Two New Approximations for Variable-Order Fractional Derivatives

We introduced a parameter σ(t) which was related to α(t); then two numerical schemes for variable-order Caputo fractional derivatives were derived; the second-order numerical approximation to variable-order fractional derivatives α(t) ∈ (0, 1) and 3 − α(t)-order approximation for α(t) ∈ (1, 2) are established. For the given parameter σ(t), the error estimations of formulas were proven, which were higher than some recently derived schemes. Finally, some numerical examples with exact solutions were studied to demonstrate the theoretical analysis and verify the efficiency of the proposed methods.


Introduction
Fractional differential equations include constant-order and variable-order equations; a great quantity of natural phenomena can be modeled by variable-order fractional differential equations; the study of such problems has attracted much attention.In recent years, profound background of physical applications for the variable-order fractional calculus has been already established; the definition of variable-order operator has been investigated in [1][2][3][4].Numerous problems in mathematical physics and engineering have been modeled by variable-order fractional differential equations, such as successful applications in mechanics [5], in the simulation of linear and nonlinear viscoelasticity oscillators [6], and in other cases where the order of the derivative varies with time [7].In addition, a physical experimental investigation of variable-order operators has been considered in [8].Variable-order fractional derivatives can be used to model anomalous diffusion, as they can describe the time dependent diffusion process more specifically than fractional derivatives of constant order, just as shown in [9].The difference schemes of fractional derivatives with constant and variable order are investigated in [10].
Due to the existence of variable fractional derivative, it is usually difficult to obtain the analytical solution of such equations; therefore, it is particularly significant to give numerical solutions to these problems.Cao and Qiu [11] derived a high order numerical method for variable-order fractional ordinary differential equation by establishing a second-order numerical approximation to variable-order Riemann-Liouville fractional derivative.Fu et al. [12] adopted the method of approximate particular solutions for both constant-order and variable-order time fractional diffusion models.Several finite difference methods for variable-order fractional partial diffusion equations were proposed in [13][14][15][16][17][18].
As what we can see above, there are many different definitions of the variable-order fractional derivatives.However, different from other definitions in mathematics, only initial condition is needed for the variable-order Caputo definition which can be easily used in physical field; this definition means that the memory rate of system is determined by the current time instant and changes with time.For simplicity, only variable-order Caputo definition was discussed in the whole paper.
A good approximation of the variable-order Caputo derivatives was observed in [19], it was defined at the points  +1/2 and   ; then what will happen if it is redefined in a neighborhood at the point   ?Approximation formula for the -order (0 ≤  ≤ 1) Caputo derivatives at the point  =  −1+ was investigated in [20], where  = 1 − /2 is a superconvergence point which ensures that the convergence order improves from the local second order to the overall 3− order.In the current work, we applied the ideas in [20]; a parameter () was given, which was changed with (), and the selection of () played a vital role in the numerical results in the paper.We proposed two new approximation formulas of second-order and 3 − () accuracy for variable-order time fractional operator with orders 0 < () < 1 and 1 < () < 2, respectively.Specifically, we adopted the following definition of variable-order Caputo fractional derivatives: The paper is organized as follows.In the next section, we present two new formulas, select () = 1 − ()/2 for 0 < () < 1 and () = 3/2 − ()/2 for 1 < () < 2, respectively, and provide the corresponding error analysis.In Section 3, the numerical verifications are presented; as can be seen from the numerical results, minor changes to the selected parameters () will have a significant impact on the error estimates.Finally, in Section 4, the conclusion is drawn.

Approximation Formulas
In this section, we present the following lemma, which plays a vital role in the later analysis.
Estimating the derivative  0  ()  () with order () ∈ (0, 1) at the grid point  −1+ ,  = 0, 1, . . .,  − 1, from (1), we directly obtain  0 Evaluate the integration on each subinterval, leading to  0 For each interval [ −1 ,   ],  = 1, . . .,  − 1, denote the second-degree interpolation polynomial in the Lagrange form as follows: and for the last interval [ −1 ,  −1+ ], denote the first-degree interpolation polynomial in the Lagrange form as As an approximation formula, the following result is obtained: where where Substituting ( 8) and ( 10) into (5) yields  0 and the truncation error is Here we denote the discrete approximation formula for the variable-order derivative with order  −1+ as 0 Δ  −1+   −1+ ; thus from (12), it is easy to get the following result: where Next, the analysis for the approximation error of formula (12) was given in detail.
From (19), it can be obtained in the same method.

2.2.
Second-Order Formula for () ∈ (1, 2).Denote   0 =   ( 0 ),  = 3/2 −  −1+ /2.Computing the variable-order derivative with order ( For the interval [ 0 ,  1 ], the cubic interpolation polynomial in the Hermite form is as and it follows that the truncation error is For the interval [  ,  +1 ],  = 1, . . .,  − 2, the cubic interpolation polynomial at points  −1 ,   ,  +1 , and  +2 in the Lagrange form is as the following holds: the truncation error is Substituting ( 25), (28), and (31) into (23), there is Denote the approximation formula of the variable-order derivative with order  −1+ as 0 Δ  −1+   −1+ and obtain where Substituting the above formula into (34), thus For the first term in the right hand side of (36), this leads to For the second term in the right hand side of (36), the following holds: For the third term in the right hand side of (36), this yields Substituting (37)-( 39) into (36), we obtain the approximate difference scheme of the  0 where Similar to the proof of the Theorem 2, the truncation error of formula (40) is given by following theorem.
where  2 is a positive constant.
Proof.From (33), we obtain the truncation error From Lemma 1, the result below is natural.
Remark 4. From the second term of the right hand of the truncation error estimate in Theorem 2, we can easily obtain the fact that formula (14) does not have exact second-order accuracy; it is related to the selection of () and .This can be seen from Table 2 in the numerical example below; its accuracy is much higher than the second order.

Numerical Verification
In this section, the validity and numerical accuracy of the new presented formula ( 14) and (40) are demonstrated, respectively.Meanwhile, the corresponding computational results with the formula of [19] are given for contrast.
Example 1 (accuracy of formula ( 14) and [19]).Take () =  5 −  3 , 0 <  ≤ .Compute the Caputo fractional derivative of () at  = 1, 2 numerically.The exact solution is given by  0 From the results presented in Table 1, taking  = 1 −  1 /2 = 1/2, we found that although both formula (14) and [19] have second-order approximation accuracy, the numerical accuracy by the formula ( 14) is significantly higher than that by the formula of [19]; furthermore, the computational errors are also obviously smaller than the formula of [19].Thus, the new formula ( 14) is valid for solving complex problems that require high accuracy, and better computational results can be obtained from the formula.
In Table 2, taking  = 1 −  2 /2 = 0.9323, it shows that the convergence order is higher as compared with the theoretical result, and the error is smaller than the theoretical result, but why this happens is not clear yet; it requires further investigation.
Moreover, on the basis of Table 1, if a slight disturbance to  is added to Tables 3 and 4, the experimental results will be greatly different.It can be easily found that there is a very close relationship between the convergence order and the selection of .The experimental results show that the convergence order is the ideal result only when  = 1 −  1 /2.Tables 2,  5, and 6 also illustrate this fact.
Similar to Example 1, we also give Tables 8 and 9 as a comparison; the results show that  = 3/2 −  1 /2 is the best choice.
Remark 3. Due to the fact that the selection of  is very important, it is worth noting that different selected  leads to the different numerical accuracy.From considerable trials, we observed that only if we take  = 1−()/2 for () ∈ (0, 1) and  = 3/2−()/2 for () ∈ (1, 2), will numerical results be closer to theoretical analysis.In addition, carefully observing Table 9, although the convergence order is not higher than in Table 7, the error is lower than in Table 7; further investigation is needed on this phenomenon.

Conclusion
In this paper, two approximations to variable-order Caputo fractional derivatives were developed, and the analysis for the truncation errors of new formulas was made.In addition, numerical examples support theoretical results.In further work, we also want to give a strict proof of the selection of  and apply these new methods to solve more complex cases within a reasonable accuracy.