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Approximations for the Caputo derivative (II)

In the present paper we use the expansion formula of the polylogarithm function to construct approximations of the Caputo derivative which are related to the midpoint approximation of the integral in the definition of the Caputo derivative. The asymptotic expansion formula of the Riemann sum approximation of the beta function and the first terms of the expansion formulas of the approximations of the Caputo derivative of the power function are obtained in the paper. The induced shifted approximations of the Gr\"unwald formula and the approximations of the Caputo derivative studied in the first part of the paper are constructed and applied for numerical solution of fractional differential equations.


Introduction
Approximations of fractional integrals and derivatives have recently been an active research topic [5,8,9,19,21,24,27]. The fractional integral of order α > 0 and the Caputo derivative of order α, where 0 < α < 1 are defined as Let h = x/n, where n is a positive integer, and x m = mh, y m = y(x m ). The L1 approximation (1) for the Caputo derivative is a commonly used approximation for numerical solution of fractional differential equations.
The weights σ (α) k of the L1 approximation have the following properties where C 1 = α(α − 1) and C 2 = α − 1. When the function y has a continuous second derivative, the L1 approximation has accuracy O(h 2−α ) ( [17]). The numerical solution of the fractional relaxation equation (20) which uses the L1 approximation for the Caputo derivative is computed with [5] u n = 1 In Table 1 we compute the error and the order of numerical solution (3) for Equation I and α = 0.25, Equation II and α = 0.5 and Equation III, α = 0.75. In [5] we derived the second-order expansion of the L1 approximation By approximating y ′′ (x) using second-order backward difference we obtain the second-order approximation for the Caputo derivative The asymptotic expansions of the trapezoidal approximation for the definite integral and the integral approximations for the fractional derivative involve the values of the Riemann zeta function defined as The Riemann zeta function is a special case (x = 1) of the polylogarithm function is defined as The polylogarithm function has properties where α = 1, 2, 3, · · · and | ln x| < 2π. From (6) with x = e iωh we obtain In [6] we use the Fourier transform method to derive the asymptotic expansion formula of the trapezoidal approximation for the fractional integral In the first part of this paper [7] we obtain approximations (8) and (9) for the Caputo derivative of order 2 − α.
In section 2 we use the Fourier transform method to derive the asymptotic expansion formula (18) of approximation (19). In section 3 we use (18) we obtain approximation (10) for the Caputo derivative of order 2 − α and the second-order approximation (11).
In Figure 1 we compare the absolute values of the coefficients C 1 (α), C 8 (α), C 9 (α) and C 10 (α). In the first part of this paper [7] we derive approximations (8) and (9) for the Caputo derivative and we apply the approximations for computing the numerical solution of the fractional relaxation equation. The accuracy of the numerical solution of Equation I, Equation II and Equation III which uses approximation (9) is higher than the accuracy of the numerical solutions using (1) and (8). In Table 1 and Table 3 we compute the errors of the numerical solutions of Equation I, Equation II and Equation III which use approximations (1) and (10). The numerical solution which uses approximation (10) for the Caputo derivative has a higher accuracy than the numerical solution using the L1 approximation (1). The improvement is 30% for Equation I and α = 0.25, 15% for Equation II and α = 0.5 and 4.7% for Equation III with α = 0.75. In section 4 we construct a finite difference scheme for the fractional subdiffusion equation with accuracy O (τ 2−α + h 2 ), using approximation (10) for the Caputo derivative and we analyse the convergence of the scheme.
2 Approximation for the Caputo Derivative of Order 4 − α The construction of approximation (8) is based on the trapezoidal approximation for the fractional integral and the second-order backward difference approximation for the first derivative on each subinterval. The coefficient C 8 (α) of the term y ′′ n h 2−α in the expansion of approximation (8) is greater than the coefficient C 1 (α) of the L1 approximation. The midpoint approximation for the definite integral is around twice more accurate than the trapezoidal approximation. In this section we use the Fourier transform method to derive approximations (10) and (11) for the Caputo derivative of order 2 − α and 2 based on the midpoint approximation for the fractional integral. The coefficient C 10 (α) of approximation (10) is smaller than C 1 (α). The accuracy of the numerical solutions of Equation I, Equation II and Equation III which use approximation (10) for the Caputo derivative is higher than the accuracy of the numerical solutions which use the L1 approximation (1). The exponential Fourier Transform of the function y is defined as The Fourier transform has properties F [y(x − b)](w) = e iwbŷ (w) and From the midpoint approximation for the fractional integral in the definition of the Caputo derivative The generating function of an approximation is related to the Fourier transform of the approximation. The Fourier transform/generating function method is used by Ding and Li [8], Lubich [15], Tian et. al. [24] for constructing approximations for the fractional derivative. By applying Fourier transform to (12) and letting n → ∞ From the Taylor expansion formula of the exponential function From (7) the function Li α e iwh 2 has fourth-order expansion By combining (7) and (15) we obtain From (14) and (16) F By applying inverse Fourier transform we obtain the expansion formula Denote From (17) and the properties of the inverse Fourier transform we obtain the expansion formula of approximation (19) for the Caputo derivative.
The function y(x) = 1 has Caputo derivative y (α) (x) = 0. Approximation (19)    In the present section we use approximation (18) to obtain approximations for the Caputo derivative of order 2 − α and 2 and we apply the approximations for computing the numerical solution of the fractional relaxation equation In [5] we showed thatȳ Let u n be an approximation for the value of the exact solution y n = y(x n ) of equation (20). The numbers u n are computed with u 0 = y(0), u 1 =ȳ 1 and

Approximation for the Caputo derivative of order 2 − α
By approximating y ′ n in (18) with first order backward difference we obtain the approximation for the Caputo derivative . Proof.
The function y(x) = x has fractional derivative The statement of Claim 3 follows from (22) and (23).

Second-Order Approximation for the Caputo derivative
By approximating y ′ n and y ′′ n in (18) with , we obtain the approximation for the Caputo derivative whereδ Approximation (26) has accuracy O (h 2 ) when y(0) = y ′ (0) = 0. From approximation (26) we obtain the second order approximation approximation for the Caputo derivative for k = 0, 1, · · · , n − 2 and The formulas for the weights δ

Numerical Solution of the Fractional Subdiffusion Equation
The analytical and numerical solutions of the fractional subdiffusion equation have been studied extensively [5,12,14,17,18,22,28]. In this section we use approximation (10) for the Caputo derivative to construct a finite-difference scheme for the fractional subdiffusion equation Denote u m n = u(nh, mτ ) and F m n = F (nh, mτ ). By approximating the Caputo derivative in the time direction using (10) and the second derivative in the space direction by second-order central difference approximation we obtain where R 1 and R 2 are the column vectors of dimension N − 1 The numerical solution of the fractional subdiffusion equation on the first layer of the grid G is computed with the approximation [5] The numerical solution on the first layer of the grid G satisfies the system of equations Example 1: The fractional subdiffusion equation has the solution u(x, t) = e x+t .
has the solution u(x, t) = sin(πx)E α (−t α ). The first partial derivative of the solution is unbounded at t = 0. In Table 6 we compute the error and the order of numerical solution (28) for equation (31). The numerical results from Table 6 suggest that the accuracy of numerical solution (28) is O(h). A numerical analysis of the scheme using the L1 approximation for the Caputo derivative is given by Jin, Lazarov and Zhou [14]. We use the fractional Taylor polynomials of the solution in order to improve the differentiability class of the solution and the accuracy of the numerical solution. Denote by D t u(x, t) = 1 π 2 ∂ 2 u(x, t) ∂x 2 . By applying fractional differentiation of order α we obtain ∂ 2n sin x ∂x 2n = (−1) n sin(πx). Proof. Denote S m = m k=1 1/k α . From the formula for the sum of powers We have that We can further show that W m < α/ (24m 1+α ). From Claim 4 we obtain The maximum (infinity) norm of the vector V = (v i ) and the square matrix L = (l ij ) of dimension N − 1 are defined as The matrix K is a diagonally dominant tridiagonal matrix with positive elements on the main diagonal and negative elements on the diagonals below and above the main diagonal. The matrix K −1 is a positive matrix. From the Ahlberg-Nilson-Varah bound [2,25] The numbers σ Let e m n = u m n − U m n be the error of numerical solution (28). The error vector E m = (e m n ) on the m-th layer of the grid G satisfies the system of equations and A m n τ 2−α + B m n h 2 is the approximation error at the point (nh, mτ ). Let A be a positive constant such that |A m n | < A and |B m n | < A, for all m and n. The numerical solution on the first layer of G is computed using approximation (29), which is obtained from the the L1 approximation when n = 1. The convergence of the numerical solution of the time-fractional subdiffusion equation which uses the L1 approximation for the Caputo derivative is studied in [17,18]. We can assume that the number A is large enough, such that (36) holds for m = 1.