A LAPLACE TRANSFORM BASED KERNEL

A LAPLACE TRANSFORM BASED KERNEL REDUCTION SCHEME FOR FRACTIONAL DIFFERENTIAL EQUATIONS DANIEL BAFFET∗‡ AND JAN S. HESTHAVEN† ‡ Abstract. The nonlocal nature of the fractional integral makes the numerical treatment of fractional differential equations expensive in terms of computational effort and memory requirements. In this paper we propose a method to reduce these costs while controlling the accuracy of the scheme. This is achieved by splitting the fractional integral of a function f into a local term and a history term. Observing that the history term is a convolution of the history of f and a regular kernel, we derive a multipole approximation to the Laplace transform of the kernel. This enables that the history term be replaced by a linear combination of auxiliary variables defined as solutions to standard ordinary differential equations. We derive a priori error estimates, uniform in f , and obtain estimates on the number of auxiliary variables required to satisfy an error tolerance. The resulting formulation is discretized to produce a time-stepping method. The method is applied to some test cases to illustrate the performance of the scheme.

1. Introduction. The nonlocal nature of fractional differential equations (FDEs) makes their numerical treatment considerably more difficult than that of standard differential equations. Standard methods for FDEs require that the entire solution history is stored and used throughout the computation. This may be expensive both in terms of computational cost and memory requirements. In this paper we propose a method to address this difficulty. Only a few methods for treating this issue have been proposed. Worth mentioning due to its simplicity is the fixed memory principle [1]. The use of nested meshes [2] has also been proposed to this end. Both these methods rely on storing past information and the direct evaluation of the convolution. A different approach is proposed in [3], where nonlocal information is approximated by auxiliary variables which are advanced in time as the scheme progresses. These auxiliary variables correspond to the coefficients of an expansion in a weighted L2 space. In this paper we take a different approach, more commonly used in the derivation of absorbing or nonreflecting boundary conditions (see, e.g., [4] and the references therein). In this context, the method is designed to approximate a convolution ! t wδ (t − τ ) f (τ ) dτ , (1.1) wδ ∗ f (t) = 0

by a linear combination of solutions ψ1 , . . . , ψJ to standard ordinary differential equations (ODEs). Here wδ (t) = w(t + δ) and w(t) = t−1+α /Γ(α). In the following we refer to ψ1 , . . . , ψJ as the auxiliary variables of the scheme. However, in the context of the fractional integral in which w is the kernel, its singularity prevents the application of the existing approach directly to w0 ∗ f = w ∗ f . Indeed, the condition δ > 0 is necessary, and the resulting approximation is not uniform in δ. In our approach, we take δ to be the step-size ∆t, but for the sake of generality, we distinguish the two. For the proposed method to be cost effective, the number of auxiliary variables J must be small. To measure this, it is natural to compare J to the number of steps N = T /∆t. The analysis, however, is performed at the continuous level and provides uniform in f error estimates. The main result of this paper implies the following a priori estimate on J: for T > 0, δ > 0, and an error tolerance ε > 0, there exists an approximation operator Iδ,r with ∗ daniel.baff[email protected] ‡ MATHICSE,

´ Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland 1

A Laplace Transform Based Kernel Reduction Scheme for FDEs

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J = pm, " # 1−α p = O log δ −1 T + log log ε such that

% $ m = O log(αε)−1

∥Iδ f − Iδ,r f ∥L2 (0,T ) ≤ ε ∥Iδ f ∥L2 (0,T )

∀ f ∈ L2 (0, T ) .

(1.2)

(1.3)

Moreover, the analysis prescribes the approximation operator Iδ,r explicitly, up to the evaluation of some integrals. The rest of the paper is structured as follows. In §2 we present an overview of the method and introduce the required notation. The main results of the paper and their proofs are presented in §3. In §4 we provide details on the implementation of the kernel reduction method and the fully discrete stepping methods used in the numerical tests whereas §5 presents numerical results for two examples including a fractional Van der Pol equation. We conclude with some remarks in §6. Appendix A contains some results regarding the upper incomplete gamma function, while Appendix B states and proves a lemma on the multipole approximation of functions. 2. Overview. Consider the fractional differential equation Dα u = f (t, u) ,

(2.1a)

in (0, T ) subject to an initial condition u(0) = u0 ,

(2.1b)

where α ∈ (0, 1), f : [0, T ] × Π → Rd , Π ⊂ Rd , Dα is the Caputo α-derivative, D α u = I α u′ , and I α is the fractional integral I α f (t) =

!

1 Γ(α)

t 0

(t − τ )−1+α f (τ ) dτ .

(2.2)

(2.3)

By applying I α to (2.1a) and substituting (2.1b), we get u = u0 + I α (f ◦ u) ,

(2.4)

where f ◦ u(t) = f (t, u(t)), and in fact (2.1) is equivalent to (2.4). In the following we develop a method for the localization of the history part of the fractional integral of f ◦ u. 2.1. Kernel reduction. In what follows we present the main idea of the method and introduce the notation used subsequently. For simplicity, let f : (0, T ) → Rd and δ > 0. We begin the derivation by splitting the integral I α f (t + δ) into a local term Jδ f (t) =

!

Iδ f (t) =

!

t+δ

t

wδ (t − τ ) f (τ ) dτ

(2.5)

wδ (t − τ ) f (τ ) dτ ,

(2.6)

and a history term

0

t

A Laplace Transform Based Kernel Reduction Scheme for FDEs

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where wδ (t) = w(t + δ)

w(t) =

t−1+α ; Γ(α)

(2.7)

namely I α f (t + δ) = Jδ f (t) + Iδ f (t) .

(2.8)

The benefit of splitting the fractional integral this way is that wδ is smooth in [0, ∞). This implies that its Laplace transform w ˆδ has an appropriate asymptotic behavior as |z| → ∞, allowing for a uniform approximation by a rational function in the half plane Re z ≥ η. This is in contrast to w, whose Laplace transform w, ˆ given by w(λ) ˆ = λ−α , which does not allow for such approximation. The drawback of the decomposition is that the resulting approximation is not uniform in δ. The following relies on [5]. A summary of relevant material is provided in Appendix A. For λ ∈ C with Re λ > 0, we have ! ∞ ! ∞ e−λt w(t + δ) dt = eδλ e−λt t−1+α dt w ˆδ (λ) = 0 δ (2.9) −α δλ Γ(α, δλ) =λ e , Γ(α) where Γ(α, z) is the upper incomplete Gamma function ! ∞ Γ(α, z) = e−s sα−1 ds .

(2.10)

z

It is convenient to focus our study on $ % Γ(α, z) . ˆδ δ −1 z = z −α ez vˆ(z) = δ −α w Γ(α)

(2.11)

The main results, however, are stated for wδ . Observing that Γ(α, z) /Γ(α) = 1 − z α γ ∗ (α, z), where γ ∗ is an entire function, we find that $ % vˆ(z) = z −α − γ ∗ (α, z) ez (2.12) is analytic and single valued in C \ (−∞, 0]. Note that for Re z > 0, vˆ is the Laplace transform of v(t) = w1 (t) = δ 1−α wδ (δt) =

1 −1+α (1 + t) . Γ(α)

(2.13)

Utilizing an asymptotic expansion of the upper incomplete Gamma (A.7), we obtain vˆ(z) = Kα

N &

% $ (−1)n Γ(n + 1 − α) z −n−1 + O |z|−N −2

n=0

|z| → ∞

(2.14)

in | arg z| < π, where Kα =

sin(πα) 1 = . Γ(α) Γ(1 − α) π

(2.15)

A Laplace Transform Based Kernel Reduction Scheme for FDEs

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This suggests that vˆ can be approximated uniformly in Re z ≥ η by a sum of poles r(λ) =

J & j=1

σj . λ − λj

(2.16)

Let r' be a multipole approximation to vˆ of the form (2.16). Then r(λ) = δ α r'(δλ) is a multipole approximation to w ˆδ . Note that r can be also written in the form (2.16). It thus follows that w ˆδ fˆ ≈ rfˆ =

J &

ψˆj = (λ − λj )−1 fˆ ,

σj ψˆj

j=1

(2.17)

which yields the approximation Iδ f ≈ Iδ,r f , where Iδ,r f =

J &

σj ψj

(2.18a)

j=1

and Ψ = (ψ1T , . . . , ψPT )T is the solution to Ψ′ = ΛΨ + F

Ψ(0) = 0 .

(2.18b)

Here, F = (f T , . . . , f T )T , and Λ is a direct sum of the complex scalar matrices λj Id , j = 1, . . . , J. Replacing the history term Iδ f in (2.8) by Iδ,r f , we recover I α f (t + δ) ≈ Jδ f (t) + Iδ,r f .

(2.19)

It is left to specify the multipole approximation r to w ˆδ and derive error estimates on the approximation Iδ f ≈ Iδ,r f of the history term. These issues are addressed in the following section. 3. Kernel reduction. For simplicity, the analysis of this section is performed on vˆ (given by (2.11)), while the main results are stated for w ˆδ . One way to obtain a rational approximation to vˆ is to represent it as an integral of the form ! ρ(ζ) dζ (3.1) vˆ(λ) = λ −ζ C over a contour C in the complex plane, and to, subsequently, approximate this integral by some quadrature. This implies that different approximations may be obtained by choosing different contours and different quadratures. In the following we develop estimates on the error produced by a specific integral representation of vˆ and quadrature. This representation and quadrature are proposed in [4] for the approximation of the logarithmic derivative of a Hankel function and is based on the following representation of the incomplete Gamma function ! ∞ −α −x z α e−z x e Γ(α, z) = dx , (3.2) Γ(1 − α) 0 z+x which leads to vˆ(z) = Kα

!

∞ 0

x−α e−x dx . z+x

(3.3)

The second ingredient to the multipole approximation is a quadrature. Here we use the one proposed by Lemma 3.5 of [4]. Although different, the lemma used in this paper, presented with its proof in Appendix B, is inspired by that previous result.

A Laplace Transform Based Kernel Reduction Scheme for FDEs

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3.1. Statement of main results. The main result is as follows. Theorem 3.1. Suppose 0 < α < 1, η > 0 and δ > 0. Then, w ˆδ admits an approximation r(λ) =

p m−1 & & k=1 j=0

σkj λ − λkj

λkj = ck + rk ω j ,

ω=e

2πi m

(3.4)

such that ( ) |w ˆδ (λ) − r(λ)| ≤ Am (α) + Bp (α, δη) |w ˆδ (λ) |

(3.5a)

holds in Re λ ≥ η, where

Am (α) = A0 α−1 3−m

Bp (α, η) = B0

p η2p + 1 (1 − α) e−η(2 −1) (η2p )1+α

(3.5b)

The constants A0 and B0 are positive and independent of α, η, δ, p and m. Theorem 3.1 remains valid if Am in (3.5) is replaced by Am (α, η) = A0 Γ(α, η) 3−m .

(3.6)

Since for η > 0, Γ(α, η) < Γ(α), this is a slightly stronger result. However, this is not expected to be significant for the proposed numerical scheme, as typically δη is very small: in the present implementation we set δη = ∆t/T = N −1 , where T is the final simulation time and ∆t is the step size. Thus, Γ(α, η) is very close to Γ(α) = α−1 Γ(1 + α). Theorem 3.1 implies an estimate on the error e = Iδ f − Iδ,r f . Let δ > 0. By Theorem 3.1, there exists a sum of J = pm poles r such that (3.5) holds for all Re λ = η. Suppose f ∈ L2 (0, T ). We extend f to (0, ∞) by setting f = 0 in (T, ∞). By applying the Laplace transform L to e, we get eˆ = L(Iδ f ) − L(Iδ,r f ) = w ˆδ fˆ −

J &

σj ψˆj .

(3.7)

j=1

ˆ = (λI − Λ)−1 Fˆ and thus obtain eˆ = (w ˆδ − r) fˆ. Due to (3.5), By (2.18b), we recover Ψ we have ) ( ˆδ (λ) f (λ) | Re λ ≥ η , (3.8) |ˆ e(λ) | ≤ Am (α) + Bp (α, δη) |w which again implies

( ) ∥Iδ f − Iδ,r f ∥L2 (0,T ) ≤ eηT Am (α) + Bp (α, δη) ∥Iδ f ∥L2 (0,T )

(3.9)

by Parseval’s relation.

3.2. Proof of Theorem 3.1. The proof of Theorem 3.1 relies on a similar result for vˆ, given by Lemma 3.3 below. The proof requires the following proposition. Proposition 3.2. Let 0 < α < 1 and ! a * −α −x * ! ∞ * −α −x * *x e * *x e * * * * * V1 (a, z) = Kα V2 (a, z) = Kα (3.10) * z + x * dx * z + x * dx 0 a

A Laplace Transform Based Kernel Reduction Scheme for FDEs

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for a > 0, and 0 ̸= z ∈ C, with Re z ≥ 0. There exist constants b0 , b1 , b2 , independent of α such that for each a > 0 and z ̸= 0, with Re z ≥ 0, the following estimates hold: b0 ≤ |ˆ v (z) | Γ(α) |z|α (1 + |z|1−α )

(3.11)

V1 (a, z) ≤ b1 Γ(α) |ˆ v (z) |

(3.12)

Γ(1 − α, a) |ˆ v (z) | Γ(1 − α)

(3.13)

V2 (a, z) ≤ b2 M (a)

M (a) = max(1, a−1 ) .

Proof. Fix some 0 < α < 1 and a > 0. Estimate (3.11) follows directly from Lemma A.2. To prove (3.12), by (3.11), it suffices to show V1 (a, z) ≤

|z|α (1

b1 , + |z|1−α )

(3.14)

for some b1 , independent of α. For x > 0 and Re z ≥ 0, we have |z + x|2 = |z|2 + x2 + 2x Re z ≥ |z|2 + x2 . The Cauchy-Schwarz inequality implies |z|2 + x2 ≥

(|z| + x)2 , 2

and hence √ 1 2 ≤ . |z + x| x + |z|

(3.15)

Thus, V1 (a, z) ≤

√ 2 V1 (a, |z|) .

(3.16)

This implies that to show (3.14) for all Re z > 0, it suffices to show it is satisfied for z = y real and positive. For y > 0, we have ! a −α −x x e dx ≤ vˆ(y) , (3.17) V1 (a, y) = Kα y+x 0 and hence V1 (a, y) ≤

y α (1

Γ(α, y) 1 (1 + y 1−α )ey . 1−α +y ) Γ(α)

(3.18)

By Lemma A.3, there exists a constant b1 , independent of α, such that $ % Γ(α, y) 1 + y 1−α ey ≤ b1 . Γ(α)

(3.19)

Fig. 1. The setup in the complex plane: the half plane Re z ≥ η and the contour C covered by the disks Dk centered at ck of radii rk .

Combining (3.18) and (3.19), we find (3.14) which implies (3.12). To prove (3.13), it suffices, due to (3.11), to show that V2 (a, z) ≤

b2 Kα M (a) Γ(1 − α, a) |z|α (1 + |z|1−α )

(3.20)

holds. For x ≥ a, by (3.15), we obtain

√ √ 1 2 2 M (a) ≤ ≤ . |z + x| a + |z| 1 + |z|

(3.21)

This implies V2 (a, z) ≤

Kα M (a) Γ(1 − α, a) . 1 + |z|

(3.22)

Observing 1 |z|α + |z| 1 2 = ≤ α , 1 + |z| 1 + |z| |z|α (1 + |z|1−α ) |z| (1 + |z|1−α )

(3.23)

we recover (3.20), and the proof is complete. We are now in a position to prove an estimate on the error of the multipole approximation to vˆ. The approximation is based on the integral representation (3.3) of vˆ and Lemma B.1. In the following we sketch the construction of a multipole approximation and the argument justifying the application of Lemma B.1. We refer to Figure 1 for an illustration of the setup. For p ∈ N, define xp = xp (η) = η(2p − 1) and ! xp −α −x ! ∞ −α −x x e x e vˆ1 = Kα dx vˆ2 = Kα dx . (3.24) z + x z+x xp 0 Note that vˆ1 has the form vˆ1 (z) =

!

C

ρ(ζ) dζ , z−ζ

where ρ(ζ) = −Kα (−ζ)−α eζ , and the contour C is the segment (−xp , 0) in the complex plane directed from ζ = 0 to ζ = −xp . Also note that C is covered by the disks , + k = 1, . . . , p , (3.25a) Dk = ζ ∈ C : |ζ − ck | ≤ rk

A Laplace Transform Based Kernel Reduction Scheme for FDEs

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where rk = 2k−2 η

$ % ck = −η 3 · 2k−2 − 1 .

(3.25b)

The reader may verify that for each k = 1, . . . , p and z ∈ C with Re z ≥ η, there holds Re(z − ck ) ≥ 3rk . Lemma B.1 states that there exists an approximation (3.4) to vˆ1 , such that |ˆ v1 (z) − r(z) | ≤

3m

2 V1 (xp , z) −1

(3.26)

holds for all Re z ≥ η, where V1 is given by (3.10). We leave it to the reader to verify this. Lemma 3.3. Suppose 0 < α < 1, and η > 0. Then vˆ admits an approximation (3.4) such that ) ( v (z) | (3.27) |ˆ v (z) − r(z)| ≤ Am (α) + Bp (α, η) |ˆ

holds for all Re z ≥ η, where Am and Bp are given by (3.5b). The constants A0 and B0 are positive and independent of α, η, p and m. Proof. Let 0 < α < 1, and η > 0. For p ∈ N define xp = xp (η) = η(2p − 1), and let vˆ1 and vˆ2 be defined by (3.24). Also let r be the multipole approximation (3.4) to vˆ1 as prescribed by Lemma B.1. By the triangle inequality, we have v2 (z) | . |ˆ v (z) − r(z) | ≤ |ˆ v1 (z) − r(z) | + |ˆ

(3.28)

Next we estimate the relative contributions of each of the terms on the right hand side of (3.28). We begin with the contribution of vˆ2 . Noting |ˆ v2 (z) | ≤ V2 (xp , z) ,

(3.29)

where V2 is given by (3.10), we employ (3.13) to obtain |ˆ v2 (z)| ≤ bM (xp )

Γ(1 − α, xp ) |ˆ v (z) | Γ(1 − α)

Due to the identity (1 − α)Γ(1 − α) = Γ(2 − α), and the estimate ! ∞ −xp Γ(1 − α, xp ) = xβ−1 e−x dx ≤ x−α p e

(3.30)

(3.31)

xp

we recover v (z) | , |ˆ v2 (z)| ≤ Bp (α, η) |ˆ

(3.32)

where Bp is given by (3.5b) and B0 > 0 is independent of α, η, p and m. Next we estimate the contribution of |ˆ v1 (z) − r(z) | to estimate (3.28). By Lemma B.1, (3.26) holds in Re z ≥ η. Thus we evoke (3.12), to obtain |ˆ v1 (z) − r(z)| ≤

b Γ(α) |ˆ v (z) | , 3m − 1

(3.33)

which yields |ˆ v1 (z) − r(z)| ≤ Am (α) |ˆ v (z) | ,

(3.34)

A Laplace Transform Based Kernel Reduction Scheme for FDEs

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where Am is given by (3.5b), and A0 > 0 is independent of α, η, p and m. This concludes the proof. This now allows us to complete the proof of Theorem 3.1. Proof. (Theorem 3.1). Let η' = δη. By Lemma 3.3, there exists r' of the form (3.4), such that (3.27) holds for all Re z ≥ η'. Namely, ( ) |ˆ v (δλ) − r'(δλ) | ≤ Am (α) + Bp (α, δη) |ˆ v (δλ) | (3.35) holds for all Re λ ≥ η. We multiply (3.35) by δ α and use w ˆδ (λ) = δ α vˆ(δλ) to obtain (3.5), with r(λ) = δ α r'(δλ) and the theorem follows.

4. Implementation. The scheme has two main components, comprising the kernel reduction scheme to deal with the time history and the local time-stepping scheme. We shall discuss these separately in the following. 4.1. Kernel reduction scheme. See Figure 1 for an illustration of the setup. We have ! δ α sin (πα) ∞ x−α e−x dx w ˆδ (λ) = δ α vˆ(δλ) = π δλ + x 0 (4.1) ! sin (πα) ∞ x−α e−δx dx = π λ+x 0 For η > 0, and p ∈ N, define xk (η) = η(2k − 1)

k = 0, . . . , p ,

(4.2)

Ik = (xk−1 (η) , xk (η))

k = 1, . . . , p .

(4.3)

and

Let rk and |ck | (ck < 0), denote the radius and center of Ik . The resulting expansion is given by r(λ) =

p m−1 & & k=1 j=0

σkj λ − λkj

λkj = ck + rk ω j

(4.4a)

where ω = e2πi/m , for each k = 1, . . . , p and j = 0, . . . , m − 1, m−1 1 & −jl ω Qkl m l=0 " #l ! −x + |ck | sin(πα) = x−α e−δx dx π rk Ik

σkj =

(4.4b)

Qkl

(4.4c)

and rk = 2k−2 η

$ % |ck | = η 3 · 2k−2 − 1 .

(4.4d)

In our implementation, each Qkl is approximated by an appropriate Gauss-Jacobi quadrature. To reduce the number of auxiliary variables we use the fact that the poles are

A Laplace Transform Based Kernel Reduction Scheme for FDEs

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distributed symmetrically around the real line. Thus we can remove poles with negative imaginary part and real duplicate poles to recover p m−1 & &

σkj ψkj = Re

k=1 j=0

P &

θj ψj .

(4.5)

j=1

This may be done by Algorithm 1. Remark 1. Algorithm 1 treats the cases where m is even or odd differently. The resulting number of poles P in the current implementation is given by P = pm/2 + 1 if m is even and P = p(m+1)/2 if m is odd. This has an implication: let m be an odd number. The accuracy of the scheme is determined by the effective number of poles J = pm, and the computational cost is that of computing P = p(m+1)/2 auxiliary variables. However, replacing m by m + 1 provides the accuracy obtained by J1 = p(m + 1) = J + p poles at the cost of computing only P1 = p(m + 1)/2 + 1 = P + 1 auxiliary variables. That is, when m is odd, the added computational cost of increasing m by one is small compared to the potential added accuracy. To avoid the appearance of different behaviors of errors with respect to P and the switching between the two different algorithms (when m is odd or even), in this paper, the numerical results are obtained with odd m. 4.2. Time-stepping scheme. Consider the initial value problem (2.1). We write the solution u at t+∆t as (2.4), split the fractional integral I α as in (2.8), and replace the history term I∆t by an approximation I∆t,r to obtain the semi-discrete approximation ν(t + ∆t) = J∆t (f ◦ ν)(t) + u0 + I∆t,r (f ◦ ν)(t) .

(4.6)

In the following we discuss a method for the discretization of (4.6). We denote by v and Φ = (ϕj ) as the fully discrete approximations of u and Ψ = (ψj ), respectively. To approximate J∆t (f ◦ ν), we use a 1-step implicit scheme, thus v n+1 = ∆tα

1 &

ak f n+k + H n

H n = u0 + Re

P &

θj ϕnj ,

(4.7)

j=1

k=0

where a0 =

α Γ(2 + α)

a1 =

1 . Γ(2 + α)

(4.8)

This scheme requires the solution of an algebraic equation; this is done by a Newton solver. Since some of the λj , chosen for the approximation of the history term, have large negative real parts, we use a 1-step A-stable method to approximate (2.18b). Note that coupling in (2.18b) may occur only through F . Also note that F is only a function of v, and v n+1 depends only on Φn . Thus, applying 1-step implicit schemes to (2.18b) is simple (recall that Λ is diagonal). In this paper we test the application of trapezoidal rule, #−1 -" # . " % ∆t ∆t $ n+1 ∆t Λ Λ Φn + F (4.9) + Fn Φn+1 = I − I+ 2 2 2 and the backward Euler scheme

−1

Φn+1 = (I − ∆tΛ)

$ n % Φ + ∆tF n+1 ,

(4.10)

A Laplace Transform Based Kernel Reduction Scheme for FDEs

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Algorithm 1 Removing Poles Input: λkj = ck + rk ω j and σkj defined by (4.4b) (k = 1, . . . , p, j = 0, . . . , m − 1). Output: θ = (θ1 , . . . , θP ) and λ = (λ1 , . . . , λP ) 1: if m is even then 2: Let m = 2µ. 3: σ (tmp) = (σkµ : 1 ≤ k ≤ p) 4: λ(tmp) = (λkµ : 1 ≤ k ≤ p) 5: Remove poles with negative imaginary parts: λ ← (λkj : 1 ≤ k ≤ p ,

0 ≤ j ≤ µ − 1)

Remove corresponding weights:

6:

σ ← (σkj : 1 ≤ k ≤ p ,

0 ≤ j ≤ µ − 1)

Correct weights corresponding complex poles:

7:

σkj ← 2σkj

1 ≤ k ≤ p,

1≤j ≤µ−1

Correct weights corresponding real poles:

8:

(tmp)

σk0 ← σk0 + σk−1 9: 10: 11: 12: 13: 14: 15:

2≤k≤p

σ ← reshape(σ, 1, pµ) (Rearrange the matrix σ = (σkj ) as a line vector (σj ).) λ ← reshape(λ, 1, pµ) (tmp) Define θ = (σ, σp ) (P = pm/2 + 1) (tmp) λ ← (λ, λp ) else (m is odd) Let m = 2µ − 1. Remove poles with negative imaginary parts: λ ← (λkj : 1 ≤ k ≤ p ,

16:

Remove corresponding weights: σ ← (σkj : 1 ≤ k ≤ p ,

17:

0 ≤ j ≤ µ − 1)

0 ≤ j ≤ µ − 1)

Correct weights corresponding complex poles: σkj ← 2σkj

18: Define θ = reshape(σ, 1, pµ) 19: λ ← reshape(λ, 1, pµ) 20: end if

1 ≤ k ≤ p,

1≤j ≤µ−1

(P = p(m + 1)/2)

where F n = F (tn , v n ) .

(4.11)

Remark 2. The fact that the backward Euler scheme can be used to compute the auxiliary variables is of particular importance. It suggests that diagonally implicit Runge-Kutta methods may be employed to obtain high-order multi-stage methods. To achieve this, a high-order multi-stage method must also be used instead of (4.7). 5. Numerical tests. We seek approximate solutions to an initial value problem (2.1) in an interval (0, T ). In the following, ∆t is the step size, N = T /∆t, p is the number

A Laplace Transform Based Kernel Reduction Scheme for FDEs α=0.5 u0=1

−1

α=0.5 u0=1

−1

10

12

10

−2

∆t=1.0e−03 , ∆t=1.0e−03 , ∆t=1.0e−03 , ∆t=1.0e−03 , ∆t=1.0e−03 ,

−2

10

10

−3

−3

10

10

−4

P=60 (p=15 , m=7) P=75 (p=15 , m=9) P=90 (p=15 , m=11) P=105 (p=15 , m=13) P=120 (p=15 , m=15)

−4

eh

10

eh

10

−5

−5

10

10

−6

10

−7

10

−8

10

0

−6

∆t=1.0e−02 , ∆t=1.0e−02 , ∆t=1.0e−02 , ∆t=1.0e−02 , ∆t=1.0e−02 , 1

P=22 P=33 P=44 P=55 P=66

(p=11 , (p=11 , (p=11 , (p=11 , (p=11 ,

10

m=3) m=5) m=7) m=9) m=11)

2

3

−7

10

−8

4

10

5

0

1

2

t

3

4

5

t

Fig. 2. Accuracy tests for problem (5.1); the local error eh as a function of t; auxiliary variables discretized by the trapezoidal rule.

of circles, m is the number of poles per circle, and P is the number of auxiliary variables used in the calculation, excluding unnecessary poles. Unless mentioned otherwise, the scheme used for approximation of the auxiliary variables is the trapezoidal rule (4.9). 5.1. The Mittag-Leffler function. We consider the initial value problem Dα u = −u

u(0) = u0 = 1

(5.1)

in (0, T ). Its solution is given by u(t) = Eα (−tα )

Eα (t) =

∞ &

k=0

tk . Γ(αk + 1)

(5.2)

The results are obtained with α = 0.5, T = 5, and η = 1/T = 0.2. Figures 2 show the local error eh = |u − v| as a function of time. The results are obtained by discretizing the auxiliary variables by the trapezoidal rule (4.9). Figure 2 shows that as P grows, the error is reduced until the error is saturated by the local discretization error. Then, the error can only be reduced by refining the step-size. Figure 3 shows the local error eh as a function of t, where the auxiliary variables are discretized by the backward Euler scheme (4.10), showing a similar behavior of the error to the one illustrated in Figure 2. Here, however, the errors are larger and become saturated by the discretization error at a significantly smaller P . In Figure 4 we retain only the graphs corresponding to the errors obtained with the highest number of auxiliary variables, and compare the results obtained when discretizing the auxiliary variables by the trapezoidal rule (solid lines) and backward Euler scheme (dashed lines). To test the validity of the theoretical estimates on p and m we set ∆t = 10−4 and compute the 2-norm of the error ∥eh ∥2 , where ∥v∥22 = h

N &

n=0

|v n |2 .

Figure 5 shows ∥eh ∥2 as a function of m and p. The dashed lines correspond to the theoretical estimate (3.9). Here, 'm = α−1 3−m A

'p = (1 − α) yp + 1 e−yp B yp1+α

yp = N −1 2p

(5.3)

A Laplace Transform Based Kernel Reduction Scheme for FDEs α=0.5 u =1

α=0.5 u =1

0

−1

13

0

−1

10

10

−2

∆t=1.0e−03 , ∆t=1.0e−03 , ∆t=1.0e−03 , ∆t=1.0e−03 , ∆t=1.0e−03 ,

−2

10

10

−3

−3

10

10

−4

P=60 (p=15 , m=7) P=75 (p=15 , m=9) P=90 (p=15 , m=11) P=105 (p=15 , m=13) P=120 (p=15 , m=15)

−4

eh

10

eh

10

−5

−5

10

10

−6

10

−7

10

−8

10

0

−6

∆t=1.0e−02 , ∆t=1.0e−02 , ∆t=1.0e−02 , ∆t=1.0e−02 , ∆t=1.0e−02 , 1

P=22 P=33 P=44 P=55 P=66

(p=11 , (p=11 , (p=11 , (p=11 , (p=11 ,

10

m=3) m=5) m=7) m=9) m=11)

2

−7

10

−8

3

4

10

5

0

1

2

t

3

4

5

t

Fig. 3. Accuracy tests for problem (5.1); the local error eh as a function of t; auxiliary variables discretized by the backward Euler scheme. α=0.5 u0=1

−1

10

TR TR BE BE

−2

10

∆t=1.0e−02 , ∆t=1.0e−03 , ∆t=1.0e−02 , ∆t=1.0e−03 ,

P=66 (p=11 , m=11) P=120 (p=15 , m=15) P=66 (p=11 , m=11) P=120 (p=15 , m=15)

−3

10

−4

eh

10

−5

10

−6

10

−7

10

−8

10

0

1

2

3

4

5

t

Fig. 4. Accuracy tests for problem (5.1); the local error eh as a function of t. A comparison of the errors produced by the two methods.

$ % substitute for Am (α) and Bp α, ∆tT −1 , respectively. We remark that estimate (3.9) is of the 2-norm of (I∆t − I∆t,r )f (on the continuous level), while Figure 5 shows the error of the (discrete) numerical solution. Therefore the results shown in Figure 5 should be compared to estimates derived from a convergence analysis, which may differ from the present estimates. Nevertheless, Figure 5 shows that (3.9) predicts the error’s decay rate well. 5.2. Fractional Van der Pol equation. Consider the nonlinear fractional differential equation % $ (Dα )2 x − ε 1 − x2 Dα x + x = 0

(5.4a)

in (0, T ), with initial conditions

x(0) = x0

Dα x(0) = y0 .

(5.4b)

Here ε is a non-negative constant, and x0 , y0 ∈ R. For α = 1, (5.4a) is reduced to the classical Van der Pol equation. In this case it can be shown to have a stable periodic solution. To apply the scheme we write (5.4a) as a system by substituting y = Dα u.

A Laplace Transform Based Kernel Reduction Scheme for FDEs α=0.5 u =1 ∆t=1.0e−04

α=0.5 u =1 ∆t=1.0e−04

0

−1

14

0

0

10

10 p=10 p=11 p=12 p=13 p=14 p=15 p=16 p=17 p=18 p=19

−2

10

−3

||eh||2

10

−1

m= 3 m= 5 m= 7 m= 9 m=11 m=13 m=15 m=17 m=19

!p B

10

−2

10 ||eh||2

!m A

−3

10

−4

10

−4

10 −5

10

−5

10

−6

10

−6

0

5

10 m

15

20

10

10

12

14

16

18

20

p

Fig. 5. Accuracy tests for problem (5.1) – verification of estimate (1.2). α=0.8 u =(2 0)T ∆t=1.0e−04 0

α=0.8 u =(2 0)T 0

−2

10

6 4

−3

10

2

−4

10 0

−5

10 −2

−6

10 −4

−7

10

−6 −8 0

∆t=1.0e−02 ∆t=1.0e−03

−8

2

4

6 t

8

10

12

10

0

2

4

6 t

8

10

12

Fig. 6. Fractional Van der Pol equation (5.5); on the left is a reference solution vref computed with p = m = 25 and ∆t = 10−4 ; on the right, numerical solutions computed with p = m = 25 and different ∆t are compared with vref .

Thus, we have Dα x = y

$ % D y = ε 1 − x2 y − x , α

(5.5a) (5.5b)

in (0, T ) subject to the initial condition x(0) = x0

y(0) = y0 .

(5.5c)

In the tests below, we fix α = 0.8, ε = 4, T = 12 and η = 1/T . First we look at the affect of the step-size on the error. Hence, we choose p and m sufficiently large, and compare numerical solutions computed with different ∆t. In this test we use p = 25 and m = 25. Figure 6 shows, on the left the reference solution computed ∆t = 10−4 , and on the right, the difference between the reference solution and other numerical solutions. In Figure 7, numerical solutions computed with different parameters p and m are compared to the reference solution computed with p = 25, m = 25 and ∆t = 10−4 . The figure shows E = ∥v − vref ∥2 as a function of m; the different graphs correspond to different p.

A Laplace Transform Based Kernel Reduction Scheme for FDEs α=0.8 u0=(2 0)T ∆t=1.0e−02

0

E

!m A

p=10 p=11 p=12 p=13 p=14 p=15 p=16 p=17 p=18 p=19

−1

10

−2

10

E

−1

−2

10

p=10 p=11 p=12 p=13 p=14 p=15 p=16 p=17 p=18 p=19

10

10

α=0.8 u0=(2 0)T ∆t=1.0e−03

0

10

15

!m A

−3

10

−4

10 −3

10

−5

10

−4

10

0

−6

5

10 m

15

10

20

0

5

10 m

15

20

Fig. 7. Fractional Van der Pol equation (5.5); numerical solutions computed with ∆t = 10−2 (left) and ∆t = 10−3 (right) are compared with vref . Here E = ∥v − vref ∥2 , where vref is a reference solution computed with p = m = 25 and ∆t = 10−4 . α=0.8 u0=(2 0)T ∆t=1.0e−02

0

p=10 p=11 p=12 p=13 p=14 p=15 p=16 p=17 p=18 p=19

−4

E

10

p=10 p=11 p=12 p=13 p=14 p=15 p=16 p=17 p=18 p=19

−2

10

−4

10 E

−2

10

−6

−6

10

10

−8

−8

10

10

−10

−10

10

α=0.8 u0=(2 0)T ∆t=1.0e−03

0

10

10

0

5

10 m

15

20

10

0

5

10 m

15

20

Fig. 8. Fractional Van der Pol equation (5.5); numerical solutions computed with ∆t = 10−2 (left) and ∆t = 10−3 (right) are compared with vref . Here E = ∥v − vref ∥2 , where vref is a reference solution computed with p = m = 25 and the same step-size as v, i.e., ∆t = 10−2 on the left and ∆t = 10−3 on the right.

Finally, we look at how the error of the history term behaves for numerical solutions. To eliminate the discretization error we compare the numerical solutions to reference solutions computed with the same step-size but large p and m. We repeat the test for two step-sizes ∆t = 10−2 and ∆t = 10−3 . The reference solutions are computed with p = 25 and m = 25. Figure 8 shows E as a function of m; the different graphs correspond different values of p. 6. Concluding remarks. In this work we propose a method for the localization of the convolution (1.1).in the fractional differential equation. We present a priori error estimates, showing that for a given time interval (0, T ), δ > 0 and error tolerance ε > 0, the number of auxiliary variables required to achieve (1.3) is J = pm, with p and m satisfying (1.2). We have tested the performance of the scheme on two problems: corresponding to the Mittag-Leffler function, and a fractional Van der Pol equation. The results illustrate the strength of the method, i.e., its ability to reduce the memory and computational costs. This will be of particular importance for problems requiring long time integration. For applications, however, methods are required to possess some additional features such as adaptive step-size and high-order convergence. We leave the

A Laplace Transform Based Kernel Reduction Scheme for FDEs

16

treatment of these issues to future work. Appendix A. The incomplete Gamma function. The analysis of the paper relies on properties of the upper incomplete Gamma function. Some properties used are derived in the literature for the more general confluent hypergeometric functions. In this section we state some results regarding the incomplete Gamma function, and confluent hypergeometric functions. At the end of this section, we prove some lemmas required for the proof of Theorem 3.1. The following is based on [5] and [6]. The upper incomplete Gamma function Γ(α, z) can be written in terms of a confluent hypergeometric function [5], Γ(α, z) = e−z U (1 − α, 1 − α, z)

(A.1)

(in [6], the function U is denoted by Ψ). For a > 0 and Re z > 0, U is given by ! ∞ 1 e−zx xa−1 (1 + x)b−a−1 dx (A.2a) U (a, b, z) = Γ(a) 0 ! ∞ ez a−1 b−a−1 = e−zx (x − 1) x dx . (A.2b) Γ(a) 1 Combining (A.1) and (A.2a), for α ∈ (0, 1) and Re z > 0,

! ∞ −α −zx e−z x e dx Γ(α, z) = Γ(1 − α) 0 1+x ! ∞ −α −x z α e−z x e = dx . Γ(1 − α) 0 z+x

(A.3a) (A.3b)

There holds Γ(α, z) = Γ(α) (1 − z α γ ∗ (α, z)) ,

(A.4)

where γ ∗ (α, z) = e−z

∞ 1 & (−z)n zn = Γ(α + n + 1) Γ(α) n=0 (α + n)n! n=0 ∞ &

(A.5)

is entire and single valued. The upper incomplete Gamma function satisfies the recurrence relation Γ(1 + α, x) = αΓ(α, x) + xα e−x .

(A.6)

In | arg z| < 3π/2, the incomplete Gamma function admits the asymptotic expansion z 1−α ez Γ(α, z) =

N &

(−1)n

n=0

% $ Γ(n + 1 − α) −n z + O |z|−N −1 Γ(1 − α)

|z| → ∞ .

(A.7)

By [7], (which relies on [6, pp. 287, Eq. (21)]) and (A.1), for 0 ≤ α < 2 and z ̸= 0, we have ! ∞ Γ(α, z) Γ(α, z) = µ(α, z, t) dt , (A.8) 0

A Laplace Transform Based Kernel Reduction Scheme for FDEs

17

where . * e−t t1−α 1 1 − α, 1 − α * t(t + 2 Re z) µ(α, z, t) = * 2 F1 2−α Γ(2 − α) |t + z|2(1−α) |t + z|2

and

∞ * . & a, b * µn (a, b, c) z n *z = 2 F1 c

-

(A.9)

(A.10)

n=0

is a hypergeometric series. The particular values of the coefficients µn are not important in this context, only that µ0 = 1, µn (a, a, c) ≥ 0, provided c > 0, and that (A.8) is well defined. Following [7], we note that for Re z ≥ 0, α ∈ [0, 2) and t > 0, there holds µ(α, z, t) > 0, and hence Γ(α, z) Γ(α, z) > 0 .

(A.11)

This implies that Γ(α, ·) has no roots z ̸= 0 with Re z ≥ 0. Since z = 0 is also not a root of Γ(α, ·), we conclude that Γ(α, ·) has no roots z with Re z ≥ 0. Lemma A.1. There exist constants c > 0 and δ > 0, such that for each α ∈ [0, 1] and 0 ̸= z ∈ C satisfying Re z ≥ 0 and |z| ≤ δ, there holds |Γ(α, z) | > c .

(A.12)

Proof. Let 0 < α ≤ 1. Owing to (A.4) and (A.5), for |z| → 0, we have Γ(α, z) = Γ(α) − Thus,

$ % zα + O |z|1+α . α

* * * z α ** * − c0 |z|1+α |Γ(α, z) | ≥ *Γ(α) − α*

(A.13)

(A.14)

holds for some constant c0 > 0. Let

ρ(α, x) = Γ(α) −

Γ(1 + α) − xα xα = . α α

(A.15)

It is readily verified that |ρ(α, z) | ≥ |ρ(α, |z|) | .

(A.16)

Thus, in the following, we consider real nonnegative x. Define x0 (β) = Γ(1 + β)1/β , for β ∈ [0, 1]. Clearly, x0 is continuous (up to a removable singularity at zero) and positive in ∂ρ β ∈ [0, 1]. In addition, for each 0 < x < x0 (β), there holds ρ(β, x) ≥ 0 and ∂x (β, x) < 0. Now define x1 = inf

β∈[0,1]

x0 (β) . 2

(A.17)

Due to the above, 0 < x1 < x0 (β), for all β ∈ [0, 1]. As x1 < x0 , we have that for each 0 < x ≤ x1 , there holds ρ(α, x) ≥ ρ(α, x1 ) ≥ Γ(1 + α)

2α − 1 ≥ c1 , α2α

(A.18)

A Laplace Transform Based Kernel Reduction Scheme for FDEs

18

where c1 is independent of α. It follows that for each |z| ≤ x1 , there holds |Γ(α, z) | ≥ ρ(α, |z|) − c0 |z|1+α ≥ c1 − c0 |z|1+α .

(A.19)

Thus, for sufficiently small δ, (A.12) holds for all |z| ≤ δ. To obtain (A.12) also for α = 0, we rely on the continuity of Γ(·, z) for all z ̸= 0 with Re z ≥ 0. Lemma A.2. Let % $ (A.20) ν(α, z) = 1 + z 1−α ez Γ(α, z) . There exists c > 0, such that for each α ∈ [0, 1] and 0 ̸= z ∈ C with Re z ≥ 0, there holds |ν(α, z) | > c .

(A.21)

Proof. To prove this lemma, we show that (A.21) holds in three regions of the complex half plane Re z ≥ 0: a half disk |z| < r, an unbounded half annulus |z| > R, and the half annulus r ≤ |z| ≤ R. For each Re z ≥ 0 there holds $ % (A.22) |ez 1 + z 1−α | ≥ eRe z |1 + Re z| ≥ 1 , and we have

|ν(α, z) | ≥ |Γ(α, z) | .

(A.23)

Lemma A.1 states that there exist c > 0 and r > 0 such that (A.21) holds for all α ∈ [0, 1] and |z| ≤ r. Next we show that there exists R > r, such that (A.21) with c = 1/2 holds for all |z| > R with Re z ≥ 0. Suppose 0 < α ≤ 1. By (A.7), for |z| → ∞, %$ $ %% $ ν(α, z) = 1 + z α−1 1 + O |z|−1

| arg z| ≤

3π . 2

(A.24)

It follows that there exists R > r, independent of α, such that |ν(α, z) | ≥ 1/2 holds for all |z| > R, with | arg z| ≤ π/2. Due to the continuity of ν(·, z) at each z with Re z ≥ 0 and |z| > R, (A.21) also holds for α = 0. Finally, we use the continuity of |ν| in the compact set [0, 1] × T , with / π0 , T = z ∈ C : r ≤ |z| ≤ R , | arg z| ≤ 2

to obtain (A.7). By (A.11), which holds in [0, 1] × T , we recover |ν| > 0 in [0, 1] × T . Since [0, 1] × T is compact, and |ν| is continuous on [0, 1] × T , there exists c > 0 such that (A.21) holds in [0, 1] × T . The rest of the proof is straightforward, thus completing the result. Lemma A.3. Let µ be defined by (A.20). There exists a constant c > 0, such that for each α ∈ (0, 1] and 0 < y ∈ R, 0 < ν(α, y) ≤ cΓ(α) .

(A.25)

Proof. By (A.7), for |z| → ∞, ν(α, y) = 1 + y −1+α + O(y) .

(A.26)

A Laplace Transform Based Kernel Reduction Scheme for FDEs

19

Thus, there exist constants c1 > 0 and y0 , independent of α, such that (A.25) with c = c1 holds for all y ≥ y0 . In y ∈ [0, y0 ], owing to 0 < Γ(α, y) ≤ Γ(α)

(A.27)

the conclusion is clear, and we have the lemma. Appendix B. Multipole approximation. The following is inspired by Lemma 3.5 of [4]. Lemma B.1. Suppose C is a contour in the complex plane, 1 for each k = 1, . . . , p, Dk is a closed disk of radius rk > 0 centered at ck ∈ C, Uk = j≤k Dj (k = 0, . . . , p), C ⊂ Up , Ck = C ∩ (Dk \ Uk−1 ), and ! ρ(ζ) dζ , (B.1) f (z) = C z−ζ where ρ is integrable. Let a > 1 and g(z) =

p m−1 & & k=1 j=0

σkj , z − (ck + rk ω j )

(B.2a)

where ω = e2πi/m , σkj =

m−1 1 & −jl ω Qkl m

Qkl =

!

ρ(ζ)

Ck

l=0

"

ζ − ck rk

#l

dζ .

(B.2b)

Then, the estimate |f (z) − g(z) | ≤

2F (z) , am − 1

(B.3)

with F (z) =

* ! * * ρ(ζ) * * * * z − ζ * d|ζ|

(B.4)

C

holds for every z satisfying Re(z − ck ) ≥ ark for all k = 1, . . . , p. Proof. Let f=

p &

fk

fk (z) =

Ck

k=1

and similarly, F =

2

k

Fk , g =

2

!

k

ρ(ζ) dζ z−ζ

(B.5)

gk . Fix some 1 ≤ k ≤ p. By the identity

m−1 & ys 1 ym 1 + = , x−y xs+1 xm x − y s=0

(B.6)

we have fk (z) =

m−1 & s=0

rks Qks + (z − ck )s+1

"

rk z − ck

#m !

Ck

ρ(ζ) z−ζ

"

ζ − ck rk

#m

dζ .

(B.7)

A Laplace Transform Based Kernel Reduction Scheme for FDEs

20

Similarly, gk (z) =

m−1 & s=0

" #m m−1 m−1 & & rk rks σkj ω jm js . σ ω + kj s+1 (z − ck ) z − ck z − (ck + rk ω j ) j=0 j=0

(B.8)

Observing that ω m = 1 and m−1 &

σkj ω js = Qks ,

(B.9)

j=0

we find gk (z) =

m−1 & s=0

rks Qks + (z − ck )s+1

"

rk z − ck

#m

gk .

(B.10)

Therefore, |fk (z) − gk (z) | ≤ a

−m

Since, *! * * *

Ck

we get

ρ(ζ) z−ζ

"

ζ − ck rk

*! * * *

Ck

#m

ρ(ζ) z−ζ

"

ζ − ck rk

* ! * dζ ** ≤

Ck

#m

* * dζ ** + a−m |gk | .

* * * ρ(ζ) * * * * z − ζ * d|ζ| = Fk (z)

|fk (z) − gk (z) | ≤ a−m (Fk (z) + |gk (z) |)

≤ a−m Fk (z) + a−m (|fk (z) − gk (z) | + |fk (z) |) .

(B.11)

(B.12)

(B.13)

As |fk (z) | ≤ Fk (z), we get |fk (z) − gk (z) | ≤

2Fk (z) , am − 1

(B.14)

and hence the conclusion. REFERENCES [1] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [2] N. J. Ford, A. C. Simpson, The Numerical Solution of Fractional Differential Equations: Speed Versus Accuracy, Numer. Algorithms, Vol. 6, No. 4, 333-346. [3] D. Baffet, J. S. Hesthaven, High-Order Accurate Local Schemes for Fractional Differential Equations, J. Sci. Comput., DOI: 10.1007/s10915-015-0089-1. [4] B. Alpert, L. Greengard, T. Hagstrom, Rapid Evaluation of Nonreflecting Boundary Kernels for Time-Domain Wave Propagation, SIAM J. Numer. Anal., Vol. 37, No. 4, pp. 1138-1164. [5] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Tenth Printing, 1972, National Bureau of Standards. [6] A. Erd´ elyi Higher Transcendental Functions, Vol. 1, McGraw-Hill, 1953. [7] J. Wimp, On the Zeros of a Confluent Hypergeometric Function, Proc. Amer. Math. Soc., Vol. 16, No. 2 (April., 1965), pp. 281-283.