Fractional Sturm-Liouville eigenvalue problems, II M. Dehghan1 and A. B. Mingarelli2 1 Department
arXiv:1712.09894v1 [math.CA] 28 Dec 2017
2 School
of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran. of Mathematics and Statistics, Carleton University, Ottawa, Canada
Abstract We continue the study of a non self-adjoint fractional three-term Sturm-Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left-Riemann-Liouville fractional integral under Dirichlet type boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter α, 0 < α < 1, there is a finite set of real eigenvalues and that, for α near 1/2, there may be none at all. As α → 1− we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm-Liouville problem with the composition of the operators becoming the operator of second order differentiation. Keywords: Fractional,, Sturm-Liouville,, Fractional calculus, Laplace transform,, Mittag-Leffler function,, Eigenvalues,, Asymptotics. 2010 MSC: 26A33, 34A08, 33E12, 34B10
1. Introduction This is a continuation of [6] where the results therein are extended to three-term Fractional Sturm-Liouville operators (with a potential term) formed by the composition of a left Caputo and left-Riemann-Liouville fractional integral. Specifically, the boundary value problem is of the form, −c D0α+ ◦ D0α+ y(t) + q(t)y(t) = λy(t), with boundary conditions
I01−α y(t)|t=0 = 0, +
0 < α < 1,
, 0 ≤ t ≤ 1,
and I01−α y(t)|t=1 = 0, +
where q ∈ L2 (0, 1), is a real valued unspecified potential function. We note that these are not self-adjoint problems and so there is non-real spectrum, in general. For the analogue of the Dirichlet problem described above we study the existence and asymptotic behavior of the real eigenvalues and show that for each α, 0 < α < 1, there is a finite set of real eigenvalues and that, for α near 1/2, there may be none at all. As α → 1− we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm-Liouville problem with the composition of the operators becoming the operator of second order differentiation acting on a suitable function space. 1
[email protected] 2
[email protected]
Preprint submitted to Elsevier
December 29, 2017
2. Preliminaries We recall some definitions in Fractional Calculus. We refer the reader to our previous paper [6] for further details. Definition 2.1. The left and the right Riemann-Liouville fractional integrals Iaα+ and Ibα− of order α ∈ R+ are defined by Z t 1 f (s) Iaα+ f (t) := ds, t ∈ (a, b], (2.1) Γ(α) a (t − s)1−α and
Ibα− f (t) :=
1 Γ(α)
Z
b
t
f (s) ds, (s − t)1−α
t ∈ [a, b),
(2.2)
respectively. Here Γ(α) denotes Euler’s Gamma function. The following property is easily verified. Property 2.1. For a constant C, we have Iaα+ C =
(t−a)α Γ(α+1)
· C.
Definition 2.2. The left and the right Caputo fractional derivatives c Daα+ and c Dbα− are defined by c
Daα+ f (t)
:=
Ian−α +
1 ◦ D f (t) = Γ(n − α)
and c
Dbα− f (t)
n
:= (−1)
Ibn−α −
n
Z
t
a
(−1)n ◦ D f (t) = Γ(n − α) n
f (n) (s) ds, (t − s)α−n+1
Z
b
t
t > a,
f (n) (s) ds, (s − t)α−n+1
t < b,
(2.3)
(2.4)
respectively, where f is sufficiently differentiable and n − 1 ≤ α < n.
Definition 2.3. Similarly, the left and the right Riemann-Liouville fractional derivatives Daα+ and Dbα− are defined by Z t 1 dn f (s) Daα+ f (t) := D n ◦ Ian−α f (t) = ds, t > a, (2.5) + Γ(n − α) dtn a (t − s)α−n+1 and
Dbα− f (t) := (−1)n D n ◦ Ibn−α f (t) = −
(−1)n dn Γ(n − α) dtn
Z
t
b
f (s) ds, (s − t)α−n+1
t < b,
(2.6)
respectively, where f is sufficiently differentiable and n − 1 ≤ α < n. 2.1. The Mittag-Leffler function The function Eδ (z) defined by Eδ (z) :=
∞ X
k=0
zδ , Γ(δk + 1)
(z ∈ C, ℜ(δ) > 0),
was introduced by Mittag-Leffler [16]. In particular, when δ = 1 and δ = 2, we have √ E1 (z) = ez , E2 (z) = cosh( z). 2
(2.7)
(2.8)
The generalized Mittag-Leffler function Eδ,θ (z) is defined by Eδ,θ (z) =
∞ X
k=0
zk , Γ(δk + θ)
(2.9)
where z, θ ∈ C and Re (δ) > 0. When θ = 1, Eδ,θ (z) coincides with the Mittag-Leffler function (2.7): Eδ,1 (z) = Eδ (z).
(2.10)
Two other particular cases of (2.9) are as follows: ez − 1 E1,2 (z) = , z
√ sinh( z) √ E2,2 (z) = . z
(2.11)
Further properties of this special function may be found in [7]. Property 2.2. If 0 < δ < 2 and µ ∈ ( δπ 2 , min(π, δπ)), then function Eδ,θ (z) has the following exponential expansion as |z| → ∞ (see [9]) ( P 1 1 1−θ 1 1 1 δ exp(z δ ) − N | arg(z)| ≤ µ, k=1 Γ(θ−δk) z k + O( z N +1 ), δz Eδ,θ (z) = (2.12) PN 1 1 1 µ ≤ | arg(z)| ≤ π. − k=1 Γ(θ−δk) zk + O( zN +1 ), 2.2. Laplace transform Definition 2.4. The Laplace transform of a function f (t) defined for all real numbers t ≥ 0, t stands for the time, is the function F (s) which is a unilateral transform defined by Z ∞ e−st f (t)dt, F (s) = L{f (t)} := 0
where s is the frequency parameter. Definition 2.5. The inverse Laplace transform of a function F (s) is given by the line integral f (t) = L{F (s)} :=
1 lim 2πi T →∞
Z
γ+iT
est F (s)ds
γ−iT
where the integration is done along the vertical line ℜ(s) = γ in the complex plane such that γ is greater than the real part of all singularities of F (s). Definition 2.6. The convolution of f (t) and g(t) supported on only [0, ∞) is defined by Z t f (s)g(t − s)ds, f, g : [0, ∞) → R. (f ∗ g)(t) = 0
Property 2.3. If ℜ(q) > −1, then L{tq } =
Γ(q+1) sq+1 .
Property 2.4. If ℜ(q) > −1, then L−1 {sq } =
1 tq+1 Γ(−q) .
3
Property 2.5. If f (t) is assumed to be a differentiable function and its derivative to be of exponential type, then L{f ′ (t)} = sL{f (t)} − f (0). Property 2.6. L{(f ∗ g)(t)} = L{f (t)} × L{g(t)}. δ−θ
Property 2.7. L−1 { ssδ +λ } = tθ−1 Eδ,θ (−λtδ ). Property 2.8. According to the definition of the left fractional integral (??), we can write I0α+ f (t) =
1 1 (f (t) ∗ 1−α ). Γ(α) t
So, by virtue of Properties 2.3 and 2.6, we have 1 1 L{f (t)} × L{ 1−α } Γ(α) t 1 = α L{f (t)}. s
L{I0α+ f (t)} =
Property 2.9. According to the definition of the left Caputo fractional derivative (2.3), we can write for 0 < α < 1 by using Properties 2.8 and 2.5 L{c D0α+ f (t)} = L{I01−α Df (t)} + 1 = 1−α L{Df (t)} s 1 = 1−α (sL{f (t)} − f (0)) s = sα L{f (t)} − sα−1 f (0). Property 2.10. Accoring to the definition of the left Riemann-Liouville fractional derivative (2.5), we can write for 0 < α < 1 by using Properties 2.8 and 2.5 L{D0α+ f (t)} = L{DI01−α f (t)} + = sL{I01−α f (t)} − I01−α f (t)|t=0 + + 1 = s 1−α L{f (t)} − I01−α f (t)|t=0 + s = sα L{f (t)} − I01−α f (t)|t=0 . +
3. Fractional Sturm-Liouville problem Consider the following fractional Sturm-Liouville equation −c D0α+ ◦ D0α+ y(t) + q(t)y(t) = λy(t),
0 < α < 1, t > 0.
(3.1)
where q(t) ∈ L2 (0, 1) is a potential function. Let’s define h(t) := q(t)y(t). By taking the Laplace transform on both sides of equation (3.1) and using Property 2.9, we have −sα L{D0α+ y(t)} + sα−1 D0α+ y(t)|t=0 = λL{y(t)} − L{h(t)}, 4
and then using Property 2.10, we get y(t)|t=0 + sα−1 D0α+ y(t)|t=0 = λL{y(t)} − L{h(t)}. −sα sα L{y(t)} − I01−α +
It can be readily obtained
L{y(t)} =
sα−1 1 sα I01−α y(t)|t=0 + D α+ y(t)|t=0 + 2α L{h(t)}. + 2α λ+s λ + s2α 0 s +λ
(3.2)
On the other hand, from Property 2.7 we have sα 1 } = 1−α E2α,α (−λt2α ), λ + s2α t sα−1 L−1 { } = tα E2α,α+1 (−λt2α ), λ + s2α 1 } = t2α−1 E2α,2α (−λt2α ). L−1 { λ + s2α
(3.3)
y(t) = c1 tα−1 E2α,α (−λt2α ) + c2 · tα E2α,α+1 (−λt2α ) + t2α−1 E2α,2α (−λt2α ) ∗ h(t),
(3.4)
L−1 {
So, we can write from (3.2), (3.3)
in which c1 = I01−α y(t)|t=0 and c2 = D0α+ y(t)|t=0 are given constants and ∗ is the convolution symbol. + The boundary value problem (3.5) is now considered on (3.1) subject to the Dirichlet type boundary conditions in (3.5) below, I01−α y(t)|t=0 = 0, and I01−α y(t)|t=1 = 0. (3.5) + + Since, for any z 6= 0,
Eδ,δ (z) =
1 Eδ,0 (z), z
we get
1 E2α,0 (−λt2α ). λt Hence, using (3.4) and (3.6) we can write down the general solution for (3.1) in the form Z t E2α,0 (−λ(t − s)2α ) y(t) = c1 tα−1 E2α,α (−λt2α ) + c2 tα E2α,α+1 (−λt2α ) − q(s)y(s) ds. λ(t − s) 0 t2α−1 E2α,2α (−λt2α ) = −
(3.6)
(3.7)
This is the Volterra integral equation equivalent of (3.1). Remark 1:
When α → 1, the integral equation (3.7) becomes √ √ Z t √ sin( λt) sin( λ(t − s)) ′ √ y(t) = y(0) cos( λt) + y (0) √ + q(s)y(s)ds, λ λ 0
(3.8)
which is exactly the integral equation equivalent of the classical Sturm-Liouville equation −y ′′ + q(t)y = λy for λ > 0. Remark 2:
Observe that, for each α, E2α,0 (−λ(t − s)2α ) = lim − λ(t − s) s→t−
0, if α ∈ (1/2, 1], 1, if α = 1/2.
and so, for each 1/2 < α < 1, the kernel appearing in (3.7) is uniformly bounded on [0, 1]. This agrees with the equivalent result for the classical case (3.8). 5
4. Asymptotic distribution of the eigenvalues In this section we give the asymptotic distribution of the eigenvalues when α is either very close to 1/2 from the right or very close to 1 from the left. It must be mentioned that the integral term in (3.7) tends to zero as λ → ∞, due to the fact that the kernel of this integral namely (1 − E2α,0 (−λ(t − s)2α ))/(λ(t − s)) vanishes when λ goes to infinity. Indeed, specifying the constants in (3.7) by setting, without loss of generality, c1 = 0 and c2 = 1, we get, Z t E2α,0 (−λ(t − s)2α ) y(t) = tα E2α,α+1 (−λt2α ) − q(s)y(s) ds. (4.1) λ(t − s) 0 The asymptotic estimates in Property 2.2 show that tα E2α,α+1 (−λt2α ) = O(λ−1/2 ) for all sufficiently large λ. In addition, since y is continuous on [0, 1] there is an M > 0 such that |y(t)| ≤ M , t ∈ [0, 1]. Furthermore, since q ∈ L2 (0, 1) we have q ∈ L1 (0, 1). Finally the kernel appearing in (4.1) is uniformly bounded by 1 on account of Remark 2 and the definition of the E-function. Combining these results we find 1 1 1 1 . + M ||q||1 = O √ +O y(t) = O √ λ λ λ λ where, for each α, the O-terms are uniform for t ∈ [0, 1]. Thus, by imposing the boundary conditions (3.5) in (4.1) and using the fact that I01−α {tα E2α,α+1 (−λt2α )} = tE2α,2 (−λt2α ), the eigenvalues of the original + problem (3.1)-(3.5) become the zeros of a transcendental equation of the form, 1 E2α,2 (−λ) + O √ = 0. λ For each α < 1, and close to 1, and for large λ, the real zeros of the preceding equation approach those of E2α,2 (−λ) and spread out towards the end-points of intervals of the form (4.2). For α close to 1/2 there are no zeros, the first two zeros appearing only when α ≈ 0.7325. For α larger than this critical value, the zeros appear in pairs and in intervals of the form (4.2), below. Recall that for α < 1 there are only finitely many such zeros, their number growing without bound as α → 1. We are concerned with the asymptotic behavior of these real zeros. Recall the distribution of the real zeros of E2α,2 (−λ) in [6]. There we showed that, for each n = 0, 1, 2, . . . , N ∗ − 1, where N ∗ depends on α, the interval 2α 2α ! 1 1 (2n + 23 + 2α )π )π (2n + 21 + 2α , (4.2) , In (α) := π π ) ) sin( 2α sin( 2α always contains at least two real zeros of E2α,2 (−λ). For α → 1, these intervals approach the intervals (2n + 1)2 π 2 , (2n + 2)2 π 2 ,
whose end-points are each eigenvalues of the Dirichlet problem for the classical equation −y ′′ = λ y on [0, 1]. Since each interval In contains two zeros we can denote the first of these two zeros by λ2n (α). Equation (4.2) now gives the a-priori estimate
1 (2n + 12 + 2α )π π sin( 2α )
2α
≤ λ2n (α) ≤
1 (2n + 23 + 2α )π π sin( 2α )
2α
.
(4.3)
For α close to 1 we find the approximation λ2n (α) ≈
(2n + 2)π π sin( 2α )
2α
,
from which, in the case where α → 1, we derive the classical eigenvalue asymptotics, λn ∼ n2 π 2 as n → ∞. 6
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