On the sectorial property of the Caputo derivative operator

Accepted Manuscript On the sectorial property of the Caputo derivative operator Kazufumi Ito, Bangti Jin, Tomoya Takeuchi PII: DOI: Reference:

S0893-9659(15)00087-7 http://dx.doi.org/10.1016/j.aml.2015.03.001 AML 4741

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Applied Mathematics Letters

Received date: 29 January 2015 Revised date: 1 March 2015 Accepted date: 1 March 2015 Please cite this article as: K. Ito, B. Jin, T. Takeuchi, On the sectorial property of the Caputo derivative operator, Appl. Math. Lett. (2015), http://dx.doi.org/10.1016/j.aml.2015.03.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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On the Sectorial Property of the Caputo Derivative Operator Kazufumi Ito∗

Bangti Jin†

Tomoya Takeuchi‡

March 1, 2015

Abstract In this note, we establish the sectorial property of the Caputo fractional derivative operator of order α ∈ (1, 2) with a zero Dirichlet boundary condition. Keywords: Caputo fractional derivative, sectorial property

1

Introduction

We consider the following Sturm-Liouville problem with a left-sided Caputo fractional derivative in the leading term: find u and λ ∈ C such that α −C 0 Dx u + qu = λu

u(0) = u(1) = 0,

in D ≡ (0, 1),

(1.1)

α where α ∈ (1, 2) is the order of the derivative, q ∈ L∞ (D) and C 0 Dx u is the left-sided Caputo derivative of α order α defined in (2.1) below. In case of α = 2, the fractional derivative C 0 Dx u coincides with the usual 00 second-order derivative u , and the model (1.1) recovers the classical Sturm-Liouville problem, which has been extensively studied [9]. The interest in the model (1.1) mainly stems from modeling superdiffusion processes, in which the mean squares variance grows faster than that in a Gaussian process. It arises in e.g., subsurface flow and magnetized plasma [1, 2]; see also [3, Chapter 11] for an application in harmonic analysis. Analytically, very little is known about the Sturm-Liouville problem (1.1). In the case of q ≡ 0, the eigenvalues {λj } ⊂ C are zeros of the Mittag-Leffler function Eα,2 (−λ), where the two parameter Mittag-Leffler function Eα,β (z) is defined by (with Γ(·) being the Gamma function)

Eα,β (z) =

∞ X

k=0

zk , Γ(kα + β)

and the corresponding eigenfunctions are given by xEα,2 (−λj xα ) [7]. We refer to [10] for detailed discussions on the zeros of Mittag-Leffler functions. The numerical experiments in [7] indicate that the eigenvalues to problem (1.1) lie in a sector in the complex plane, i.e., the Caputo fractional derivative is a sectorial operator; see also [5] for a Galerkin finite element method for computing the eigenvalues. In this work we shall rigorously establish the sectorial property of the Caputo derivative operator. This property plays an important role in spectral analysis, and underlies the use of semigroup theory for studying the related time-dependent space fractional diffusion equation [4]. The rest of the paper is organized as follows. In Section 2, we describe preliminaries of fractional calculus. In Section 3, we establish the sectorial property of the Caputo derivative operator. ∗ Department

of Mathematics, North Carolina State University, Raleigh, NC 27695, USA. ([email protected]) of Computer Science, University College London, Gower Street, London WC1E 6BT, UK. ([email protected]) ‡ Institute of Industrial Science, The University of Tokyo, Tokyo, Japan. ([email protected]) † Department

1

2

Preliminaries on fractional calculus

Now we recall preliminaries on fractional calculus. For any positive non-integer real number α with n − 1 < α < n, n ∈ N, the (formal) left-sided Caputo fractional derivative of order α is defined by (see, e.g., [8, p. 92])  n  C α n−α d φ , (2.1) D φ = I 0 x 0 x dxn and the (formal) left-sided Riemann-Liouville fractional derivative of order α is defined by [8, p. 70]:   dn n−α R α φ . (2.2) 0Ix 0Dx φ = dxn Here 0Ixγ for γ > 0 is the left-sided Riemann-Liouville integral operator of order γ defined by Z x 1 ( 0Ixγ φ)(x) = (x − t)γ−1 φ(t)dt. Γ(γ) 0 The right-sided versions of fractional-order integrals and derivatives are defined analogously by Z 1 1 γ (xI1 φ)(x) = (t − x)γ−1 φ(t) dt, Γ(γ) x and C α x D1 φ

= (−1)n xI1n−α



 dn φ , dxn

R α xD1 φ

= (−1)n

dn dxn



n−α φ xI1



.

The integral operator 0Ixγ satisfies a semigroup property, i.e., for γ, δ > 0, there holds [8, Lemma 2.3, p. 73] γ+δ φ = 0Ixγ 0Ixδ φ ∀φ ∈ L2 (D), (2.3) 0Ix

and it holds also for the right-sided Riemann-Liouville integral operator xI1γ . We also recall a useful change of integration order formula [8, Lemma 2.7, p. 76]: (0Ixβ φ, ψ) = (φ, xI1β ψ) ∀φ, ψ ∈ L2 (D).

3

(2.4)

Sectorial property

α Now we establish the sectorial property of the Caputo derivative operator A = − C 0 Dx + q, i.e., Z x 1 Au(x) = − (x − t)1−α u00 (t)dt + q(x)u(x), Γ(2 − α) 0

with a zero Dirichlet boundary condition. First we recall the concept of sectorial operators [4, Definition 3.8, p. 97]. For θ ∈ (0, π/2] and κ ∈ R, we define a sector Σθ,κ by Σθ,κ = {z ∈ C : z 6= κ, | arg(z − κ)| < θ + π/2}. Then a densely defined operator A on a Banach space X is called a sectorial operator on X if there exists numbers θ ∈ (0, π/2] and κ ∈ R such that (i) the sector Σθ,κ ⊂ ρ(A), the resolvent set of A, and (ii) for any  ∈ (0, θ) there exists a constant γ = γ() > 0 with k(A + zI)−1 kL(X) ≤

γ() , |z − κ|

¯ ,κ , z 6= κ. ∀z ∈ Σ

By Theorem 3.6 of [4], the sectorial property of an operator follows from the V -coercivity of the associated bilinear form, i.e., it is continuous and satisfies a G¨ arding type inequality. We shall use this result to prove the desired sectorial property. 2

We first consider the special case q ≡ 0, and denote the Caputo derivative operator with q ≡ 0 by A0 , i.e., Z x 1 (x − t)1−α u00 (t)dt. A0 u(x) = − Γ(2 − α) 0

We appeal to the adjoint technique, since the spectrum of the adjoint operator A∗0 (with respect to L2 (D)) is conjugate to that of A0 . The domain dom(A0 ) of the derivative operator A0 is given by  dom(A0 ) = u ∈ L2 (D) : u00 ∈ dom(0 Ixα ) with u(0) = u(1) = 0 . Next we derive a representation of the adjoint A∗0 . Using (2.4) and integration by parts, we deduce −(A0 u, φ) = (0 Ix2−α u00 , φ) = (u00 , x I12−α φ)

= (u0 x I12−α φ)|1x=0 − (u0 , (x I12−α φ)0 )

= −u0 (0)(x I12−α φ)(0) − (u0 , x I12−α φ0 )

= −u0 (0)(x I12−α φ)(0) + (u, (x I12−α φ0 )0 ). α−1 α−1 Here the third line follows from the fact that R φ= C φ for φ ∈ H 1 (D) with φ(1) = 0 [5, Lemma xD1 xD1 ∗ 4.1]. Therefore, the adjoint operator A0 of A0 is given by

A∗0 φ(x) = −

d dx

Z

1

x

(t − x)1−α 0 φ (t)dt Γ(2 − α)

with its domain dom(A∗0 ) given by  dom(A∗0 ) = φ ∈ H 1 (D) : φ(1) = 0; (x I12−α φ)(0) = 0,

2−α 0 φ x I1

∈ H 1 (D) .

It is easy to see that the nonlocal condition (x I12−α φ)(0) = 0 is explicitly given by (x1−α , φ) = 0. Since dom(A0 ) 6= dom(A∗0 ), the operator A0 is not self-adjoint. The operator A∗0 involves a nonlocal constraint in its domain dom(A∗0 ), and hence it is inconvenient for direct treatment. To this end, we introduce an operator A˜0 defined by Z 1 d (s − x)1−α ˜0 A˜0 φ˜ = − φ (s) ds dx x Γ(2 − α) with its domain

n o ˜ ˜ dom(A˜0 ) = φ˜ ∈ L2 (D) : x I12−α φ˜0 ∈ H 1 (D) with φ(0) = φ(1) =0 .

The operator A˜0 is the right-sided Riemann-Liouville fractional derivative, since in view of [5, Lemma ˜ 0 . Hence it is coercive and continuous in the space H α/2 (D) [5], and 4.1], there holds x I12−α φ˜0 = (x I12−α φ) 0 hence it is sectorial (see [4, Theorem 3.6] or [6, Lemma 2.1]). Next we construct the perturbation term. ˜ ˜ Given any φ ∈ dom(A∗0 ), we set φ(x) = φ(x) − φ(0)(1 − x) ∈ dom(A˜0 ), i.e., φ(x) = φ(x) + φ(0)(1 − x) ∈ R 1 1−α 2−α ∗ ∗ ˜ ((1 − x)φ(0) + φ)dx = 0, dom(A0 ). The nonlocal condition (x I1 φ)(0) = 0 in dom(A0 ) implies 0 x R1 1−α ˜ ˜ i.e., the constant φ(0) can be uniquely constructed from φ by φ(0) = −cα 0 φ(x)x dx, with cα = Γ(4 − α)/Γ(2 − α). Hence, the operator A∗0 can be expressed using A˜0 as Z 1 (s − x)1−α 0 d (s − x)1−α ˜0 φ (s) ds = − (φ (s) − φ(0)) ds Γ(2 − α) dx x Γ(2 − α) x Z 1 ˜ ˜ 1−α dt, = A˜0 φ(x) + c0α (1 − x)1−α φ(t)t

A∗0 φ = −

d dx

Z

1

0

3

R1 ˜ 1−α dt, which is with c0α = cα /Γ(2 − α). Namely, the perturbation term is given by c0α (1 − x)1−α 0 φ(t)t a rank-one operator. By the standard Sobolev embedding theorem, there holds Z 1 Z 1 Z 1 1−α 1−α 1−α ˜ ˜ ˜ ˜ (1 − x)1−α φ(x)dx| φ(t)t dt φ(t)t dt, φ)| = | |((1 − x) (3.1) 0 0 0 ˜ γ α/2 , ˜ 2−γ ˜ 2p k φk ≤ ck φk ≤ ckφk 2 L (D) L (D) H (D) where p > 1/(2 − α) and some γ ∈ (0, 2). This together with Young’s inequality indicates that the associated bilinear form for A∗0 satisfies a G¨arding type coercivity inequality, and clearly also the continuity α/2 on the space H0 (D). This and [4, Theorem 3.6] imply that the operator A∗0 has the sectorial property and so does the operator A0 . In the general case q 6= 0, we proceed like before to deduce Z 1 d (s − x)1−α 0 A∗ φ = − φ (s) ds + qφ dx x Γ(2 − α) Z 1 (s − x)1−α ˜0 d ˜ − x)) (φ (s) − φ(0)) ds + q(φ˜ − cα (x1−α , φ)(1 =− dx x Γ(2 − α) ˜ ˜ − cα q(1 − x)(x1−α , φ), ˜ = A˜φ(x) + c0 (1 − x)1−α (x1−α , φ) α

˜ = dom(A˜0 ), the domain of the operator A˜0 . From this relation and the where the domain dom(A) estimate (3.1), we deduce that the associated bilinear form satisfies a G¨ arding type coercivity inequality, and thus the operator A has the sectorial property. By summarizing the preceding discussions, we arrive at the main result of the note. α Theorem 3.1. For any q ∈ L∞ (D), the operator A = − C 0 Dx + q is sectorial.

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