Spectral approximation methods and error estimates for Caputo

Spectral approximation methods and error estimates for Caputo fractional derivative with applications to initial-value problems Beiping Duan∗, Zhoushun Zheng and Wen Cao School of Mathematics and statistics, Central South University, Changsha 410083, P.R. China. Abstract In this paper, we revisit two spectral approximations, including truncated approximation and interpolation for Caputo fractional derivative. The two approaches have been studied to approximate Riemann-Liouville(R-L) fractional derivative by Chen et al. and Zayernouri et al. respectively in their most recent work. For truncated approximation the reconsideration partly arises from the difference between fractional derivative in R-L sense and Caputo sense: Caputo fractional derivative requires higher regularity of the unknown than R-L version. Another reason for the reconsideration is that we distinguish the differential order of the unknown with the index of Jacobi polynomials, which is not presented in the previous work. Also we provide a way to choose the index when facing multi-order problems. By using generalized Hardy’s inequality, the gap between the weighted Sobolev space involving Caputo fractional derivative and the classical weighted space is bridged, then the optimal projection error is derived in the non-uniformly Jacobi weighted Sobolev space and the maximum absolute error is presented as well. For the interpolation, analysis of interpolation error was not given in their work. In this paper we build the interpolation error in non-uniformly Jacobi weighted Sobolev space by constructing fractional inverse inequality. With combining collocation method, the approximation technique is applied to solve fractional initial-value problems(FIVPs). Numerical examples are also provided to illustrate the effectiveness of this algorithm. Mathematics Subject Classification(2010): 26A33 (main), 65M70, 39A70

KeywordFractional derivative; Jacobi polynomials; Spectral method; Fractional ODEs

1

Introduction

The history of fractional calculus is almost as long as the integer ones, which goes back to the Leibniz’s note in his list to L’Hospital, dated 30 September 1695. After a long exploration of many mathematicians(including Euler, Laplace, Fourier, e.t., see [1] for the history), presentday definitions of fractional order are given. Compared to R-L version, fractional derivative in Caputo sense is more likely to be used in physical problems for its advantage in dealing with initial condition [2]. In fact the Caputo fractional derivative is an integro-differential operator, defined by a convolution of a weak singular kernel and classical derivative of a given function, which can be used to model non-Markovian behavior of spatial or temporal processes. So fractional calculus has attracted considerable interest of researchers in different fields during past decades. Furthermore, ∗ Email:

beiping [email protected]

1

models established by fractional differential equations(FDEs) have been successfully applied to several branches of science, such as physics, chemistry and engineering(see e.g. [3, 4] and the references therein). Numerical methods for solving FDEs have been attracting more and more interest in recent years. The key of these methods is how to approximate the fractional derivative. In [5], fractional centered difference was defined and was proved it could represent for Riesz fractional derivative. Furthermore, in [6] this scheme was proved to be of second order accuracy for Riesz fractional derivative. Numerical algorithms were established in [7] to calculate the fractional integral and the Caputo derivative. Li et. al. [8] developed numerical method based on the piecewise polynomial interpolation to approximate the fractional integral and the Caputo derivative, and to solve FDEs. An automatic quadrature approach based on the Chebyshev polynomials was proposed in [9] for approximating the Caputo derivative. In [10], Li derived recursive formulas to compute fractional order integrals and fractional order derivatives of the Legendre, Chebyshev and Jacobi polynomials. And with using collocation method, multi-term problems could be solved. In [11], a point interpolation method (PIM) was applied to approximate R-L fractional derivative, after which space fractional diffusion equation was solved numerically. Besides, some other methods such as the L1, L2 and L2C schemes, are utilized frequently to discretize fractional order derivative in the procedure of solving FDEs(e.g. [12, 13, 14, 15, 16]). Up to now, finite difference is still the main approach to approximate fractional derivatives. However, most of these schemes’ convergence order is no more than 2. Kumar and Agrawal[17] proposed a block-by-block method for a class of FIVPs. Later Huang et. al.[18] demonstrated that the convergence order of this method is at least 3. A higher-order difference scheme was proposed by Cao and Xu in [19], which was proved to be of (3 + α)’th order for α ≤ 1 and of 4’th order for α > 1. Spectral methods have been applied to approximate fractional derivatives. However, for the difficulty in computing the fractional derivative of general Jacobi polynomials with indexes γ, δ 6= 0 analytically, most researchers prefer the Lengendre polynomials. In fact, the following two formulas are always used with Galerkin method to solve fractional diffusion or fractional reaction-diffusion equation(e.g. in [20, 21]), L µ −1 Dx Ln (x) L µ x D1 Ln (x)

=

Γ(n + 1) (1 + x)−µ Jn(µ,−µ) (x), Γ(n − µ + 1)

x ∈ [−1, 1]

(1.1)

=

Γ(n + 1) (1 − x)−µ Jn(−µ,µ) (x), Γ(n − µ + 1)

x ∈ [−1, 1]

(1.2)

(µ,−µ)

where Ln (x) and Jn (x) are the Lengendre polynomial and Jacobi polynomial of n’th degree µ L µ respectively, L −1 Dx and x D1 are the left and right R-L fractional derivatives respectively. For general Jacobi polynomials, thanks to Bateman fractional integral formula, the following formulas hold   Γ(n + δ + 1) δ (γ,δ) L µ (x) = (1 + x)δ−µ J (γ+µ,δ−µ) (x) (γ ∈ R, δ > −1) (1.3) −1 Dx (1 + x) Jn Γ(n + δ − µ + 1) n L µ x D1



 γ (1 − x) Jn(γ,δ) (x) = (1 − x)γ−µ

Γ(n + γ + 1) J (γ−µ,δ+µ) (x) Γ(n + γ − µ + 1) n

(γ > −1, δ ∈ R)

(1.4)

For further details, we refer interested readers to [33]. Zayernouri and Karniadakis studied the δ (γ,δ) function (1 − x) Jn (x)(n = 0, 1, · · · ) in [31] and pointed out it is a family of eigenfunctions of a fractional Sturm-Liouville operator. Most recently Chen et. al.[33] referred to this kind of functions as generalized Jacobi functions(GJFs) and elaborated its properties. Furthermore, spec-

2

tral approximation results for these GIFs were derived in the non-uniformly Jacobi weighted space involving fractional derivatives(we let Bpγ,−δ (I) denote the space) and Petrov-Galerkin spectral method was constructed for a class of prototypical FDEs with convergence analysis. In fact, one key property the authors utilized in [33] to obtain the estimate results in Bpγ,−δ (I) is: for R-L fractional derivative operator, it holds L δ+l −1 Dx

L δ+l x D1

δ = DlL −1 Dx ,

δ = (−D)lL x D1 .

However, this property is no longer valid for fractional derivative in Caputo sense. In this paper, we will provide two ways to approximate Caputo derivative: truncated approximation and inter(γ,δ) (γ,δ) polation. The basis we use are with the form (x − a)δ J˜n , where J˜n is the shifted Jacobi polynomial defined on [a, b]. The main theme of this paper is to discuss the approximation results p for δ = m ∈ N+ in the classical Sobolev space Bγ,−m (I)(also referred to as non-uniformly Jacobi weighted space) and give the optimal estimate. What makes it different from [33] is that the estimate holds for any α ≤ m. By establishing fractional inverse inequality, the optimal estimate for interpolation is also derived. In fact, by a modified technique, it is demonstrated that the convergence speed of the two approaches in L2 is not less than O(N α−p ) as well. Furthermore we will also prove the truncated approximation can also lead to spectral convergence with respect to the norm k·k∞ for smooth data. Corresponding to the results for δ = m, we also present the estimate of truncated approximation for δ ∈ (m − 1, m), which is actually a generalization of [33]. In addition, we will give the error estimate for the fractional interpolation proposed in [32] at the end of Section 3. This paper remaining is organized as follows: In section 2 we recall basic definitions and properties about fractional calculus and Jacobi polynomials for preparation. In section 3 we present two spectral approximation approaches for Caputo derivative: truncated approximation and interpolation and derive the error estimate results. In section 4 some numerical tests are carried out to check the theoretical estimates. With combining collocation method, the approximation technique is applied to solve fractional initial value problems. Numerical results also show the effectiveness of the method we give.

2 2.1

Fractional derivatives and Jacobi polynomials Fractional integral and derivative

There are several definitions for a fractional derivative of order α, α ∈ R+ , here we present the frequently-used ones: R-L version and Caputo version. Definition 2.1 ([22]). Let f (x) be a function defined on the interval [a, b], then the fractional integration a Ixα f (x) is defined as Z x 1 α−1 α (x − ξ) f (ξ)dξ, (2.1) I f (x) = a x Γ(α) a α And the fractional Riemann-Liouville derivative L a Dx f (x) and Caputo derivative isted, are defined as L α a Dx f (x)

=

dn n−α f (x), aI dxn x

C α a Dx f (x)

where n is a positive integer and α ∈ [n − 1, n).

3

= a Ixn−α

dn f (x), dxn

C α a Dx f (x),

if ex-

(2.2)

Lemma 2.1 ([22]). For α > 0 and x ∈ (a, b] , we have L α α a Dx a Ix f (x)

= f (x).

(2.3)

Lemma 2.2 ([22]). Assume f (x) ∈ C n [a, b], then the fractional Riemann-Liouville derivative and Caputo derivative can be correlated by L α a Dx f (x)

α =C a Dx f (x) +

n−1 X

f (i) (a)

i=0

Γ(i + 1) i−α (x − a) Γ(i + 1 − α)

α ∈ [n − 1, n).

(2.4)

So if f (i) (a) = 0 for i = 0, 1, · · · , n − 1, then L α a Dx f (x)

α =C a Dx f (x).

Lemma 2.3. Let α ∈ [n−1, n), n ∈ N+ , x = c1 t+c2 (c1 > 0) and t ∈ (t1 , t2 ]. Denote a = c1 t1 +c2 , b = c1 t2 + c2 , then the following formula holds α t1 Dt f (c1 t

α + c2 ) = cα 1 a Dx f (x)

(2.5)

where t1 Dtα represents Riemann-Liouville fractional derivative operator or Caputo fractional derivative operator, and f is a function defined on the interval [a, b]. Proof. Consider the fractional integral of order β(β > 0), of the function f Z x 1 β−1 β (x − ξ) f (ξ)dξ. I f (x) = a x Γ(β) a Let ξ = c1 τ + c2 then we have β a Ix f (x) =

=

1 Γ(β) cβ1 Γ(β)

Z

t

c1 (c1 (t − τ )) t1 Z t

(t − τ )

β−1

β−1

f (c1 τ + c2 )dτ

f (c1 τ + c2 )dτ = cβ1 · t1 Itβ f (c1 t + c2 )

t1

Combining definition 2.1, we have L α a Dx f (x)

=

dn dxn

n−α f (x) a Ix




=

dn cn1 dtn

n−α cn−α f (c1 t + c2 ) t1 I t 1




1L α t1 Dt f (c1 t + c2 ), cα 1  n   n  d d n−α n−α C α n−α f (x) = c1 t1 It f (c1 t + c2 ) a Dx f (x) = a Ix dxn cn1 dtn 1 = αC Dα f (c1 t + c2 ). c1 t1 t =

Hence Eq.(2.5) holds.

4

2.2

Jacobi polynomials

 (γ,δ) ∞ The well-known Jacobi polynomials Jn (t) 0 , which are mutually orthogonal with the weight ω γ,δ (x) = (1 − x)γ (1 + x)δ (γ, δ > −1) are defined on the interval [−1, 1] and are usually computed via the following three-term recurrence formula: (γ,δ)

J0 and (γ,δ)

Ji

(t) =

(γ,δ)

(t) = 1,

J1

(t) =

γ+δ+2 γ−δ t+ , 2 2

(γ + δ + 2i − 1){γ 2 − δ 2 + t(γ + δ + 2i)(γ + δ + 2i − 2)} (γ,δ) Ji−1 (t) 2i(γ + δ + i)(γ + δ + 2i − 2) (γ + i − 1)(δ + i − 1)(γ + δ + 2i) (γ,δ) − Ji−2 (t), i = 2, 3, · · · i(γ + δ + i)(γ + δ + 2i − 2)

In order to approximate functions defined on the interval [a, b], we defined the so-called shifted (γ,δ) Jacobi polynomials J˜i (x) as (γ,δ) (γ,δ) J˜i (x) = Ji (t)

(i = 0, 1, · · · )

where t = 2 x−a b−a − 1, with the weight function ω γ,δ (x) = (b − x)γ (x − a)δ (γ, δ > −1). And

b

Z



(2.6)

2 (γ,δ) J˜i (x) ω γ,δ (x)dx = hγ,δ i ,

(2.7)

a

where hγ,δ = i

(b − a)γ+δ+1 Γ(i + γ + 1)Γ(i + δ + 1) . (2i + γ + δ + 1)Γ(i + 1)Γ(i + γ + δ + 1)

(2.8)

(γ,δ) The k’th order derivative of J˜n (x) satisfies (γ,δ) ∂xk J˜i (x) =

Γ(γ + δ + i + k + 1) ˜(γ+k,δ+k) J (x), (b − a)k Γ(γ + δ + i + 1) i−k

i≥k

(2.9)

which are mutually orthogonal with the weight function ω γ+k,δ+k (x). Furthermore, b

Z



(γ,δ)

∂xk J˜i

(x)

a

where

2

ω γ+k,δ+k (x)dx = hγ,δ k,i ,

(2.10)

γ+δ+1

(b − a) Γ(i + γ + 1)Γ(i + δ + 1)Γ(i + k + γ + δ + 1) . (2i + γ + δ + 1)Γ(i − k + 1)Γ2 (i + γ + δ + 1)

hγ,δ k,i =

(2.11)

(γ,δ) δ (γ,δ) Theorem 2.1. Denote P¯n (x) = (x − a) J˜n (x). Then the Riemann-Liouville fractional (γ,δ) derivative of P¯n (x) of order α is L α a Dx



 P¯n(γ,δ) (x) =

Γ(n + δ + 1) ¯ (γ+α,δ−α) P (x). Γ(n + δ − α + 1) n

where γ ∈ R, δ > α − 1. 5

(2.12)

Proof. Analogous formula can be found, e.g. (2.36) in [33], which is derived based on the following formula(see [26]) with using the definitions of fractional calculus (γ−α,δ+α)

δ+α

(1 + t)

Jn

(t)

Γ(δ + α + 1) = (δ+α,γ−α) Γ(δ + 1)Γ(α) Jn (1)

Z

(γ,δ) (τ ) δ Jn (t (δ,γ) Jn (1)

t

(1 + τ ) −1

α−1

− τ)

dτ

(2.13)

(γ,δ)

where Jn (t) represents Jacobi polynomial of degree n with corresponding weight. Noticing the interval here is [a, b], with using the mapping t = 2 x−a b−a − 1 from t ∈ [−1, 1] onto x ∈ [a, b] and lemma 2.3, Eq.(2.12) can be verified.  (γ,δ) ∞ Remark: P¯n (x) n=0 are referred to as generalized Jacobi functions in [33], which are eigenfunctions of defined fractional Sturm-Liouville problems. In fact take γ = δ = 0 then Eq.(2.12) reduces to Eq.(1.1). Furthermore, corresponding to Eq.(2.12), the following formula holds L α x Db



 ¯ (γ,δ) (x) = Q n

Γ(n + γ + 1) ¯ (γ−α,δ+α) Q (x), Γ(n + γ − α + 1) n

(2.14)

α ¯ (γ,δ) (x) = (b − x)γ J˜n(γ,δ) (x) where L x Db denotes right R-L fractional derivative operator (see [22]), Qn α and γ > α − 1, δ ∈ R. In fact Eq.(2.12) and Eq.(2.14) still hold for α < 0 where L x Db should be −α understood as R-L fractional integral operator x Ib .

3

Approximation schemes for Caputo fractional derivative with error estimates

Before we present the approximation schemes and give the error estimates, we first recall some useful definitions and lemmas. Definition 3.1. Suppose I = [a, b] and L2α,β (I) is the space of all functions defined on the interval I with corresponding norm kvkL2 (I) < ∞. The inner product and norm are defined as follows α,β

Z (v1 , v2 )L2α,β (I) =

v1 (x)v2 (x)ω α,β (x)dx,

1/2

kvkL2

α,β (I)

I

= (v, v)L2

α,β (I)

.

n And let Bα,β (I) denote the non-uniformly (or anisotropic) Jacobi-weighted Sobolev space:

 n Bα,β (I) : v : ∂xk v ∈ L2α+k,β+k (I), 0 ≤ k ≤ n ,

n ∈ N,

equipped with the inner product and norm (v1 , v2 )B n

α,β (I)

=

n X

∂xk v1 , ∂xk v2

k=0


L2α+k,β+k (I)

,

kvkB n

α,β (I)

1/2

= (v, v)B n

α,β (I)

.

α,β Definition 3.2. Let PN (I) be the space of polynomials with degree at most N on I and πN be α,β the orthogonal projection from L2α,β (I) onto PN (I). It means that for any f1 ∈ L2α,β (I) , πN f1 belongs to PN (I) and satisfies

∀f2 ∈ PN (I),

α,β (f1 − πN f1 , f2 )L2α,β (I) = 0.

6

Lemma 3.1 (Generalized Hardy’s Inequality [28]). For all measurable functions v ≥ 0, the following generalized Hardy’s inequality !1/q

b

Z

q

|(Kv) (x)| w1 (x)dx

Z

!1/p

b

p

|v(x)| w2 (x)dx

≤λ

(3.1)

a

a

holds if and only if Z a −1 and j = 0, 1, · · · , s − 1 we have Rn (b) = 0. Integrating by parts and noticing that z (j) (a) = 0 for j = 0, 1, · · · , m − 1, with Eq.(2.14) we have o  Rb n s−α ¯ (γ+α,δ−α) 1 s γ+α,α−δ bn = hγ+α,δ−α z (−∂) I P (x)ω dx n x x b a n  Rb L α γ+α ˜(γ+α,δ−α) 1 (3.19) = hγ+α,δ−α a z x Db (b − x) Jn (x) dx n R b (γ,δ) γ Γ(n+γ+α+1) 1 = γ+α,δ−α z (b − x) J˜n (x)dx hn

Γ(n+γ+1)

a

s−α α s where L is the right R-L fractional derivative operator. x Db = (−∂)x x Ib

10

Looking back to Eq.(2.8), the definition of hγ+α,δ−α , a simple manipulation leads to n Γ(n + γ + α + 1) Γ(n + δ + 1) 1 = γ,δ Γ(n + γ + 1) Γ(n + δ − α + 1) hn

1 hγ+α,δ−α n

(3.20)

As (b − x)γ J˜n(γ,δ) (x) = P¯n(γ,δ) (x)ω γ,−δ (x), taking a look at Eq.(3.7) we arrive at the relation between bn and an . The theorem above provides an important information about the relation between the Caγ,δ γ,δ puto fractional derivative operator Dα and the projection operator π ˆN , that is Dα π ˆN z = γ+α,δ−α α π ˆN D z. Now let us give the following two theorems respectively for δ = m and δ ∈ (m − 1, m). p Theorem 3.3. Suppose z ∈ Bγ,−m (I) where γ > −1, m, p ∈ N. Then for α ∈ [0, m], there exists a constant C independent of N such that if m ≤ p ≤ N + 1 + m γ,m kDα (z − π ˆN z)kL2

γ+α,α−m

≤ CN α−p k∂xp zkL2

(3.21)

γ+p,p−m

and if p > N + 1 + m γ,m kDα (z − π ˆN z)kL2

≤C

γ+α,α−m

Proof. It is known that

¯ (γ+α,m−α) 2 (x) 2

Pn

Lγ+α,α−m

N α−m−1/2 −N e ∂xm+1+N z L2 . γ+1+m+N,N +1 N!

2

= J˜n(γ+α,m−α) (x) 2

Lγ+α,m−α

(3.22)

= hnγ+α,m−α

 (γ,δ) ∞ where hnγ+α,m−α is defined by Eq.(2.8). Thanks to theorem 3.2 and the orthogonality of P¯n (x) n=0 in L2γ,−δ (I) it holds 2

γ,m kDα (z − π ˆN z)kL2 γ+α,α−m

∞

2

P

(γ+α,m−α) Γ(n+m+1) ¯

= a (x) P n Γ(n+m−α+1) n

n=N +1 L2γ+α,α−m   ∞ 2 P Γ(n+m+1) = a2n Γ(n+m−α+1) hnγ+α,m−α n=N +1   2 γ+α,m−α  ∞ P hn Γ(n+m+1) ≤ max · a2n hzn . Γ(n+m−α+1) hz n≥N +1

n

where hzn

Z =

b



(3.23)

n=N +1

2 m ∂xp (x − a) J˜n(γ,m) (x) ω γ+p,p−m dx.

(3.24)

a (γ,m)

Notice the R-L fractional derivative L Dα P¯n

(γ,m)

(x) will reduce to ∂xm P¯n

11

(x) when α = m, so

applying Eq.(2.12) we have hzn =



Γ(n + m + 1) Γ(n + 1)

2 Z



Γ(n + m + 1) Γ(n + 1)

2

=

b



2 ∂xp−m J˜n(γ+m,0) (x) ω γ+p,p−m dx

a

·

(3.25)

γ+m,0 hp−m,n

where hγ+m,0 p−m,n is defined by Eq.(2.11). An implicit restriction for p must be pointed out here is p − m ≤ n for each hγ+m,0 p−m,n , n = N + 1, N + 2, · · · , which in fact requires p ≤ N + m + 1 for a given N . So if p > N + m + 1, we take p = N + m + 1 to get the best result. As is well known, the quotient of two gamma functions Γ(s+a) Γ(s+b) has the following expansion [5, 29, 30] " # N X Γ(s + a) = sa−b 1 + ck s−k + O(s−N −1 ) Γ(s + b)

(3.26)

k=1

a−b for s  1. Furthermore, for s + a > 1 and s + b > 1 lemma 2.1 in [34] which yields Γ(s+a) Γ(s+b) ' s presents an upper bound to restrict this quotient:

Γ(s + a) a−b ≤ κa,b s s Γ(s + b) where κa,b s

 = exp

(3.27)

a−b 1 (a − 1)(a − b) + + 2(s + b − 1) 12(s + a − 1) s

 .

Applying Eq.(3.27) to the last line of Eq.(3.23), we have for n ≥ 1 and p ≤ n + m 

Γ(n + m + 1) Γ(n + m − α + 1)

Notice that

2

hγ+α,m−α Γ(n + γ + α + 1) Γ(n + m − p + 1) n α,p 2(α−p) = ≤ Cγ,m n . (3.28) hzn Γ(n + γ + p + 1) Γ(n + m − α + 1) ∞ X

a2n hzn ≤

∞ X

2

a2n hzn = k∂xp zkL2

(3.29)

γ+p,p−m

n=0

n=N +1

and n ≥ N + 1 ≥ 1 in Eq.(3.23), then we arrive at the estimate for m ≤ p ≤ N + m + 1 γ,m kDα (z − π ˆN z)kL2

γ+α,α−m

≤ CN α−p k∂xp zkL2

γ+p,p−m

(3.30)

where C is independent of N . For the case p > N + m + 1, take p = N + m + 1 in the procedures above then we get the following estimate

γ,m kDα (z − π ˆN z)kL2 ≤ CN α−1−m−N ∂xm+1+N z L2 . (3.31) γ+α,α−m

γ+1+m+N,N +1

Using Stirling’s formula we obtain γ,m kDα (z − π ˆN z)kL2

γ+α,α−m

≤C

N α−m−1/2 −N e ∂xm+1+N z L2 . γ+1+m+N,N +1 N!

12

(3.32)

To get the estimate for the case δ ∈ (m−1, m) we introduce the non-uniformly Jacobi weighted space involving fractional derivatives, which is defined in [33]  Bqγ,−δ (I) = L Dδ+l ∈ L2γ+δ+l,l (I), l = 0, 1, · · · , q . m Theorem 3.4. Suppose z ∈ Bγ,−m (I) ∩ Bqγ,−δ (I), where γ > −1, δ ∈ (m − 1, m) and m ∈ N+ , q ∈ N. Then for any α ∈ [0, m], there exists a constant C independent of N such that if α − δ ≤ q ≤N +1



α γ,δ (3.33) ≤ CN α−δ−q L Dδ+q z L2 ˆN z) 2

D (z − π Lγ+α,α−δ

γ+δ+q,q

and if q > N + 1

α γ,δ ˆN z)

D (z − π

L2γ+α,α−δ

≤C

N α−δ−1/2 −N . e L Dδ+N +1 z L2 γ+δ+N +1,N +1 N!

(3.34)

(γ,δ) (γ,δ) Proof. Applying lemma 2.2 it can be verified Dα P¯n (x) = L Dα P¯n (x) for α ∈ [0, m) and (γ,δ) (γ,δ) (γ,δ) δ ∈ (m − 1, m). For α = m it is known both Dα P¯n (x) and L Dα P¯n (x) reduce to ∂xm P¯n (x). ¯ z to Analogous to the process of proving theorem 3.3 where the difference is we use the following h n z replace hn in Eq.(3.23)

2 (x) ω γ+δ+q,q dx a   2 Rb (γ,δ) = a ∂xq L Dδ P¯n (x) ω γ+δ+q,q dx  2 2 R  b q ˜(γ+δ,0) = Γ(n+δ+1) (x) ω γ+δ+q,q dx J ∂ n x Γ(n+1) a  2 γ+δ,0 = Γ(n+δ+1) hq,n Γ(n+1)

¯z = h n

R b L

(γ,δ)

Dδ+q P¯n

(3.35)

is defined by Eq.(2.11). Also it is worth to point out that here the restriction for q where hγ+δ,0 q,n is q ≤ n for each n = N + 1, N + 2, · · · , namely q ≤ N + 1. Following the same procedures of demonstrating theorem 3.3, from Eq.(3.26) to (3.32) and noticing that ∞ X

¯z a2n h n

≤

n=N +1

∞ X

¯ z = L Dδ+q z 2 2 a2n h n L

(3.36)

γ+δ+q,q

n=0

we can arrive at Eq.(3.33) and Eq.(3.34). We can see that when δ = m, the estimates above reduce to Eq.(3.21) and Eq.(3.22). To ensure the approximation for smooth enough data is also convergent in the sense of k · k∞ , we provide the following two theorems respectively for the case when δ = m and δ ∈ (m − 1, m). p Theorem 3.5. Suppose z ∈ Bγ,−m (I) ∩ C s (I) where γ > −1, I = [a, b] and m, p ∈ N. Denote κ = max{γ + α, m − α} where 0 ≤ α ≤ m and s = dαe, then there exists a constant C independent of N such that for κ + α + 1 < p ≤ N + 1 + m γ,m kDα (z − π ˆN z)k∞ ≤ CN κ+α+1−p k∂xp zkL2

γ+p,p−m

13

(3.37)

and for p > N + 1 + m γ,m kDα (z − π ˆN z)k∞ ≤ C

N κ+α+1/2−m −N . e ∂xN +1+m z L2 γ+N +1+m,N +1 N!

(3.38)

Proof. Thanks to theorem 3.2 we know Dα z has the following expansion in L2γ+α,α−m (I) Dα z =

∞ X

Γ(n + m + 1) an P¯n(γ+α,m−α) (x). Γ(n + m − α + 1) n=0

(3.39)

As z ∈ C s (I), it can be verified Dα z ∈ C(I). Classical analysis tells us if the series on the right hand (γ+α,m−α) side of Eq.(3.39) is convergent uniformly, then it converges to Dα z because every P¯n (x) is continuous on I. Applying Cauchy-Schwarz inequality to the remainder of the expansion series above we have ∞ ∞ P (γ+α,m−α) (γ+α,m−α) m−α P Γ(n+m+1) Γ(n+m+1) ¯ ˜ (x) = (x − a) (x) Γ(n+m−α+1) an Pn Γ(n+m−α+1) an Jn n=N +1 n=N +1 1/2  ∞ 2 1/2  P  ∞ P Γ(n+m+1) ˜(γ+α,m−α) 2 z z −1 ¯ (x) · ≤ CI an hn (hn ) Γ(n+m−α+1) Jn n=N +1

n=N +1

(3.40) where C¯I = (b − a)m−α . From [27](P.78) we know if γ + α ≥ m − α, max J˜n(γ+α,m−α) (x) ≤ J˜n(γ+α,m−α) (b) =

a≤x≤b

Γ(n + γ + α + 1) Γ(γ + α + 1)Γ(n + 1)

(3.41)

Γ(n + m − α + 1) . Γ(m − α + 1)Γ(n + 1)

(3.42)

and if γ + α ≤ m − α max J˜n(γ+α,m−α) (x) ≤ J˜n(γ+α,m−α) (a) =

a≤x≤b

So if κ + α + 1 < p ≤ N + 1 + m, for γ + α ≥ m − α we have ∞ P

(hzn )

−1



n=N +1

2 ∞ P Γ(n+m+1) ˜(γ+α,m−α) α,p 2γ+4α−2p+1 (x) ≤ n C¯γ,m Γ(n+m−α+1) Jn n=N +1 R ∞ α,p 2γ+4α−2p+1 ≤ C¯γ,m s=N s ds ≤ CN 2γ+4α−2p+2

(3.43)

and for γ + α ≤ m − α it holds ∞ P n=N +1

(hzn )

−1



2

Γ(n+m+1) ˜(γ+α,m−α) (x) Γ(n+m−α+1) Jn α,p ≤ C˜γ,m

R∞

≤

∞ P

α,p 2m−2p+1 C˜γ,m n

n=N +1

(3.44)

s2m−2p+1 ds ≤ CN 2m−2p+2 . s=N

Recalling Eq.(3.29), with combining Eq.(3.40), Eq.(3.43) and Eq.(3.44), we get the estimate for p>κ+α+1 ∞ X Γ(n + m + 1) (γ+α,m−α) an P¯n (x) ≤ CN κ+α+1−p k∂xp zkL2 . (3.45) γ+p,p−m Γ(n + m − α + 1) n=N +1

∞

Take p = N + m + 1 in the estimate above if p > N + m + 1. So if p > κ + α + 1, the expansion

14

on the right hand side of Eq.(3.39) converges uniformly to Dα z which implies ∞ X

γ,m Dα (z − π ˆN z)(x) =

n=N +1

Γ(n + m + 1) an P¯n(γ+α,m−α) (x) Γ(n + m − α + 1)

(3.46)

holds for any x ∈ [a, b]. Thus Eq.(3.37) is verified. For the case p > N + 1 + m take p = N + 1 + m in Eq.(3.37) then we can obtain Eq.(3.38) with utilizing Stirling’s formula. m Theorem 3.6. Suppose z ∈ C s (I) ∩ Bγ,−δ (I) ∩ Bqγ,−δ (I) where γ > −1, δ ∈ (m − 1, m), s = dαe, + I = [a, b] and m ∈ N . Denote κ = max{γ + α − δ, −α} where 0 ≤ α ≤ m, then there exists a constant C independent of N such that for κ + α + 1 < q ≤ N + 1



α γ,δ (3.47) ˆN z) ≤ CN κ+α+1−q L Dδ+q z L2

D (z − π ∞

γ+δ+q,q

and for q > N + 1

α γ,δ ˆN z)

D (z − π

∞

≤C

N κ+α+1/2−δ −N . e L Dδ+1+N z L2 γ+δ+N +1,N +1 N!

(3.48)

Proof. This theorem can be demonstrated by following the same steps of proving theorem 3.5. To ¯ z in Eq.(3.40) avoid repetition we just present a brief sketch for the proof process. Replace hzn by h n then we can obtain the estimate for κ + α + 1 < q ≤ N + 1 ∞ X

Γ(n + δ + 1) an P¯n(γ+α,δ−α) (x) ≤ CN κ+α+1−q L Dδ+q z L2 (3.49) γ+δ+q,q Γ(n + δ − α + 1) n=N +1

∞

Thus we complete the proof of Eq.(3.47). Taking q = N + 1 in the proof process we can get Eq.(3.48) with using Stirling’s formula.

3.3

Error estimates for interpolation

Next we shall consider the interpolation error involving fractional derivative. To begin with, we present the following theorem. Theorem 3.7 (Fractional Inverse Inequality). Suppose ϕδN (δ > −1) is a poly-fractonomial with the form (x − a)δ ϕN where ϕN ∈ PN . Then for 0 ≤ α < δ + 1 or α = δ + l, l = 0, 1, · · · we have

L α δ



D ϕN 2 ≤ CN α ϕδN L2 . (3.50) L γ+α,α−δ,

where γ > −1 and C is independent of N . Proof. Expand ϕN by shifted Jacobi polynomials as ϕN =

N X

˜(γ,δ) (x). ϕˆγ,δ n Jn

n=0

15

γ,−δ

Consider the case α < δ + 1. With applying theorem 2.1 we have

2 N

L α P

δ γ,δ ˜(γ,δ)

D (x − a) ϕ ˆ J = (x) n n



L α δ 2

D ϕ 2

N L γ+α,α−δ

n=0

L2

γ+α,α−δ

N

2

P γ,δ Γ(n+δ+1)

δ−α ˜(γ+α,δ−α)

ϕˆn Γ(n+δ−α+1) (x − a) = Jn (x)

2 n=0 Lγ+α,α−δ   N 2
P Γ(n+δ+1) γ+α,δ−α γ,δ 2 hn = ϕˆn Γ(n+δ−α+1) n=0  2 γ+α,δ−α  P N
hn Γ(n+δ+1) γ,δ 2 γ,δ ≤ max hn ϕ ˆ γ,δ n Γ(n+δ−α+1) h

0≤n≤N

n

(3.51)

n=0

Thanks to the estimate Eq.(3.27), for n ≥ 1 it holds 

Γ(n + δ + 1) Γ(n + δ − α + 1)

2

hγ+α,δ−α n hγ,δ n

α n2α . ≤ Cγ,δ

(3.52)

And for n = 0, after a simple manipulation we get 

Γ(n + δ + 1) Γ(n + δ − α + 1)

2

hnγ+α,δ−α hγ,δ n

=

Γ(γ + α + 1)Γ(δ + 1) Γ(γ + 1)Γ(δ − α + 1)

(3.53)

which is also bounded. So

L α δ 2

D ϕN 2

Lγ+α,α−δ

≤ CN 2α

N X

ϕˆγ,δ n


2

2α δ 2 ϕN L2 hγ,δ n = CN

.

(3.54)

γ,−δ

n=0

For the case α = δ + l, noticing L Dδ+l = ∂xl L Dδ we have L

 Γ(n + δ + 1) Γ(n + γ + δ + l + 1)  (γ+α,l) δ J˜ (x). Dα (x − a) J˜n(γ,δ) (x) = Γ(n + 1) (b − a)l Γ(n + γ + δ + 1) n−l

(3.55)

(γ+α,l) For l > n, J˜n−l (x) = 0. Analogous to the process from Eq.(3.51)-Eq.(3.53), we can get

L α δ 2

D ϕN 2

Lγ+α,α−δ

α ≤ C˜γ,δ N 2α

N X

ϕˆγ,δ n


2

˜ α 2α ϕδ 2 2 hγ,δ n = Cγ,δ N N L

.

(3.56)

γ,−δ

n=0

Hence Eq.(3.50) is verified. Using lemma 2.2 it is easy to verify that L Dα ϕδN = Dα ϕδN for 0 ≤ α ≤ dδe and δ ≥ 0. So we have the following corollary: Corollary. Suppose ϕδN (δ ≥ 0) is a poly-fractonomial with the form (x − a)δ ϕN where ϕN ∈ PN . Then for 0 ≤ α ≤ dδe it holds

α δ



D ϕN 2 ≤ CN α ϕδN L2 . (3.57) L γ+α,α−δ,

γ,−δ

where γ > −1 and C is independent of N . Utilizing the corollary above, now let us present the estimate for the case when δ = m.

16

p 1 Theorem 3.8. Suppose z(x) satisfies z ∈ Bγ,−m (I) and (x−a)−m z = w ∈ Bγ,m (I), where γ > −1 and m, p ∈ N. Then for 0 ≤ α ≤ m, if max{1, m} ≤ p ≤ N + 1 + m it holds

α γ,m ≤ CN α−p k∂xp zkL2 , (3.58)

D (z − IˆN z) 2 γ+p,p−m

Lγ+α,α−m

and if p > N + 1 + m the estimate satisfies

α γ,m

D (z − IˆN z)

L2γ+α,α−m

≤C

N α−m−1/2 −N e ∂xN +1+m z L2 . γ+1+m+N,N +1 N!

(3.59)

Proof. As the proof process of theorem 3.3 we shall firstly demonstrate Eq.(3.58). Notice the fact γ,m γ,m γ,m IˆN (ˆ πN z) = π ˆN z, we have



α γ,m γ,m γ,m γ,m ≤ kDα (z − π . (3.60) ˆN z)kL2 + Dα IˆN (z − π ˆN z) 2

D (z − IˆN z) 2 Lγ+α,α−m

Lγ+α,α−m

γ+α,α−m

Thanks to theorem 3.3, for m ≤ p ≤ N + 1 + m the first term on right hand satisfies γ,m z)kL2 kDα (z − π ˆN

γ+α,α−m

≤ CN α−p k∂xp zkL2

γ+p,p−m

.

(3.61)

For the second term, applying Eq.(3.57) it holds



α ˆγ,m

γ,m γ,m γ,m (z − π ˆN z) ˆN z) 2 ≤ CN α IˆN

D IN (z − π

L2γ,−m

Lγ+α,α−m

.

(3.62)

Recall the estimate for Jacobi-Gauss interpolation(see P131 in [27]) γ,m kIN v(x)kL2

γ,m

. kv(x)kL2γ,m + N −1 k∂x v(x)kL2

γ+1,m+1

and notice z = (x − a)m w then we have

ˆγ,m γ,m γ,m γ,m ˆN z) 2 = kIN (w − πN w)kL2

IN (z − π Lγ,−m

γ,m

γ,m . kw − πN wkL2

γ,m

(3.63)

γ,m + N −1 k∂x (w − πN w)kL2

.

γ+1,m+1

Take α = 0 in Eq.(3.21) then we obtain the estimate for the first term on right hand side γ,m kw − πN wkL2

γ,m

γ,m = kz − π ˆN zkL2

γ,−m

≤ CN −p k∂xp zkL2

γ+p,p−m

.

(3.64)

For the second term it holds 2

γ,m k∂x (w − πN w)kL2

=

∞ X

γ+1,m+1

a2n hγ,m 1,n ≤ max

n≥N +1

n=N +1

∞ o X n z −1 hγ,m (h ) a2n hzn n 1,n

(3.65)

n=N +1

z where hγ,m 1,n and hn are defined in Eq.(2.11) and Eq.(3.24) respectively. As it is stated below Eq.(3.25), the inequality above implies p ≤ N + 1 + m. Thanks to Eq.(3.27) and p ≥ max{1, m} we have for n ≥ 1 z −1 ≤ Cˆγ,m n2−2p hγ,m 1,n (hn )

17

where Cˆγ,m is independent of n. So γ,m k∂x (w − πN w)kL2

γ+1,m+1

≤ CN 1−p k∂xp zkL2

γ+p,p−m

.

(3.66)

Combining Eq.(3.60)-Eq.(3.62), Eq.(3.63)-Eq.(3.64) and Eq.(3.66), Eq.(3.58) can be verified. From Eq.(3.61), Eq.(3.64) and Eq.(3.65) we can see taking p = N + 1 + m leads to the best result for p > N + 1 + m. So Eq.(3.59) is verified. At the end of this section we shall discuss the fractional Lagrange interpolation proposed in [32] and give the error estimate for this type of interpolation at Jacobi-Gauss points. The fractional Lagrange interpolation basis are with the form hδj (x)

 =

x−a xj − a

δ Y  N  x − xk , xj − xk

0≤j≤N

(3.67)

k=0 k6=j

(γ,δ) Take the roots of J˜N (x) as the interpolation nodes then we can verify

γ,δ IˆN z(x) =

N X

z(xj )hδj (x).

(3.68)

j=0 m 1 Theorem 3.9. Suppose z(x) satisfies z ∈ Bγ,−δ (I)∩Bqγ,−δ (I) and (x−a)−δ z = w ∈ Bγ,δ (I), where + γ > −1, δ ∈ (m − 1, m) and m ∈ N , q ∈ N. Then for 0 ≤ α ≤ m, if max{α − δ, 1 − δ} ≤ q ≤ N + 1 it holds



α γ,δ ≤ CN α−δ−q L Dδ+q z L2 . (3.69)

D (z − IˆN z) 2 Lγ+α,α−δ

γ+δ+q,q

and if q > N + 1 the estimate satisfies

α γ,δ

D (z − IˆN z)

L2γ+α,α−δ

≤C

N α−δ−1/2 −N e L Dδ+1+N z L2 . γ+δ+N +1,N +1 N!

Proof. Following the same procedures from Eq.(3.60) to Eq.(3.63), we obtain



α

γ,δ γ,δ ≤ Dα (z − π ˆN z) 2

D (z − IˆN z) 2 Lγ+α,α−δ Lγ+α,α−δ 



γ,δ γ,δ +CN α w − πN w 2 + N −1 ∂x (w − πN w) 2 Lγ,δ

(3.70)

(3.71)

 .

Lγ+1,δ+1

Consider the first and second term on the right hand side of Eq.(3.71). Applying theorem 3.4, we have if α − δ ≤ q ≤ N + 1 the first term satisfies



α γ,δ ˆN z) 2 ≤ CN α−δ−q L Dδ+q z L2 (3.72)

D (z − π Lγ+α,α−δ

γ+δ+q,q

and if 0 ≤ q ≤ N + 1 the second term satisfies



γ,δ γ,δ ˆN z

w − πN w 2 = z − π Lγ,δ

L2γ,−δ

18

≤ CN −δ−q L Dδ+q z L2

γ+δ+q,q

(3.73)

Similar to Eq.(3.65), the last term on the right hand side of Eq.(3.71) satisfies

2

γ,δ

∂x (w − πN w) 2

Lγ+1,δ+1

=

∞ X

a2n hγ,δ 1,n ≤ max

n≥N +1

n=N +1

∞ n
o X ¯ z −1 ¯z hγ,δ h a2n h n n 1,n

(3.74)

n=N +1

¯ z is defined in Eq.(3.35). Using Eq.(3.27) we have for n ≥ 1 where h n ¯z hγ,δ 1,n hn


−1

q ≤ Cγ,δ n2(1−δ−q) .

(3.75)

So if q ≥ 1 − δ,



γ,δ

∂x (w − πN w)

L2γ+1,δ+1

≤ CN 1−δ−q L Dδ+q z L2

(3.76)

γ+δ+q,q

Combining Eq.(3.71)–Eq.(3.76) we can conclude that for max{α − δ, 1 − δ} ≤ q ≤ N + 1, Eq.(3.69) is valid. For the case q > N + 1, taking q = N + 1 in the analysis procedures above and utilizing Eq.(3.34), Eq.(3.69) can be demonstrated.

4

Applications of the algorithms

The MATLAB 7.10 software is used for all computations in this section. In the first two examples, we present the approximation results for fractional derivatives. And in example 4.3 and example 4.4 the algorithm is applied to solve FIVPs with combining collocation method. In all these examples, the maximum absolute error k·k∞ is obtained from 1000 equispaced points in I, and the norm k·kL2 is computed by Jacobi-Gauss quadrature with 200 points. Since γ+α,α−m γ,m γ,m z) are similar, here we only show the results the behaviors of Dα (z − π ˆN z) and Dα (z − IˆN  N for truncated approximation, where an n=0 is computed by Jacobi-Gauss quadrature with 200 nodes.

4.1

Approximation results

Example 4.1. Let u(x) = (x + 1)3.6 and I = [−1, 1]. Now we utilize (3.6) and (3.11) to compute the fractional derivative −1 Dxα (x + 1)3.6 in Caputo sense, which has the analytical expression Γ(4.6) 3.6−α . We set γ = 0 to compute, with m = 1 for α ∈ (0, 1) and m = 2 for α ∈ (1, 2). Γ(4.6−α) (x + 1) Fig.1 presents the convergence order for α ∈ (0, 1) and α ∈ (1, 2) with respect to the norm 7 6 k · kL2γ+α,α−m . It can be verified that z(x) ∈ B0,−1 (I) for α ∈ (0, 1) and z(x) ∈ B0,−2 (I) for α−7 α ∈ (1, 2). So the approximate solution uN should converge to u at the rate of O(N ) when α ∈ (0, 1), and of O(N α−6 ) when α ∈ (1, 2). From the slopes of the dashed lines in Fig.1 we can see the numerical results are well consistent with the theoretical analysis. In terms of approximating given functions, with a modified technique we shall show scheme (3.11) can be proved convergent in L2 (I) with convergence rate not less than O(N α−p ). In fact, to p approximate Dα u(x) on I = [a, b], we do it on the substituted interval I l = [a, b+l] if z ∈ Bγ,−m (I l ), then choose the part on [a, b] as what we need. After a simple manipulation, we get the error on [a, b] satisfies q m−α −(γ+α) γ,m γ,m kDα (z − π ˆN z)kL2 (I) ≤ (b − a) l kDα (z − π ˆN z)kL2 (4.1) (I l ) . γ+α,α−m

19

−5

−3

10

10 α=0.2 α=0.4 α=0.6 α=0.8

−6

10

−7

α=1.2 α=1.4 α=1.6 α=1.8

−4

10

10

−5

||Dα(uN−u)||ω

−9

10

α

||D (uN−u)||ω

10 −8

10

−10

10

−6

10

−7

10

−8

−11

10

−12

10

10

−9

10

−13

−10

10

1

10

2

10

1

10

2

10

N

10

N

(a) α ∈ (0, 1)

(b) α ∈ (1, 2)

Fig.1: kDα (u − uN )kL2α,α−m (I) of example 4.1 under different N and α

−4

−3

10

10

α=0.2 α=0.4 α=0.6 α=0.8

−6

10

α=1.2 α=1.4 α=1.6 α=1.8

−4

10

||Dα(uN−u)||L2

||Dα(uN−u)||L2

−5

−8

10

−10

10

10

−6

10

−7

10

−12

10

−8

10

−9

−14

10

1

10

2

10

1

2

10

10

10

N

N

(b) α ∈ (1, 2)

(a) α ∈ (0, 1)

Fig.2: kDα (u − uN )kL2 (I) of example 4.1 under different N and α where C is independent of N and l. Thus it makes sure that the approximate expression Dα uN (x) converges to Dα u(x) in L2 (I) with the rate as in L2γ+α,α−m (I l ). We will show that this technique could reduce the magnitude of oscillation on the right boundary of I when approximating Dα u(x) by scheme (3.11). In addition, what we want to state is that l usually is a small number relative to the length of the interval I. Eq.(4.1) shows that as l increases, the factor l−(γ+α) decreases while k∂xp zkL2 tends l γ+p,p−m (I ) to grow. So taking l a small number relative to the length of I is a simple way to balance the two factors. Fig.2 shows the results of the modified technique mentioned above, in which u(x) is approximated on I1 = (−1, 2] then cut off the part on the interval (1, 2] and take the remaining part as the approximate expression on (−1, 1]. From the slopes of the dashed lines in Fig.2 we can see the numerical results are well consistent with the theoretical analysis. For a visual compare of the scheme with and without combining the modified technique, we plot these results in Fig.3, from which we can see the modified technique removes the oscillation on the right boundary and makes the approximation more uniform. Fig.3(b)-(d) also show that 20

although different l leads to different result, the global behavior of Dα (uN − u) is similar.

−7

12

−9

x 10

1.5

x 10

−7

x 10 10

10

0.5

6

Dα(uN−u)

Dα(uN−u)

1

8

8 6

4

4

2

0

−0.5 0

2

0.96

0.98

1

−1

0 −2 −1

−0.5

0

0.5

−1.5 −1

1

−0.5

0.5

1

(b) With the modified technique l = 1/3

(a) Without the modified technique −9

1.5

0

x

x

−9

x 10

2.5

x 10

2 1 1.5 1

Dα(uN−u)

Dα(uN−u)

0.5

0

−0.5

0.5 0 −0.5 −1

−1 −1.5 −1.5 −1

−0.5

0

0.5

−2 −1

1

−0.5

x

0

0.5

1

x

(c) With the modified technique l = 2/3

(d) With the modified technique l = 1

Fig.3: Error of example 4.1 without(a) and with(b-d) modified technique, α = 1.2, N = 100 Example 4.2. Let u = sin(x), x ∈ [0, 10] and we calculate the Caputo fractional derivative α α 0 Dx u(x). The analytical expression of 0 Dx sin(x) is given by  ∞ P  (−1)k x2k  0 < α < 1,  0 Dxα sin(x) = x1−α Γ(2k+2−α) , k=0

∞ P  α 3−α   0 Dx sin(x) = x

k=0

(−1)k+1 x2k Γ(2k+4−α) ,

1 < α < 2.

The maximum absolute errors are displayed in Tab.1 and Tab.2 with α ∈ (0, 1) and α ∈ (1, 2) respectively. Since sin(x) is analytical, we can see Dα uN converges to Dα u exponentially with respect to the ∞-norm. From Tab.2 we can see that the modified technique is helpful to reduce the maximum absolute error which is usually reached on the right boundary. Compared with Table 2 in [10], the convergence rate of ours after modifying is higher, especially for α ∈ (1, 2). For convenience, we list the details of that Table 2 in Tab.3 below.

21

Tab.1: Maximum absolute errors of example 4.2 under different N with γ = −0.5

N

Without Modified Technique α = 0.2 α = 0.5 α = 0.8

With Modified Technique α = 0.2 α = 0.5 α = 0.8

5 10 15 20 25

1.17E-01 4.56E-04 3.01E-07 1.02E-11 8.51E-12

2.02E-01 5.05E-04 1.47E-07 1.32E-11 8.70E-12

2.51E-01 1.29E-03 1.08E-07 8.76E-11 3.62E-12

4.64E-01 3.14E-03 3.32E-07 3.49E-11 9.15E-12

2.64E-01 7.56E-04 2.54E-07 1.50E-11 3.47E-12

3.48E-01 1.41E-03 4.16E-07 3.30E-11 9.10E-12

Tab.2: Maximum absolute errors of example 4.2 under different N with γ = −0.5

N

Without Modified Technique α = 1.2 α = 1.5 α = 1.8

With Modified Technique α = 1.2 α = 1.5 α = 1.8

5 10 15 20 25

1.44E-00 9.05E-04 1.06E-06 1.17E-10 3.30E-10

3.73E-01 7.50E-04 2.55E-07 1.81E-11 6.21E-12

2.11E-00 1.94E-03 2.66E-06 2.99E-10 9.59E-10

2.92E-00 3.92E-03 6.29E-06 7.03E-10 2.63E-09

4.19E-01 1.04E-03 3.94E-07 2.19E-11 9.35E-12

4.58E-01 1.41E-03 6.01E-07 3.41E-11 2.03E-11

Tab.3: Maximum absolute errors in [10]

4.2

N

α = 0.2

α = 0.5

α = 0.8

α = 1.2

α = 1.5

α = 1.8

10 20 40 80

2.23e-07 2.12e-09 2.15e-11 2.20e-13

1.48e-06 2.11e-08 3.22e-10 5.00e-12

7.44e-06 1.62e-07 3.77e-09 8.89e-11

1.40e-04 5.37e-06 2.17e-07 8.92e-09

3.23e-04 1.86e-05 1.14e-06 7.10e-08

4.55e-04 3.95e-05 3.66e-06 3.45e-07

Solving FIVPs

Next we shall combine the approximation technique with collocation method to get the approximate solution of FIVPs with the following form  k X    ci (x)Dαi u(x) = f (x), (4.2) i=0    (i) u (a) = di , i = 0, 1, · · · , m − 1. where Dαi is usually interpreted as Caputo fractional derivative operator. For convenience, here we assume 0 ≤ α0 < α1 < · · · < αk ≤ m, m ∈ N+ . Substituting (3.11) to Eq.(4.2), combining theorem 2.1 we can get k P i=0

ci (x)

m−1 P j=dαi e

+

u(j) (a) Γ(j+1) j! Γ(j−αi +1) (x N P

n=0

j−αi

− a)

(4.3)



(γ+αi ,m−αi ) Γ(n+m+1) an Γ(n+m−α P¯n (x) i +1)

22

= f (x)

Denote g(x) = f (x) −

k X

ci (x)

i=0

m−1 X j=dαi e

u(j) (a) Γ(j + 1) j−αi (x − a) , j! Γ(j − αi + 1)

(4.4)

then Eq.(4.3) can be put in the form N X

an

n=0

k X i=0

ci (x)

Γ(n + m + 1) ¯ (γ+αi ,m−αi ) P (x) = g(x). Γ(n + m − αi + 1) n

(4.5)

Suppose {xj }N j=0 is the set of collocation points. Substitute these nodes in Eq.(4.5) then we get a system of linear equations N X n=0

an

k X

ci (xj )

i=0

Γ(n + m + 1) ¯ (γ+αi ,m−αi ) P (xj ) = g(xj ), Γ(n + m − αi + 1) n

j = 0, 1, · · · , N.

(4.6)

Example 4.3. Consider the following Baglay-Torvik equation [23] 1.5 u(2) (x) + C 0 Dx u(x) + u(x) = f (x),

x ∈ (0, 1], u(0) = 0, u(1) (0) = c,

with the exact solution u(x) = sin(ct). Tab.4: Maximum absolute errors of our method and SCT in [23] N

c

Method(4.6)

SCT[23]

c

Method(4.6)

SCT[23]

4 8 16 32 48 64

1 1 1 1 1 1

4.07e-06 6.20e-12 1.11e-16 2.22e-16 1.11e-16 1.11e-16

3.4e-04 4.3e-07 1.8e-08 7.1e-10 9.9e-11 2.4e-11

4π 4π 4π 4π 4π 4π

5.16e+01 3.55e+00 6.49e-05 1.13e-11 1.00e-11 6.66e-12

3.9e-00 4.7e-01 3.5e-05 1.4e-06 1.9e-07 4.8e-08

We choose Jacobi-Gauss points with the weight ω γ,m as the collocation points then obtain uN via solving the linear equation system (4.6). The maximum absolute errors of spectral-collocation method (4.6) and SCT method in [23] are listed in Tab.4. The parameters in our method are γ = 0 and m = 2. Numerical results show that ours plays better. Example 4.4. Let us consider the fractional oscillation equation appearing in applied problems C α 0 Dt u(t)

+ u(t) = 0,

t ∈ (0, T ], α ∈ (1, 2)

with the initial condition u(0) = 1, u(1) (0) = 0. This kind of equations were previously studied, e.g. by Blank in [24], Edwards et. al. in [25] and Li in [8]. By using Laplace transform we can get the analytical solution u(x) = Eα,1 (−tα ). We give the approximate solution for α = 1.3, 1.5, 1.8, 1.95(the same with theirs) in Fig.4 with γ = 0, m = 2 and N = 500, where Jacobi-Gauss points with the weight ω γ,m are chosen as collocation points. Correspondingly, we present uN (t) − u(t) in Fig.5, from which we can see the method is reliable.

23

1

1.2

0.8

α=1.3

0.8

0.6

0.6

0.4

uN(t)

uN(t)

1

0.4

0.2

0.2

0

0

−0.2

−0.2 0

5

10

15

α=1.5

−0.4 0

20

5

10

t

15

20

t

(a) α = 1.3, T = 20

(b) α = 1.5, T = 20

1

1

0.8

0.8 α=1.8

α=1.95

0.6

0.6

0.4

0.4

uN(t)

uN(t)

0.2 0.2 0

0 −0.2

−0.2

−0.4

−0.4

−0.6

−0.6 −0.8 0

−0.8 10

20

30

40

−1 0

50

t

20

40

60

80

100

t

(c) α = 1.8, T = 50

(d) α = 1.95, T = 100

Fig.4: Numerical results for example 4.4 with different α

Conclusion In this paper we revisited two approaches to approximate Caputo factional derivative and γ,δ γ,δ studied the behavior of Dα (z − π ˆN z) and Dα (z − IˆN z). For δ = m, thanks to the generalized Hardy’s inequality, we get the optimal error estimates of the two approaches in the non-uniformly p Jacobi weighted Sobolev space Bγ,−m (I). For δ ∈ (m − 1, m), by utilizing the fractional weighted q Sobolev space Bγ,−δ (I), optimal estimates were also obtained. Furthermore, we also presented γ,m γ,δ the truncated error kDα (z − π ˆN z)k∞ , following which kDα (z − π ˆN z)k∞ could also be obtained. Combining collocation method, the algorithm was applied to solve FIVPs. Numerical examples show the algorithm was effective.

Acknowledgments The work of the authors was supported by the National Natural Science Foundation of China (51174236). We would like to thank the two anonymous referees’ constructive comments of JCP and suggestions which improve this paper a lot.

24

−5

9

−5

x 10

7

8

6

α=1.5

α=1.3

7

5

6

4

5

uN(t)−u(t)

uN(t)−u(t)

x 10

4 3

3 2 1

2

0

1

−1

0 −1 0

5

10

15

−2 0

20

5

10

t

(a) α = 1.3, T = 20

20

(b) α = 1.5, T = 20

−5

10

15

t

−5

x 10

6

x 10

8 4

α=1.8

α=1.95

6

uN(t)−u(t)

uN(t)−u(t)

2 4 2 0

0

−2 −2 −4 −4 −6 0

10

20

30

40

−6 0

50

20

40

t

60

80

100

t

(c) α = 1.8, T = 50

(d) α = 1.95, T = 100

Fig.5: uN (t) − u(t) of example 4.4 with different α

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