u α z ξ t 2k 3! t2 2h 3 j t u α du C3 α tα h3, which shows 3.5. Similarly we can show 3.6. u t 2k u t 2k u t 2k2 du Lemma 3.5 [ Let < α 2 and m and n be positive integers. Let a k, and b k,,k,,2,...,, j,2,...,m be introduced in 3. and 3.2, respectively. Then we have and k a k,, b k,, k,,2,..., a k, α T α, k Further, there exist constants D α and Dα 2 b k, α T α. such that j,2,...,m. a, D α 2 hα, b, D α hα, j,2,...,m.

19 Higher order numerical methods for solving fractional differential equations 9 Lemma 3.6 Let < α 2. Let c k,,k,,2,...,, j,2,...,m be introduced in 3.8. Then we have and Further there exists a constant D α 3 Proof We first show that c k,, k,,2,...,, 3.8 k c k, α T α. 3.9 such that c, D α 3 hα, j,2,...,m. 3.2 F l, l,,2,..., j. 3.2 It is easy to show that F l 2 2l α2 α 2 2l α 2l 2 α2 α 2 2l 2 α, l,,2,..., j. Further, after some direct calculations, we can show that γ n 2 γ γ n γ n γ n 2 γ, n Z, γ >. By putting n 2l 2 and γ α, we get 3.2. Next we show It is easy to show that F l F 2 l, l,2,..., j F l F 2 l 2 2l 2 α2 α 2 2l 2 α 6α 2 2l α 2 2l 2 α2 α 2 2l 2 α. Further, after some direct calculations, we can show that 2n4 α2 α 2n4 α 6α 2n2 α 2n α2 α 2n α, n Z Hence 3.22 follows from Finally we can also show F and F 2 j. Therefore we get 3.8. Further 3.9 follows from k t c k, t u α du α tα α T α. For 3.2, we have, by Lemma 3., c, 2 F 2 j D α 3 hα, with the suitable constant D α 3. Together these estimates complete the proof of Lemma 3.6.

20 2 Yubin Yan et al. Proof Proof of Theorem 3. We first consider the case where < α 2. We will use mathematical induction. Note that, by assumptions, yt y, yt y. Assume that yt k y k C h 3, 3.24 is true for k,,2,...,, j,2,...,m. We must prove that this also holds for k. In fact, we have, with j,2,...,m, t t Γ α yt y t t t u α f u,yudu k c k, f t k,y k c, f t,t2 P j t2 t u α j f u,yudu t u α P 2 udu t u α P 2 udu t u α f u,yudu c k, f t k,yt k f t k,y k k I II III. k t c k, f t k,y k c, f t,t2 P j t u α P 2 udu c, f t,yt f t,t2 P j For I, we have, by Lemma 3.4, I t t2 t u α j f u,yudu t u α P 2 udu C3 α T α h 3. For II, we have, by Lemma 3.6 and the Lipschitz condition 3., II k c k, f t k,yt k f t k,y k α T α L max yt k y k. k k c k, L yt k y k For III, we have, by Lemma 3.6 and the Lipschitz condition, III c, f t,yt f t,y P D α 3 hα L yt y P. Now let us consider the bound for yt y P. We have

21 Higher order numerical methods for solving fractional differential equations 2 Γ α yt y P t t u α f u,yudu t k t2 t u α j f u,yudu a k, f t k,yt k f t k,y k k I 2 II 2 III 2. For I 2, we have, by Lemma 3.3, I 2 t a k, f t k,y k a, f t,t2 PP j t u α P udu a, f t,yt f t,t2 PP j t2 t u α j f u,yudu t u α P udu C2 α T α h 2. For II 2, we have, by Lemma 3.5 and the Lipschitz condition 3., II 2 k a k, f t k,yt k f t k,y k α T α L max yt k y k. k k a k, yt k y k For III 2, we have, by Lemma 3.5 and Lipschitz condition 3., III 2 a, f t,yt f t,y PP D α 2 hα L yt y PP. We also need to consider the bound for yt y PP. We have Γ α yt y PP t t u α f u,yudu k t t u α f u,yudu k b k, f t k,yt k f t k,y k b k, f t k,yt k I 3 II 3. k For I 3, we have, by Lemma 3.2, I 3 C α T α h. For II 3, we have, by Lemma 3.5 and Lipschitz condition 3., b k, f t k,y k II 3 α T α L max yt k y k. k

22 22 Yubin Yan et al. Together these estimates, we have Γ α yt y C3 α T α h 3 α T α L max yt k y k k D α 3 hα L C2 α Γ α T α h 2 α T α L max yt k y k k D α 2 hα L [ C α Γ α T α h α T α L max yt k y k k [C 3 α T α h 3 Dα 3 LCα 2 T α h 2α Dα 3 Dα 2 L2 C αt α h 2α Γ α Γ α 2 [ α T α L Dα 3 L2 α T α h α Dα 3 Dα 2 α T α L 3 h 2α Γ α Γ α 2 max yt k y k. k By mathematical induction 3.24, we have [ C α yt y 3 T α h 3 Γ α [ Dα 3 LCα 2 T α h 2α Γ α 2 Dα 3 Dα 2 L2 C α T α h 2α Γ α 3 Γ α T α L Dα 3 L2 α T α h α Γ α Γ α Dα 3 Dα 2 α T α L 3 h 2α Γ α Γ α 2 C h We first choose T sufficiently small, see Lemma 3. in [ such that Γ α T α L 2. Then we fix this value for T and make the sum of the remaining terms in the right hand side of 3.25 smaller than C 2 h 3 for sufficiently small h by choosing C sufficiently large. Hence we obtain, for < α 2, yt y C 2 h3 C 2 h3 C h We also need to show that if 3.24 is true for k,,2,..., with j,2,...,m, then it also holds for k. In fact, we have, with j,2,...,m, t Γ α yt y t u α f u,yudu t t u α f u,yudu t t u α f u,yudu t t t k c k, f t k,y k c, f t,y P k c k, f t k,y k c, f t,y P t2 t u α j f u,yudu t u α Q 2 udu t c k, f t k,yt k f t k,y k c, k Using the same arguments as proving 3.26, we can show yt y C h 3, j,2,...,m. f t,yt f t,y P

23 Higher order numerical methods for solving fractional differential equations 23 Hence we complete the proof for the case where < α 2. Next we consider the case where < α. Note that, by the assumptions, yt y, yt y. Assume that yt k y k C h 2α, 3.27 for k,,2,...,, j,2,...,m. We must prove that this also holds for k. In fact, by using the same arguments as showing 3.26, we get [ C α yt y 3 T α h 3 Γ α [ Dα 3 LCα 2 T α h 2α Γ α 2 Dα 3 Dα 2 L2 C α T α h 2α Γ α 3 Γ α T α L Dα 3 L2 α T α h α Γ α Γ α Dα 3 Dα 2 α T α L 3 h 2α Γ α Γ α 2 C h 2α As in the case for < α 2, we first choose T sufficiently small such that Γ α T α L 2. Then we fix this value for T and make the sum of the remaining terms in the right had side of 3.28 smaller than C 2 h 2α for sufficiently small h by choosing C sufficiently large. Hence we obtain, for < α, yt y C 2 h2α C 2 h2α C h 2α Similarly we can show that if 3.27 is true for k,,2,..., with j,2,...,m, then it is also true for k. Together these estimates complete the proof of Theorem 3.. Remark 3.2 In Theorem 3., we require that C Dα t yt is in C 3 [,T which implies that the solution has the form, see Page 46 in [, yt ct α smoother terms. Since yt has low regularity at t, it may be necessary to use some high order methods and a small step size to approximate the solutions near t and to use a big step size to calculate the approximate solutions at other nodes to get the required accuracy. We will investigate this interesting issue in our future work. 4 Numerical simulations Example 4. [7 Consider C D α t yt βyt f t, t [,, 4. y y, 4.2 where y, < α <, β and f t t 2 2t 2 α /Γ 3 αt 3 3!t 3 α /Γ 4 α. The exact solution is yt t 2 t 3.

24 24 Yubin Yan et al. The main purpose is to check the order of convergence of the numerical method with respect to the fractional order α. For various choices of α,, we computed the errors at t. We choose the step size h /5 2 l,l,2,...,7, i.e, we divided the interval [, into n /h small intervals with nodes t < t < < t n. Then we compute the error et n yt n y n. By Theorem 2.22, we have et n yt n y n Ch 3 α, 4.3 To observe the order of convergence we shall compute the error et n at t n for the different values of h. Denote e h t n the error at t n for the step size h. Let h l h /5 2 l for a fixed l,2,...,7. We then have e hl t n e hl t n Ch3 α l 2 3 α, Ch 3 α l which implies that the order of convergence satisfies 3 α log 2 ehl t n e hl t n. In Table 4., we compute the orders of convergnce for the different values of α. The numerical results are consistent with the theoretical results. n ERC α.25 ERC α.5 ERC α Table 4. Numerical results at t for β and f t t 2 2t 2 α/γ 3 α t 3 3!t 3 α /Γ 4 α In Figure 4., we plot the order of the convergence. We have from 4.3 log 2 et n log 2 C 3 αlog 2 h. Let y log 2 et n and x log 2 h. In Figure 4., we plot the function y yx for the different values of x log 2 h where h /5 2 l,l,2,...,7. To observe the order of convergence, we also plot the straight line y 3 αx, where α.75. We see that these two lines are exactly parallel which means that the order of convergence of the numerical method is Oh 3 α. Example 4.2 [ Our second example deals with the nonlinear fractional differential equation where the unknown solution y has a smooth derivative of order α. Specifically we shall look at the equation [ C Dt α yt 432 Γ 5 α/2 Γ 9 α t8 α 3 Γ 5 α/2 t4 α/ Γ α 2 tα/2 t 4 3 [yt 3/2.

25 Higher order numerical methods for solving fractional differential equations 25 log 2 et log 2 h Fig. 4. The experimentally determined orders of convergence EOC at t in Example 4.2 with α.75 The initial conditions were chosen to be homogeneous y, y ; the latter only in the case < α < 2. This equation has been chosen because it exhibits a difficult nonlinear and nonsmooth right-hand side, and yet we are able to find its exact solution, thus allowing us to compare the numerical results for this nontrivial case to the exact results. Indeed, the exact solution of this initial value problem is and hence yt t 8 3t 4α/2 9 4 tα, C Dt α yt 432 Γ 5 α/2 Γ 9 α t8 α 3 Γ 5 α/2 t4 α/2 9 Γ α, 4 which implies C Dα t y C 3 [,T for arbitrary T > and < α 2, and thus the conditions of Theorem 3. are fulfilled. For various choices of α,2, we compute the errors at t n. We choose the step size h /5 2 l,l,2,...,7, i.e, we divided the interval [, into n /h small intervals with nodes t < t < < t n. Then we compute the error et n yt n y n. By Theorem 3., we have max yt k y k k 2m { C h 2α, if < α, C h 3, if < α 2. In Table 4.2, we compute the orders of convergence for different values of α. We observe that the order of convergence is Oh 2α for < α. But the observed order of convergence is higher than 3 for < α 2 in this example. For example, when α.25, the experimentally determined order is 3.5. When α.75, the experimentally determined order is almost 4. In Figure 4.2, we plot the order of convergence. We have log 2 et n log 2 C 2αlog 2 h. Let y log 2 et n and x log 2 h. In Figure 4.2, we plot the function y yx for the different values of x log 2 h where h /5 2 l,l,2,...,7. To observe the order of convergence, we also plot the straight line y 2αx, where α

26 26 Yubin Yan et al log 2 et log h 2 Fig. 4.2 The experimentally determined orders of convergence EOC at t in Example 4.2 with α.35 5 log 2 et log 2 h Fig. 4.3 The experimentally determined orders of convergence EOC at t in Example 4.2 with α We see that these two lines are almost parallel which confirms that the order of convergence of the numerical method is Oh 2α. In Figure 4.3, we will plot the order of convergence for α.25. We plot the function y yx for the different values of x log 2 h where h /5 2 l,l,2,...,7. To observe the order of convergence, we also plot the straight line y 3x. We observe that the order of convergence is higher than 3 almost 2α. n ERC α.35 ERC α.25 ERC α Table 4.2 Numerical results at t in Example 4.2 with the different fractional order α Acknowledgements The authors wish to thank Kai Diethelm for reading the first version of this paper and making useful suggestions. The authors also wish to thank the anonymous reviewers of this paper for their careful reading and useful comments.

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