Feb 11, 2015 - k=0 kα to determine an approximation for the fractional integral of order 2 â α and the second order approximation for the Caputo derivative.
Journal of Fractional Calculus and Applications, Vol. 7(2) July 2016, pp. 175-195. ISSN: 2090-5858. http://fcag-egypt.com/Journals/JFCA/ ————————————————————————————————
A SECOND ORDER APPROXIMATION FOR THE CAPUTO FRACTIONAL DERIVATIVE YURI DIMITROV
Abstract. When 0 < α < 1, the approximation for the Caputo derivative y (α) (x) = (α)
where σ0
(α)
= 1, σn
n X 1 (α) σk y(x − kh) + O h2−α , α Γ(2 − α)h k=0
= (n − 1)1−a − n1−a and
(α)
= (k − 1)1−α − 2k1−a + (k + 1)1−α , (k = 1 · · · , n − 1), Pn α has accuracy O h2−α . We use the expansion of k=0 k to determine an approximation for the fractional integral of order 2 − α and the second order approximation for the Caputo derivative σk
y (α) (x) = (α)
where δk (α) δ0
=
(α)
= σk
(α) σ0
n X 1 (α) δ y(x − kh) + O h2 , α Γ(2 − α)h k=0 k
for 2 ≤ k ≤ n, (α)
− ζ(α − 1), δ1
(α)
= σ1
(α)
+ 2ζ(α − 1), δ2
(α)
= σ2
− ζ(α − 1),
and ζ(s) is the Riemann zeta function. The numerical solutions of the fractional relaxation and subdiffusion equations are computed.
1. Introduction Fractional differential equations are used for modeling complex diffusion processes in science and engineering [1–5]. The Caputo fractional derivatives are important as a tool for describing nature as well as for their relation to integer order derivatives and special functions. The Caputo derivative of order α, when 0 < α < 1, is defined as the convolution of the power function x−a and the first derivative of the function on the interval [0, x] Z x 1 y 0 (ξ) dα y(x) = dξ. y (α) (x) = Dα y(x) = α dx Γ(1 − α) 0 (x − ξ)α When the function y(x) is defined on the interval (−∞, x], the lower limit of the integral in the definition of Caputo derivative is −∞. The Caputo derivative of the 2000 Mathematics Subject Classification. 26A33, 34E05, 33F05, 26A33. Key words and phrases. Fractional derivative, fractional integral, approximation, numerical solution, fractional differential equation. Submitted Feb 11, 2015. 175
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constant function 1 is zero, and Dα D1−α y(x) = y 0 (x). While the integer order derivatives describe the local behavior of a function, the fractional derivative y (α) (x) depends on the values of the function on the interval [0, x]. One approach for discretizing the Caputo derivative is to divide the interval to subintervals of small length and approximate the values of the function on each subinterval with a Lagrange polynomial. Let xn = nh and yn = y(xn ) = y(nh), where h > 0 is a small number. The Lagrange polynomial for the function y 0 (x) at the midpoint xk−0.5 of the interval [xk−1 , xk ] is the value of y 0 (xk−0.5 ). Approximation (1) for the Caputo fractional derivative is a commonly used approximation for numerical solutions of ordinary and partial fractional differential equations [6-8]. Z xn n Z xk X y 0 (xk−0.5 ) y 0 (ξ) dξ ≈ dξ Γ(1 − α)y (α) (xn ) = α (xn − ξ) (xn − ξ)α 0 k=1 xk−1 Z n X y(xk ) − y(xk−1 ) kh 1 ≈ dξ h (nh − ξ)α (k−1)h =
k=1 n X
k=1
Let
(α) ρk
= (n − k + 1)
1−α
Γ(2 − α)hα yn(α) ≈
yk − yk−1 ((n − k + 1)h)1−α − ((n − k)h)1−α . h 1−α
− (n − k)1−α . n X
(α)
ρk (yk − yk−1 ) =
n X
(α)
yk ρk −
k=1
k=1
= ρ(α) n yn +
n−1 X
n X
(α)
yk−1 ρk
k=1
(α) (α) (α) yk ρk − ρk−1 − ρ1 y0 .
k=1
Then yn(α) (α)
Let σ0
1 ≈ Γ(2 − α)hα (α)
(α)
= ρn = 1, σn (α)
σk
ρ(α) n yn
+
n−1 X
(α) (α) (α) yn−k ρn−k − ρn−k+1 − ρ1 y0
! .
k=1 (α)
= −ρ1
(α)
= (n − 1)1−a − n1−a and
(α)
= ρn−k − ρn−k+1 = (k + 1)1−α − 2k 1−α + (k − 1)1−α ,
for k = 1, 2, · · · , n − 1. Denote Ah yn =
n X
(α)
σk yn−k .
k=0
We obtain the approximation for the Caputo derivative 1 yn(α) ≈ Ah yn . Γ(2 − α)hα
(1)
Approximation (1) has accuracy O(h2−α ) when y ∈ C 2 [0, xn ] ([9]). (α) The numbers σk have the following properties: (α)
σ0
> 0,
(α)
σ1
(α)
< σ2
(α)
< · · · < σk
< · · · < 0,
∞ X k=0
(α)
σk
= 0.
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Table 1. Error and order of approximation (1) for y(x) = cos x on the interval [0, 1], when α = 0.6. h 0.05 0.025 0.0125 0.00625 0.003125
Error 0.0023484 0.000878437 0.000330265 0.000124548 0.0000470549
Ratio 2.69618 2.67338 2.65979 2.65171 2.64687
Order 1.43092 1.41867 1.41131 1.40692 1.40429
Approximation (1) and its modifications have been successfully used for numerical solutions of fractional differential equations, as well as in proofs of the convergence of numerical methods. One disadvantage of (1) is that when the order of the Caputo fractional derivative α ≈ 1, its accuracy decreases to O(h). The numerical solutions of multidimensional partial fractional differential equations require a large number of computations, when the approximation has accuracy O(h). In section 4, we determine the second order approximation (4) for the Caputo derivative by modifying the first three coefficients of (1) with values of the Riemann zeta function. Approximation (4) has accuracy O h2 for all values of α between 0 and 1. The ordinary fractional differential equation y (α) + By = F (t),
(2)
is called relaxation equation when 0 < α < 1, and oscillation equation when 1 < α < 2. In section 5 we compare the numerical solutions for the relaxation and the time-fractional subdiffusion equations for discretizations (1) and (4). We observe a noticeable improvement of the accuracy of the numerical solutions using approximation (4) for Caputo derivative, especially when α ≈ 1. When y(x) is a sufficiently differentiable function, the integral in the definition of the Caputo derivative has a singularity at the endpoint x. Sidi [10] discusses approximations for integrals with singularities. The sum of the powers of the first n − 1 integers has expansion [11] n−1 ∞ X nα+1 X α + 1 Bm α , (3) k = ζ(−α) + α + 1 m=0 m nm k=1
where α 6= −1 and Bm are the Bernoulli numbers. In section 3, we use expansion (3) to determine a second order approximation (6) for the left Riemann sums and the fractional integral of order 2 − α. In section 4 we determine the second order approximation for the Caputo derivative (4) from (6), using discrete integration by parts and second order backward difference approximation for the second derivative. 2. Preliminaries In this section we introduce the basic definitions and facts used in the paper. The fractional integral of order α is defined as the convolution of the function y(x) and the power function xα−1 on the interval [0, x] Z x y(ξ) 1 α dξ, J y(x) = Γ(α) 0 (x − ξ)1−α
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where α > 0. The fractional integral of order α is often denoted as y (−α) (x). The value of the fractional integral of order α of the constant function 1 is xα /Γ(α + 1). The Caputo derivative is defined as the composition of y 0 (x) with a fractional integral of of order 1 − a. In Claim 1, we represent the Caputo derivative with the composition of the second derivative y 00 (x) and a fractional integral of order 2 − α y (α) (x) = J 1−α y 0 (x) = J 2−α y 00 (x) +
y 0 (0)x1−α . Γ(2 − α)
The composition of fractional integrals satisfies J α J β y(x) = J β J α y(x) = J α+β y(x). The composition of the Caputo derivative and the fractional integral of order α, when 0 < α < 1, has properties Dα J α y(x) = y(x),
J α Dα y(x) = y(x) − y(0).
In Theorem 3 we use the expansion of the sum of the powers of the first n−1 integers (3), to determine the second order approximation for the fractional integral of order 2−α n h2−α X 1−α y(0) ζ(α − 1) k y(x − kh) ≈ J 2−α y(x) + x1−α h + y(x)h2−α , Γ(2 − α) 2Γ(2 − α) Γ(2 − α) k=1
where h = x/n, and ζ(s) is the Riemann zeta function, defined as the analytic continuation of the function ∞ X 1 1 1 ζ(s) = 1 + s + s + · · · + s + · · · = n−s (Re(s) > 1). 2 3 n n=1 In the special case of (3), when α = −1, the sums of the harmonic series have expansion [11] n−1 ∞ X1 X 1 B2m 1 ≈ ln n + γ − − , k 2n s=1 2m n2m k=1
where γ ≈ 0.5772 is the Euler-Mascheroni constant and B2m are the Bernoulli numbers. In section 4 we use the approximation for the fractional integral (6), to determine the second order approximations for the Caputo fractional derivative y (α) (x) =
n X 1 ζ(α − 1) 00 (α) σk y(x − kh) − y (x)h2−α + O h2 , α Γ(2 − α)h Γ(2 − α) k=0
y (α) (x) =
n X 1 (α) δk y(x − kh) + O h2 . Γ(2 − α)hα k=0
(α)
The numbers δk (α) δ0
=
(α) σ0
(α)
are computed from the coefficients σk
− ζ(α − 1),
(α) δ1
(α)
δk
=
(α) σ1
(α)
= σk
+ 2ζ(α − 1),
(α) δ2
of (1) by (α)
= σ2
− ζ(α − 1),
(k = 2, 3, · · · , n).
The values of the Riemann zeta function satisfy [13] ∞ n X 1 1 X k n ζ(s) = (−1) (k + 1)−s , 1 − 21−s n=0 2n+1 k k=0
(4)
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for all s ∈ C, and the functional equation πs ζ(s) = 2s π s−1 sin Γ(1 − s)ζ(1 − s). 2 From the functional equation for the Riemann zeta function we obtain a representation of ζ(α − 1)/Γ(2 − α) πα ζ(α − 1) ζ(2 − α). = −2α−1 π α−2 cos Γ(2 − α) 2 3. Approximation for the Fractional Integral of Order 2 − α In this section we determine a second order approximation (6) for the fractional integral of order 2 − α, when 0 < α < 1 Z x 1 (x − ξ)1−α y(ξ)dξ. J 2−α y(x) = Γ(2 − α) 0 Approximation (6) uses the left Riemann sums of a uniform partition of the interval [0, x], and the values of y(0) and y(x). The Caputo derivative y (α) (x) = J 1−α y 0 (x) is defined as the composition of the fractional integral of order 1 − α and the first derivative y 0 (x). In Claim 1 we use integration by parts to express the Caputo derivative as a composition of the fractional integral of order 2 − α and the second derivative y 00 (x). Claim 1. Let y ∈ C 2 [0, x], and 0 < α < 1. Γ(2 − α)y (α) (x) = Γ(2 − α)J 2−α y 00 (x) + y 0 (0)x1−α . Proof. From the properties of the composition of fractional integrals and Caputo derivatives J 2−α y 00 (x) = J 1−α J 1 y 00 (x) = J 1−α (y 0 (x) − y 0 (0)). Then y (α) (x) = J 1−α y 0 (x) = J 2−α y 00 (x) + J 1−α y 0 (0), y 0 (0)x1−α y (α) (x) = J 2−α y 00 (x) + . Γ(2 − α) Let x = nh, where n is a positive integer. Consider the partition Ph of the (α) (α) interval [0, x] to n subintervals of length h. Denote by Ly,h and Ty,h the left Riemann sum and the Trapezoidal sum of the function (x − ξ)1−α y(ξ). (α)
Ly,h = h
n X
(x − mh)1−α y(mh) = h
m=0 (α) Ty,h
h = 2
n−1 X
(nh − mh)1−α y(mh),
m=0
(nh)
1−α
f (0) + 2
n−1 X
! (nh − mh)
1−α
y(mh) .
m=1
Substitute k = n − m (α)
Ly,h = h2−α
n X
k 1−α y(x − kh),
k=1 n−1
(α)
Ty,h =
X y(0) 1−α x h + h2−α k 1−α y(x − kh). 2 k=1
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(α)
The numbers Ly,h and Ty,h are approximations for Γ(2 − α)J (2−α) y(x) and y(0) 1−α x h. (5) 2 Now we use (3), to determine a second order approximation for the left Riemann sums of the constant function y(x) = 1. (α)
(α)
Ly,h − Ty,h =
Lemma 2. Let x = nh, where n is a positive integer. (α)
L1,h =
1 x2−α + x1−α h + ζ(α − 1)h2−α + O h2 . 2−α 2
Proof. Consider the first terms of (3) n−1 X
k
1−α
k=1 n X
n2−α n1−α = − + ζ(α − 1) + O 2−α 2
n2−α n1−α + + ζ(α − 1) + O 2−α 2
1 na
1 na
1 x2−α + x1−α h + ζ(α − 1)h2−α + O 2−α 2
k 1−α =
k=1 2−α
,
.
Multiply by h h2−α
n X
k 1−α =
k=1
h2−α nα
.
We have that h2−α /nα = h2 /xα . Hence (α)
L1,h =
x2−α 1 + x1−α h + ζ(α − 1)h2−α + O h2 . 2−α 2
In the next theorem we determine a second order approximation for the left Riemann sums of the fractional integral J 2−α y(x) when the function y(x) is a polynomial. Theorem 3. Let x = nh and y(x) be a polynomial. y(0) 1−α x h + ζ(α − 1)y(x)h2−α + O h2 . (6) 2 Proof. Let y(ξ) be a polynomial of degree m. The Taylor polynomial for y(ξ) of degree m at the point ξ = x is equal to y(ξ). (α)
Ly,h = Γ(2 − α)J 2−α y(x) +
y(ξ) = p0 + p1 (x − ξ) + · · · + pn (x − ξ)m = p0 +
m X
pk (x − ξ)k .
k=1
Denote y0 (ξ) = y(ξ) − y(x) =
m X
pk (x − ξ)k .
k=1
The function (x − ξ)1−α y0 (ξ) has a bounded derivative on the interval [0, x]. The (α) trapezoidal approximation Ty0 ,h is a second order approximation for the fractional integral Γ(2 − α)J (2−α) y0 (x). From Lemma 2 and (5) (α)
T1,h =
x2−α + ζ(α − 1)h2−α + O h2 . 2−α
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Then p0 2−α x +p0 ζ(α−1)h2−α +O h2 . 2−α 2−α 2−α We have that y(x) = p0 and J 1=x /Γ(3 − α), p0 2−α 2−α x . Γ(2 − α)J y(x) = Γ(2 − α)J 2−a y0 (x)(x) + 2−α Hence y(0) 1−α (α) (α) Ty,h = Ly,h − x h = Γ(2 − α)J 2−α y(x) + y(x)ζ(α − 1)h2−α + O h2 . 2 (α)
(α)
(α)
Ty,h = Ty0 ,h +p0 T1,h = Γ(2−α)J 2−α y0 (x)+
In Theorem 3 we showed that (6) is a second order approximation for the left Riemann sums and the fractional integral of order 2 − α, when the function y(x) is a polynomial. From the Weierstrass Approximation Theorem every sufficiently differentiable function and its derivatives on the interval [0, x] are uniform limit of polynomials. The class of functions for which Theorem 3 holds includes functions with bounded derivatives. In section 4, we present a proof for the second order approximation (4) of the Caputo derivative. Table 2. Error and order of approximation (6) for y(x) = cos x (left) and y(x) = ln(x + 1) (right) on the interval [0, 1], when α = 0.4. h 0.05 0.025 0.0125 0.00625 0.003125
Error 0.00011853 0.00003019 7.63 × 10−6 1.92 × 10−6 4.83 × 10−7
Order 1.95822 1.97331 1.98275 1.98876 1.99264
h 0.05 0.025 0.0125 0.00625 0.003125
Error 0.00020451 0.00005145 0.00001292 3.24 × 10−6 8.11 × 10−7
Order 1.98580 1.99083 1.99400 1.99606 1.99740
4. Second Order Approximation for the Caputo Derivative In this section we use approximation (6) to determine a second order discretization for the Caputo derivative of order α, by modifying the first three coefficients of approximation (1) with the value of the Riemann zeta function at the point α − 1. Denote by ∆1h yn and ∆2h yn the forward difference and the central difference of the function y(x) at the point xn = nh. ∆1h yn = yn+1 − yn , ∆2h yn = yn+1 − 2yn + yn−1 . When y(x) is a sufficiently differentiable function 0 yn+0.5 =
∆1h yn + O h2 , h
yn00 =
∆2h yn + O h2 . h2
Lemma 4. Ah yn =
n−1 X k=1
k 1−α ∆2h yn−k + n1−α ∆1h y0 .
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Proof. n X
Ah yn =
(α)
(α)
σk yn−k = σ0 yn +
k=0
=yn +
n−1 X
(α)
σk yn−k + σn(α) y0
k=1
n−1 X
(k − 1)1−α − 2k 1−α + (k + 1)1−α yn−k + σn(α) y0
k=1
=yn +
n−1 X
n−1 X
k=1
k=1
(k − 1)1−α yn−k − 2
n−1 X
k 1−α yn−k +
(k + 1)1−α yn−k + σn(α) y0 .
k=1
Substitute K = k − 1 in the first sum and K = k + 1 in the third sum n−2 X
Ah yn = yn +
K 1−α yn−K−1 − 2
K=1
n−1 X
k 1−α yn−k +
n X
K 1−α yn−K+1 + σn(α) y0 .
K=2
k=1
We have that n−2 X
k 1−α yn−k−1 =
n X
k 1−α yn−k−1 − (n − 1)1−α y0 ,
k=1
k=1
yn +
n−1 X
k 1−α yn−k+1 =
k=2
n X
k 1−α yn−k+1 =
k=1
n−1 X
k 1−α yn−k+1 + n1−α y1 .
k=1
Then Ah yn =
n−1 X
k 1−α (yn−k+1 − 2yn−k + yn−k−1 ) + n1−α (y1 − y0 ) ,
k=1
because
(α) σn
= (n − 1)1−α − n1−α .
Lemma 5. Suppose that y(x) is sufficiently differentiable function on [0, nh] n−1
X 1 2−α 00 0 A y = h k 1−α yn−k + (nh)1−α y0.5 + O h2 . h n hα k=1
Proof. From Lemma 4 n−1 n−1 X X 1 ∆2 ∆1 y0 1−α 2 1−α 1 2−α A y = k ∆ y +n ∆ y = h k 1−α 2h yn−k +n1−α h1−α h , h n h n−k h 0 α h h h k=1
k=1
n−1
X 1 2−α 00 0 A y = h k 1−α yn−k + O h2 + (nh)1−α y0.5 + O h2 , h n hα k=1
1 Ah yn = h2−α hα
n−1 X
00 0 k 1−α yn−k +(nh)1−α y0.5 +O
2
h
(nh)
1−α
k=1
2−α
+h
n−1 X k=1
The number (nh)1−α ∼ O(1) is bounded. From (3) we have h2−α
n−1 X k=1
k 1−α ∼ h2−α O n2−α ∼ O(1).
! k
1−α
.
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Therefore n−1
X 1 00 0 Ah yn = h2−α k 1−α yn−k + (nh)1−α y0.5 + O h2 . α h k=1
Theorem 6. Let y be a polynomial and x = nh. 1 Ah y(x) = Γ(2 − α)yn(α) + ζ(α − 1)y 00 (x)h2−α + O h2 . α h Proof. From Lemma 5 n−1
X 1 2−α 00 0 A y(x) =h k 1−α yn−k + (nh)1−α y0.5 + O h2 = h hα k=1
h2−α
n X
00 0 k 1−α yn−k − h2−α n1−α y000 + x1−α y0.5 + O h2 .
k=1
Then
1 (α) 0 Ah y(x) = Ly00 ,h − x1−α (hy000 − y0.5 ) + O h2 . α h From Claim 1 and Theorem 3 y 00 (0) 1−α (α) Ly00 ,h = Γ(2 − α)J 2−α y 00 (x) + x h + ζ(α − 1)y 00 (x)h2−α + O h2 , 2 Γ(2 − α)y (α) (x) = Γ(2 − α)J 2−α y 00 (x) + y 0 (0)x1−α . Then y 00 h (α) + O h2 , Ly00 ,h = Γ(2 − α)y (α) (x) + ζ(α − 1)y 00 (x)h2−α − x1−α y00 − 0 2 00 1 y0 h (α) 00 2−α 1−α 0 0 Ah y(x) = Γ(2−α)y (x)+ζ(α−1)y (x)h −x y0 + − y0.5 +O h2 . hα 2 By Taylor’s expansion y 00 h 0 = O h2 . y00 + 0 − y0.5 2 Hence 1 Ah y(x) = Γ(2 − α)y (α) (x) + ζ(α − 1)y 00 (x)h2−α + O h2 . α h In Theorem 6 we determined the second order approximation for the Caputo derivative 1 ζ(α − 1) 00 2−α yn(α) = yn h + O h2 . (7) Ah yn − α Γ(2 − α)h Γ(2 − α) Corollary 7. Let y(x) be a polynomial. yn(α) =
n X 1 (α) δk yn−k + O h2 , α Γ(2 − α)h k=0
where
(α) δk
(α) δ0
=
=
(α) σk
(α) σ0
for 2 ≤ k ≤ n and (α)
− ζ(α − 1), δ1
(α)
= σ1
(α)
+ 2ζ(α − 1), δ2
(α)
= σ2
− ζ(α − 1).
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YURI DIMITROV
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Proof. The second order backward difference approximation for the second derivative yn00 has accuracy O(h). yn − 2yn−1 + yn−2 yn00 = + O(h). h2 From approximation (7) ζ(α − 1) yn − 2yn−1 + yn−2 1 A y − + O(h) h2−α + O h2 , yn(α) = h n α 2 Γ(2 − α)h Γ(2 − α) h ζ(α − 1) 1 Ah yn − (yn − 2yn−1 + yn−2 ) + O h2 , α α Γ(2 − α)h Γ(2 − α)h ! n X (α) 1 σk yn−k − ζ(α − 1) (yn − 2yn−1 + yn−2 ) + O h2 . = α Γ(2 − α)h
yn(α) = yn(α)
k=0
Table 3. Error and order of approximation (4) for Caputo derivative of order α = 0.25 and y(x) = cos x (left), y(x) = ln(x + 1) (right) on [0, 1]. h 0.05 0.025 0.0125 0.00625 0.003125
Error Order 0.000081955 2.30047 0.000017556 2.22284 3.95 × 10−6 2.15376 9.20 × 10−7 2.10073 2.20 × 10−7 2.06368
Denote Bh y(x) =
h 0.05 0.025 0.0125 0.00625 0.003125
n X
Error 0.000029455 6.39 × 10−6 1.46 × 10−6 3.44 × 10−7 8.31 × 10−8
Order 2.31171 2.20475 2.13162 2.08272 2.05103
(α)
δk y(x − kh).
k=0
In Corollary 7 we showed that (4) is a second order approximation for the Caputo derivative of polynomials. Now we use the Weierstrass Approximation Theorem to extend the result to differentiable functions. Theorem 8. Let x = nh and y be a sufficiently differentiable function. 1 y (α) (x) = Bh y(x) + O h2 . Γ(2 − α)hα Proof. By the Weierstrass Approximation Theorem every continuous function is a uniform limit of polynomials. Let > 0 and p (x) be a polynomial such that |y 0 (t) − p (t)| < , for all t ∈ [0, x]. Define t
Z q (t) = y(0) +
p (ξ)dξ. 0
The function q (t) is polynomial, and q0 (t) = p0 (t). We have that Z t Z t Z 0 0 |y(t) − q (t)| = (y (ξ) − p (ξ)) dξ ≤ |y (ξ) − p (ξ)| dξ < 0
0
0
x
dξ ≤ x,
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for all t ∈ [0, x]. Z (α) (α) • Γ(1 − α) y (x) − q (x) =
185
x
Z x 0 y 0 (ξ) − q0 (ξ) |y (ξ) − p (ξ)| dξ ≤ dξ, α (x − ξ) (x − ξ)α 0 0 Z x x1−α (α) (x − ξ)−α dξ = . y (x) − q(α) (x) < Γ(1 − α) 0 Γ(2 − α)
Therefore lim q(α) (x) = y (α) (x).
→0
• Now we estimate Bh (y(x) − q (x)). n n n X X X (α) (α) (α) δk (yn−k − q,n−k ) ≤ |δk | |yn−k − q,n−k | ≤ x |δk |. k=0
k=0
k=0
We have that n X
(α)
|δk | ≤
k=0
n X
(α)
|σk | + 3|ζ(α − 1)| = 2 − 3ζ(α − 1).
k=0
Hence |Bh (y(x) − q (x))| ≤ (2 − 3ζ(α − 1))x, and lim Bh q (x) = Bh y(x).
→0
• From Corollary 7 q(α) (x) =
1 Bh q (x) + O h2 . Γ(2 − α)hα
By letting → 0, we obtain y (α) (x) =
1 Bh y(x) + O h2 . α Γ(2 − α)h -3
-12
-10
-8
-6
-4
-2
-2
-1
-1
-0.5
Figure 1. Graphs of the Riemann zeta function on the intervals [−12, −1] and [−3, 0]. In Table 3 we compute the error and the numerical order of approximation (4) for the Caputo derivative of the functions y(x) = cos x and y(x) = ln(x + 1) on the interval [0, 1], when α = 0.25. In Claim 9 and Lemma 10 we discuss the properties (α) (α) of the coefficients σ2 and δ2 .
186
YURI DIMITROV
JFCA-2016/7(2)
Claim 9. Let 0 < α < 1 (α)
−0.1 < σ2
< 0.
Proof. Denote (1−α)
σ(α) = −σ2
= 2α+1 − 3α − 1.
The function σ(α) has values σ(0) = σ(1) = 0, and σ 0 (α) = ln 2.2α+1 − ln 3.3α . The first derivative of σ(α) is zero when
α 2. ln 2 3 , = 2 ln 3
ln 2.2α+1 = ln 3.3α ,
2. ln 2 ln ln 3 α= ln(3/2)
≈ 0.5736.
The function σ(α) is positive and has a maximum value σ(0.5736) ≈ 0.0985 on the interval [0, 1]. The Riemann zeta function has zeroes at the negative even integers and is decreasing on the interval [−2, 1]. The value of ζ(α−1) is negative, when α is between (α) (α) 0 and 1. Then δ0 > 0 and δ1 < 0. From the properties of the coefficients of (1), (α) (α) the numbers δn = σn are negative, for n ≥ 3. (α)
Lemma 10. The number δ2
is positive when 0 < α < 1. (1−α)
Proof. From the definition of δ2 (1−α)
δ2
(1−α)
= σ2
− ζ(−α) = −σ(α) − ζ(−α) = z(α) − σ(α).
where z(α) = −ζ(−α). The function z(α) is decreasing on the interval [0, 1] with values at the endpoints z(0) = 0.5 and z(1) = 1/12 = 0.08333. The function σ(α) is increasing on the interval [0, 0.5736] and decreasing on the interval [0.5736, 1]. (1−α) Now we show that the minimum values of δ2 on the intervals [0, 0.8] and [0.8, 1] are positive. (1−α)
min δ2
>
α∈[0,0.8]
(1−α)
min δ2
min z(α) − max σ(α), α∈[0,0.8]
α∈[0,0.8]
> z(0.8) − σ(0.5736) ≈ 0.122 − 0.0985 = 0.0235.
α∈[0,0.8]
and (1−α)
min δ2
α∈[0.8,1]
(1−α)
min δ2
>
min z(α) − max σ(α), α∈[0.8,1]
α∈[0.8,1]
> z(1) − σ(0.8) ≈ 0.083 − 0.074 = 0.009.
α∈[0.8,1]
(1−α)
Therefore the numbers δ2
are positive when 0 < α < 1.
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187
5. Numerical Experiments In section 4 we showed that the approximation for the Caputo derivative yn(α) ≈
n X 1 (α) δk yn , Γ(2 − α)hα k=0
2
has accuracy O h (α) δ0
> 0,
(α) δ1
(α)
when n ≥ 2. The numbers δk
< 0,
(α) δ2
> 0,
(α) δ3
