ASYMPTOTIC BEHAVIOUR OF THE SOLUTIONS OF ... - CiteSeerX

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS SUPPLEMENT 2007

Website: www.AIMSciences.org pp. 277–285

ASYMPTOTIC BEHAVIOUR OF THE SOLUTIONS OF FRACTIONAL INTEGRO–DIFFERENTIAL EQUATIONS AND SOME TIME DISCRETIZATIONS

Eduardo Cuesta Department of Applied Mathematics E.U.P. de Valladolid C/ Francisco Mendizabal, 1 Valladolid 47014, SPAIN Abstract. We study he asymptotic behaviour as t → +∞ of the solutions of an abstract fractional equation u = u0 + ∂ −α Au + g, 1 < α < 2, where A is a linear operator of sectorial type. We also show that a discretization in time of this equation based on backward Euler convolution quadrature inherits this behaviour.

1. Introduction. In this paper we consider the linear integro-differential equation of fractional order Z t (t − s)α−2 0 Au(s) ds + f (t), 0 ≤ t ≤ T, u(0) = u0 ∈ X (1) u (t) = 0 Γ(α − 1) where 1 < α < 2, A : D(A) ⊂ X → X is a linear, densely defined operator of sectorial type on a complex Banach space normed by k·k and f : [0, T ] → X. Notice that the convolution integral in (1) is known as the Riemann–Liouville integral. In Rt β−1 fact, notice that 0 (t−s) Γ(β) g(s) ds, β > 0, defines the Riemann–Liouville integral of order β and stands for the fractional integral of order β of g. Therefore, integrating in time in both sides, problem (1) can be written as Z t Z t (t − s)α−1 u(t) = u0 + Au(s) ds + f (s) ds, 0 ≤ t ≤ T. (2) Γ(α) 0 0 Let us recall that a closed and densely defined operator A is said to be sectorial if there exist 0 < θ < π/2, M > 0 and ω ∈ R such that its resolvent exists outside the sector ω + Sθ := {ω + λ : λ ∈ C, | arg(−λ)| < θ}, (3) and M k(λ − A)−1 k ≤ , λ∈ / ω + Sθ . (4) |λ − ω| In particular, if we assume that A is sectorial with 0 ≤ θ < π(1−α/2), then problem (2) is well posed. 2000 Mathematics Subject Classification. Primary: 26A33, 45N05, 65J10; Secondary: 44A35, 65R20. Key words and phrases. Fractional equations, convolution quadratures, asymptotic behaviour, sectorial kernels. Supported by DGI-MCYT under project MTM 2004-07194 cofinanced by FEDER funds.

277

278

EDUARDO CUESTA

Due to their applications in several fields of science (see [9, 10, 18, 19]), type (1) equations are attracting increasing interest as well as their numerical treatment. Properties of the solutions of (1) have been studied in several contexts, e.g. maximal regularity [19], regularity in the framework of numerical methods [4] or positivity and contractivity [6], among many others. Time discretizations of (1), including those referred to in this paper, i.e., the numerical schemes based on convolution quadratures (see [1, 4, 5, 6, 11, 12, 14, 15, 16]), have also been widely studied. Our first contribution in this paper is Theorem 1 where we show how the asymptotic behaviour of the continuous solutions of (1) depend on ω. To be more precise, if ω ≥ 0, then the continuous solutions are bounded by an exponential of ω 1/α t type  e  and if ω < 0, then the solutions show a merely algebraic decay of order 1 as t → +∞. Our second contribution concerns the asymptotic behaviour o |ω|t α of the numerical solution provided by the backward Euler convolution quadrature method as tn → +∞. In fact in Theorem 2 below we show that this numerical solution inherits the behaviour of the continuous one. The paper is organized as follows. In Section 2 we provide the two main results concerning both the behaviour of the solutions of (1) as t → +∞ and the behaviour of the numerical solution mentioned above as tn → +∞ respectively. In Section 3 we briefly comment that some extensions of these results can be achieved for time discretizations based on variable step–size quadratures of convolution equations in a more general framework, the framework of sectorial type kernels. 2. Asymptotic behaviour of the continuous and discrete solutions. Let A be a sectorial operator with 0 ≤ θ < π(1 − α/2) and let us denote the solution of (1) under this hypothesis as u. Since equation (1) can be written by means of the Laplace transform as U (λ) =

u0 1 + α AU (λ) + F (λ), λ λ

λα ∈ / ω + Sθ ,

where U and F stand for the Laplace transform of u and f respectively, the variation–of–constants formula allows us to write the solution of (1) as Z t u(t) = Eα (t)u0 + Eα (t − s)f (s) ds, 0 ≤ t ≤ T, (5) 0

where the family of operators Eα (t), for 0 ≤ t ≤ T, on X are defined by Z 1 Eα (t) := etλ λα−1 (λα − A)−1 dλ, 0 ≤ t ≤ T, 2πi γ

(6)

and γ is a suitable path lying outside the sector ω + Sθ . We are now in a position to state the first result of the paper which concerns the asymptotic behaviour of the continuous evolution operator Eα as t → +∞. We point out that from Theorem 1 and expression (5) we may easily derive the behaviour of u. For the simplicity of the notation, hereafter we denote k · k both the norm in X and the norm in the space of linear operators on X. Theorem 1. Let A : D(A) ⊂ X → X be a sectorial operator in a complex Banach space X, satisfying hypothesis(3)–(4), for some M > 0, ω ∈ R and 0 ≤ θ
0 depending solely on θ and α, such that  1/α  CM (1 + ωtα )eω t , ω ≥ 0,   kEα (t)k ≤ CM    , ω < 0, 1 + |ω|tα for t ≥ 0. Notice that when ω < 0 the continuous evolution operator Eα decays like t−α as t → +∞, although not with exponential order since the integrand in (6) does not admit analytical prolongation to (−∞, 0]. Notice also that the product ωtα , which will appear several times throughout the paper, is dimensionless. Proof of Theorem 1. One of the key points in the proof will be a suitable choice of the integration path γ in representation (6) of Eα . In particular, let γ be a positively oriented path whose support Γ is the set of λ ∈ C such that λα lies on the boundary of B1/tα , for t > 0, where Bδ = {ω+ + δ + Sθ } ∪ Sφ , δ is a positive constant, ω+ := max{0, ω}, απ/2 < φ < (π − θ) and Sφ = {λ : λ ∈ C, | arg(−λ)| < φ}. Notice that with this definition of γ, the resolvent (λα − A)−1 is well defined and therefore representation (6) of Eα is meaningful. Our proof involves studying expression (6) of Eα divided into two parts. To be more precise, let us split γ into two paths γ1 and γ2 whose supports Γ1 and Γ2 are the sets formed by the complex λ such that λα lies on the intersection of Γ and the boundaries of ω+ + 1/tα + Sθ and Sφ respectively, i.e., Γ1 = Γ ∩ {ω+ + 1/tα + Sθ }

Γ2 = Γ ∩ Sφ .

and

Therefore Γ = Γ1 ∪ Γ2 and Eα (t) = I1 (t) + I2 (t), for t ≥ 0, where Z 1 Ij (t) = etλ λα−1 (λα − A)−1 dλ, j = 1, 2. 2πi γj We first consider ω ≥ 0 and consequently ω+ = ω. In such a case the integral I1 can be bounded by Z 1 |etλ | · |λ|α−1 k(λα − A)−1 k | dλ| kI1 (t)k ≤ 2π γ1 M ≤ 2π

Z γ1

|etλ |

|λ|α−1 | dλ|. |λα − ω|

Furthermore, there exists C > 0, which will stand in the rest of the proof for a generic constant depending solely on θ and α, such that   1/α 1 tα 1 α |etλ | ≤ Ceω t , ≤ , |λ| ≤ C ω + , (7) |λα − ω| sin(θ) tα for λ ∈ Γ1 . Therefore, α

kI1 (t)k ≤ CM (1 + ωt )e

ω 1/α t

Z γ1

1 | dλ|. |λ|

280

EDUARDO CUESTA

Since 

1 long γ1 ≤ C ω + α t

1/α ,

and 

1 |λ| ≥ sin(θ) ω + α t

1/α λ ∈ Γ1 ,

,

we have 1/α

kI1 (t)k ≤ C(1 + ωtα )eω t , t ≥ 0. (8) We now proceed to bound I2 . To this end let zt and z¯t be the intersection points of the boundaries of ω+ + 1/tα + Sθ and Sφ , for which we have, |λα − ω| ≥ sin(φ)|zt |,

λ ∈ Γ2 ,

and

  1 |zt | ≥ C ω + α , t > 0. t The same bounds are also valid for the conjugate of zt , z¯t . Thus, Z 1 M |etλ | · |λ|α−1 α | dλ| kI2 (t)k ≤ 2π γ2 |λ − ω| M ≤ 2π sin(φ)|zt | ≤

≤

Z

|etλ | · |λ|α−1 | dλ|

γ2

CM tα π sin(φ)(1 + ωtα )

+∞

Z

e−t cos φ/αs sα−1 ds

0

CM , π sin(φ) cos(φ/α)α (1 + ωtα )

which shows that there exists C > 0 such that CM kI2 (t)k ≤ , t ≥ 0. (9) 1 + ωtα Therefore, the proof of the first statement readily follows from (8) and (9). Set ω < 0. The proof in this case follows by using similar bounds as in case ω ≥ 0. In fact, since tα 1 ≤ , λ ∈ Γ1 , |λα − ω| sin(θ)(1 + |ω|tα ) we have kI1 (t)k ≤ ≤

CM tα π sin(φ)(1 + |ω|tα )

Z

+∞

e−t cos φ/αs sα−1 ds

0

CM , π sin(φ) cos(φ/α)α (1 + |ω|tα )

(10)

for t ≥ 0. On the other hand, if ω < 0, then bound (9) is valid as well, now with ω = ω+ = 0, leading merely to the boundness of I2 (t), for t ≥ 0. Moreover, Z CM CM kI2 (t)k ≤ |etλ | · |λ|α−1 |dλ| ≤ , t > 0. 2π sin(φ)|ω| γ2 π sin(φ) cos(φ/α)α |ω|tα

FRACTIONAL INTEGRO–DIFFERENTIAL EQUATIONS

Therefore, there exists C > 0 such that,   CM M ≤ , kI2 (t)k ≤ C min 1, α |ω|t 1 + |ω|tα

t ≥ 0.

281

(11)

The proof of the second statement follows from (10) and (11). The second issue addressed in this paper concerns the asymptotic behaviour of a time discretization of (1) based on a convolution quadrature, to be more precise the one based on the backward Euler method. For the convenience of the reader let us recall the definition of convolution quadratures. Let us denote δ the quotient of the generating polynomials of a linear multistep method, K the Laplace transform of the convolution kernel k, and τ > 0 the time step. Given g : [0, +∞) → X, a convolution quadrature to approximate a continuous convolution k ∗ g defines as Z tn n X k(t − s)g(s) ds ' kn−j g(tj ), (12) 0

j=1

where tj = jτ , for j ≥ 0, and the weights kj are given by their generating power series, i.e.,   +∞ X δ(z) . (13) kj z j = K τ j=0 It can be proven that weights (13) admit the following integral representation Z τ δ(τ λ)−j−1 K(λ) dλ, j ≥ 1, kj = 2πi γ where γ is a suitable path connecting −i∞ and +i∞ with increasing imaginary part. In this paper we focus on the quadrature based on the backward Euler method, α−1 for which δ(λ) = 1 − λ, and the fractional equation (2), for which k(t) = tΓ(α) and K(λ) = λ−α . Therefore the quadrature can be written as Z tn n X (α) kn−j g(tj ), (14) k(t − s)g(s) ds ' 0

j=1

(α)

where the notation kj aims to clarify that weights depend on α. Since representation (2) of problem (1) involves an integral but no derivatives in time, this representation seems to be more convenient for our purposes. In fact, convolution quadrature (14) applied in (2) leads to the numerical scheme un = u0 +

n X

(α)

kn−j Auj +

j=1

n X

(1)

kn−j f (tj ),

n ≥ 1.

(15)

j=1

Let us point out that the solvability of (15) is guaranteed if τ is small enough. In particular let ε0 be a positive constant so that ωτ α ≤ ε0 < 1.

(16)

Thus it is not difficult to prove that un = Dn(α) u0 + τ

n X j=1

(α)

Dn−j f (tj ),

n ≥ 1,

(17)

282

EDUARDO CUESTA (α)

where Dn , for n ≥ 1, stands for the discrete evolution operator. Furthermore, in (α) such a case Dn can be expressed in integral form as Z 1 r(τ λ)n λα−1 (λα − A)−1 dλ, n ≥ 1, (18) Dn(α) = 2πi γ 1 and γ stands for a suitable path connecting −i∞ and +i∞ with where r(z) := 1−z increasing imaginary part (see e.g. [3, 4, 6] for more details). The next theorem is the second contribution of this paper and shows that under (α) hypothesis (16) the discrete evolution operator Dn inherits somehow the behaviour of Eα as tn → +∞. From Theorem 2 and expression (17) one can easily derive the asymptotic behaviour of the numerical solution un .

Theorem 2. Under the hypotheses of Theorem 1 there exists C > 0 and c > 1 which do not depend on n and τ , such that  cω 1/α tn  , ω ≥ 0, CM (1 + ωtα  n )e  (α) kDn k ≤ CM    , ω < 0, 1 + |ω|tα n for n ≥ 1. Proof of Theorem 2. The main difference with the proof of Theorem 1 addresses the study of the behaviour of r(τ λ)n instead of etn λ . Let us consider γ1 , γ2 and γ, and their supports Γ1 , Γ2 and Γ as in the proof of α Theorem 1 with 1/tα n instead of 1/t . Moreover, let us denote Z 1 (n) r(τ λ)n λα−1 (λα − A)−1 dλ, j = 1, 2, Ij = 2πi γj for n ≥ 1. We first consider ω ≥ 0. Since assumption (16) holds, we have Re(τ λ) < 1 and therefore  1/α 1 1 ≥ 1 − τ ω 1/α − , |1 − τ λ| ≥ 1 − τ ω + α tn n for λ ∈ Γ1 and n large enough. Besides, if ω > 0, then there exist C > 0 and c > 1 which do not depend on n, also for λ ∈ Γ1 and n large enough, such that  −n 1/α 1 1− ≤ Ce1/(1−ε0 ) , 1/α (1 − τ ω )n and (1 − τ ω

1/α −n

)

≤e

ctn ω 1/α

1

with c =

1

!

log . 1/α 1/α ε0 1 − ε0 Thus, there exist C > 0 and c > 1 which do not depend on n and τ such that |r(τ λ)n | ≤ Cectn ω

1/α

,

and the same bounds as those achieved for λα−1 (λα − ω)−1 , λ ∈ Γ1 , in the proof of Theorem 1 lead to (n)

cω kI1 k ≤ CM (1 + ωtα n )e

If ω = 0, then the bound is straightforward.

1/α

tn

,

n ≥ 1.

(19)

FRACTIONAL INTEGRO–DIFFERENTIAL EQUATIONS

283

Let R be a large enough positive constant. Since the upper and lower branch of γ2 are defined by 1/α ρ 7→ ρe±iφ/α , ρ ≥ ρ0 , 1/α

respectively, for certain ρ0 > 0, if ρ0 ≤ |λ| ≤ R1/α , then there exists a positive constant l depending on R, α and φ such that 1 ≤ e−l cos(φ/α)τ |λ| , λ ∈ Γ2 . (20) |r(τ λ)| = |1 − τ λ| Therefore, in view of (20) and the bound of (λα − ω)−1 , for λ ∈ Γ2 , in the proof of Theorem 1, we have Z |r(τ λ)|n |λ|α−1 k(λα − A)−1 k | dλ| 1/α

γ2 , ρ 0

≤

≤

≤|λ|≤R1/α

CM tα n π sin(φ)(1 + ωtα n)

Z

R1/α

e−l cos(φ/α)tn s sα−1 ds 1/α

ρ0

CM . π sin(φ)lα cos(φ/α)α (1 + ωtα n)

The last bound and the fact that 1/|1 − τ λ|n becomes negligible for λ ∈ Γ2 , |λ| ≥ R1/α , with n and R large enough lead to (n)

kI2 k ≤

CM , 1 + ωtα n

n ≥ 1.

(21)

The first statement of the theorem follows from (19) and (21). The proof of the second statement is given by using similar arguments to those used in the proof of Theorem 1 and the first statement above. The convergence of method (15) has already been studied (see e.g [3, 6, 13]). In fact it is shown there that if f is regular enough, then method (15) exhibits a first order of convergence. To be more precise, in [3] it is proved that there exists C > 0 such that  ω 1/α tn  τ, ω ≥ 0,   CM e ku(tn ) − un k ≤

  

CM τ, 1 + ωtα n (α)

We notice that the behaviour of Eα and Dn constants.

ω < 0, is somehow reflected in the error

3. Some extensions for variable step–size schemes. Let us briefly comment on some extensions of the results stated in Section 2, in fact extensions concerning more general convolution kernels, which are also of interest in applications (see e.g. [7, 8]), and variable step length quadratures. To be more precise, we have in mind convolution equations Z t k(t − s)g(s) ds, t ≥ 0, (22) 0

where k is a given convolution kernel and g : [0, +∞) → X is known at least on a certain time mesh {tn }N n=0 .

284

EDUARDO CUESTA

Let us notice that in many practical situations k is given in terms of its Laplace transform K, which is known in engineering terminology as the transfer function. In such a case k can often be represented by means of the Bromwich formula as Z 1 k(t) = eλt K(λ) dλ, t > 0, 2πi γ where γ is a suitable path connecting −i∞ and +i∞ with increasing imaginary part. Let K be a sectorial Laplace transform, meaning here that there exist ω ∈ R, 0 < θ < π/2, M > 0 and µ > 0 such that K is analytic outside the sector ω + Sθ

(23)

as defined in Section 1, and |K(λ)| ≤

M , |λ|µ

λ∈ / ω + Sθ .

(24)

It is well known that in this context there exists C > 0 such that |k(t)| ≤ CM eωt tµ−1 ,

t > 0.

Under assumptions (23)–(24) on K, it makes sense to consider discretizations of (22) by means of discrete convolution quadratures of type (12)–(13). These procedures have been studied e.g. in [15]. In fact, in [15] it is shown that the weights defined in (13) inherit the behaviour of k, i.e., with the notation in Section 2, there exists C > 0 such that kkn k ≤ CM eωtn tnµ−1 ,

n ≥ 1.

An extension of these results concerns adaptive quadratures, i.e., variable step length quadratures as the ones considered in [2]. To be more precise, given a time mesh t0 = 0 < t1 < t2 < . . . < tN −1 < tN = T , such that 0