Approximation of Fractional Caputo Derivative of Variable Order and Variable Terminals with Application to Initial/Boundary Value Problems

Abstract

This article presents a method for the approximate calculation of fractional Caputo derivatives, including a crucial aspect of the ability to handle arbitrary—even variable—terminals and order. The proposed method involves rearranging the fractional operator as a series of higher-order derivatives considered at a specific point. We demonstrate the effect of the number of terms included in the series expansion on the solution accuracy and error analysis. The advantage of the method is its simplicity and ease of implementation. Additionally, the method allows for a quick estimation of the fractional derivative by using a few first terms of the expansion. The elaborated algorithm is tested against a comprehensive series of illustrative examples, providing very good agreement with the exact/reference solutions. Furthermore, the application of the proposed method to fractional boundary/initial value problems is included.

1. Introduction

Fractional calculus (FC) has recently become a powerful modelling tool in science cf. review papers [1], with applications found, for instance, in the modelling of nano-structures [2,3], in phenomena observed in materials [4,5] or electrical systems [6]. This increasing attractiveness of FC comes from its fundamental property, namely, that one can define infinitely many fractional (differential and integer) operators (FOs). Let us mention here some of them, like those proposed by Liouville [7], Riemann [8], Grunwald [9], Letnikov [10] or more recently by Caputo–Fabrizio [11], Lazopoulos [12], Gerasimov–Caputo [13], and Katugampola [14]. This flexibility of FC means that fractional differential equations (integral or differo/integral) (FDEs) do not have, in general, analytical solutions [15,16]—it is therefore necessary to look for robust numerical methods to obtain their solutions (especially for practical cases).

The numerical schemes for an FDE depend on its specific type and especially on the applied FO [17,18]. Examples of FDEs with the Caputo operator include methods such as the Runge–Kutta methods [19], multistep methods [20], higher order ADI numerical difference formula [21], Gauss–Jacobi quadrature rule [22], Bernstein polynomials method [23], wavelet-based collocation method [24], trapezoidal rule [25], and central difference scheme [26]. For an FDE with Riemann–Liouville, one can distinguish methods such as quadratic interpolation [27], the Adomian decomposition method [28], the finite element method [29], and the Kansa method [30]. Additionally, for other types of FDE with different FOs, tailored methods have also been developed. These include the variational finite difference method for the Riesz operator [31], a specialised method for the $\Psi$-Caputo derivative [32,33], the Laplace transform for conformable derivatives [34], and quadrature-based approaches [35,36].

However, none of these methods can be considered universal, so there is a continuous need for new concepts, especially those that enable easy implementation and the obtaining of solutions for a broad class of problems—preferably those characteristic of real-life applications (such as, for instance, the analysis of mechanical structures like nanobeams [37,38] or the modelling of epidemic spread [39,40]). Furthermore, it is usually assumed that fractional operators operate on the entire domain (to capture global interactions) and are of constant order. However, some phenomena in physics may be better described when the terminals of the fractional operator are variable [41] and/or the fractional order is variable [1,42].

With respect to the above discussion, the novelty of this work is an extension of the method presented in [43], designated for the approximate calculation of the Riemann–Liouville operator by a series of integer order derivatives evaluated at a single point, which significantly simplifies numerical implementation. Additionally, the method allows for a natural truncation of the series, providing controlled approximation. However, the method needs the input function to be analytic and relies on the evaluation of multiple higher-order derivatives at a single point. This may limit its applicability in cases where the function is not smooth or is only available in discrete form. Nevertheless, these limitations are not critical in settings where the input function is known analytically. For instance, in fractional finite element methods [1], where smooth shape functions (often polynomials) with known derivatives are employed, as well as in spectral methods [44], the need to evaluate higher-order derivatives does not pose a significant challenge. This method can be applied efficiently and accurately to such applications.

Herein, we generalise the above methodology to the case of the approximate calculation of Caputo derivatives of variable order and/or variable terminals [45], which, to the best of the authors’ knowledge, cannot be found in the literature. The method is based on the variable change and the expansion of the sub-integral function by means of the Taylor series, resulting in the transformation of the fractional operator into a series of higher-integer-order derivatives. The work is supported by illustrative examples and a demonstration of its applications in solving fractional initial/boundary value problems.

The paper is structured as follows. Section 2 presents preliminaries on fractional calculus. Section 3 is devoted to the Taylor series-based method of fractional Caputo derivative approximation. Section 4 provides illustrative examples of fractional operator approximation, and Section 5 demonstrates the application of the proposed method to initial/boundary value problems. Section 6 concludes the article.

2. Fractional Calculus

This section is devoted to presenting some required preliminaries of fractional calculus—definitions of Riemann–Liouville integrals and Caputo derivatives [15].

2.1. Riemann-Liouville Fractional Integral

The left and right Riemann–Liouville fractional integrals of order $\alpha >0$ of a given function f are defined by

$${}_a{I}_x^{\alpha }f(x)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_a^x{(x-\tau )}^{\alpha -1}f(\tau )d\tau ,\mathrm{for}x>a,$$

(1)

and

$${}_x{I}_b^{\alpha }f(x)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_x^b{(\tau -x)}^{\alpha -1}f(\tau )d\tau ,\mathrm{for}x<b,$$

(2)

respectively, where $\Gamma$ is Euler’s gamma function.

2.2. Caputo Fractional Derivative

The left and right Caputo derivatives of order $\alpha >0$ of a given function f are defined by

$${}_a{}^C{D}_x^{\alpha }f(x)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_a^x{(x-\tau )}^{n-\alpha -1}{f}^{(n)}(\tau )\mathrm{d}\tau ,\mathrm{for}x>a,$$

(3)

and

$${}_x{}^C{D}_b^{\alpha }f(x)={\displaystyle \frac{{(-1)}^n}{\Gamma (n-\alpha )}}{\int }_x^b{(\tau -x)}^{n-\alpha -1}{f}^{(n)}(\tau )\mathrm{d}\tau ,\mathrm{for}x<b,$$

(4)

respectively, where $n-1<\alpha \le n$, $n\in \mathbb{N}$. In particular, we consider an extension of the definition of the Caputo derivative by variable order $\alpha (x)$ and variable terminals ($a(x)$ and $b(x)$), then

$${}_{a(x)}{}^{C}D_x^{\alpha }f(x)={\displaystyle \frac{1}{\Gamma (n-\alpha (x))}}{\int }_{a(x)}^x{(x-\tau )}^{n-\alpha (x)-1}{f}^{(n)}(\tau )\mathrm{d}\tau ,\mathrm{for}x>a(x),$$

(5)

and

$${}_x{}^C{D}_{b(x)}^{\alpha (x)}f(x)={\displaystyle \frac{{(-1)}^n}{\Gamma (n-\alpha (x))}}{\int }_x^{b(x)}{(\tau -x)}^{n-\alpha (x)-1}{f}^{(n)}(\tau )\mathrm{d}\tau ,\mathrm{for}x<b(x).$$

(6)

Figure 1 illustrates how the left/right derivative of Caputo operates on the function $f(x)$. In the case of a fixed terminal $a/b$, it operates on the entire ‘past’/‘future’ of the function in the interval $\langle a,b\rangle$. In contrast, with a variable/fixed boundary $a(x)$ / $b(x)$, it operates on some fragment of the ‘past’/‘future’ of $f(x)$—in Figure 1, an example is shown when a fixed horizon ℓ is set, i.e., $a(x)=x-\ell$ and $b(x)=x+\ell$.

3. Taylor Series-Based Approximation

3.1. Riemann–Liouville Integral

It is assumed that $f(t)$ is an analytical function and can be represented by a convergent power series,

$$f(t)=\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{f}^{(k)}(x)}{k!}}{(x-t)}^k.$$

(7)

Substituting the representation in Equation (7) into the integral expressions in Equation (1) and (2), and applying term-by-term fractional integration, leads directly to fractional Riemann–Liouville integral expanded into a series involving integer order derivatives [43], as follows:

$${}_a{I}_x^{\alpha }f(x)={\displaystyle \frac{1}{\Gamma (\alpha )}}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k}{k!(k+\alpha )}}{(x-a)}^{k+\alpha }{f}^{(k)}(x),$$

(8)

and right

$${}_x{I}_b^{\alpha }f(x)={\displaystyle \frac{1}{\Gamma (\alpha )}}\sum _{k=0}^{\infty }{\displaystyle \frac{1}{k!(k+\alpha )}}{(b-x)}^{k+\alpha }{f}^{(k)}(x)$$

(9)

fractional integrals, respectively.

However, from the numerical point of view, one usually considers an approximation of the fractional integral that takes a finite sum of higher-integer-order derivatives, for left,

$$\begin{array}{c}\hfill {}_a{I}_x^{\alpha }f(x)\approx {\displaystyle \frac{1}{\Gamma (\alpha )}}\sum _{k=0}^{m-1}{\displaystyle \frac{{(-1)}^k}{k!(k+\alpha )}}{(x-a)}^{k+\alpha }{f}^{(k)}(x),\end{array}$$

(10)

and right,

$$\begin{array}{c}\hfill {}_x{I}_b^{\alpha }f(x)\approx {\displaystyle \frac{1}{\Gamma (\alpha )}}\sum _{k=0}^{m-1}{\displaystyle \frac{1}{k!(k+\alpha )}}{(b-x)}^{k+\alpha }{f}^{(k)}(x)\end{array}$$

(11)

Riemann–Liouville fractional integrals.

3.2. Caputo Fractional Derivative

We derive an analogous formula as presented in Section 3.1, albeit for the approximation of the Caputo derivative allowing for variable order and/or variable terminals.

First, the fractional derivatives (Equations (5) and (6)) are transformed by adopting new variables,

$$t_1={(x-\tau )}^{n-\alpha (x)}\mathrm{and}{t}_2={(\tau -x)}^{n-\alpha (x)}.$$

(12)

Then, their differentials are as follows:

$$\left\{\begin{array}{c}d{t}_1=-(n-\alpha (x)){(x-\tau )}^{n-\alpha (x)-1}d\tau \hfill \\ d{t}_2=(n-\alpha (x)){(\tau -x)}^{n-\alpha (x)-1}d\tau .\hfill \end{array}\right.$$

(13)

Next, for the Caputo derivative, we use a new variables Equation (12) and their differentials Equation (13) to rewrite Equations (5) and (6) as follows:

$$\begin{array}{cc}\hfill {}_{a(x)}{}^{C}D_x^{\alpha (x)}f(x)& ={\displaystyle \frac{1}{\Gamma (n-\alpha (x))}}{\int }_{a(x)}^x{(x-\tau )}^{n-\alpha (x)-1}{f}^{(n)}(\tau )\mathrm{d}\tau =\hfill \\ \hfill & ={\displaystyle \frac{1}{\Gamma (n+1-\alpha (x))}}{\int }_0^{{(x-a(x))}^{n-\alpha }}{f}^{(n)}(x-t_1^{{\displaystyle \frac{1}{n-\alpha (x)}}})d{t}_1,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {}_x{}^C{D}_{b(x)}^{\alpha (x)}f(x)& ={\displaystyle \frac{{(-1)}^n}{\Gamma (n-\alpha (x))}}{\int }_x^{b(x)}{(\tau -x)}^{n-\alpha (x)-1}{f}^{(n)}(\tau )\mathrm{d}\tau \hfill \\ \hfill & ={\displaystyle \frac{{(-1)}^n}{\Gamma (n+1-\alpha (x))}}{\int }_0^{{(b(x)-x)}^{n-\alpha (x)}}{f}^{(n)}(x+t_2^{{\displaystyle \frac{1}{n-\alpha (x)}}})d{t}_2.\hfill \end{array}$$

(15)

Next, we use an approximation based on the Taylor series [46,47] to expand the functions $f^{(n)}(x-t_1^{{\displaystyle \frac{1}{n-\alpha (x)}}})$ and $f^{(n)}(x+t_2^{{\displaystyle \frac{1}{n-\alpha (x)}}})$ around a point x,

$$\begin{array}{cc}\hfill f^{(n)}(x-t_1^{{\displaystyle \frac{1}{n-\alpha (x)}}})=& f^{(n)}(x)-t_1^{{\displaystyle \frac{1}{n-\alpha (x)}}}{f}^{(n+1)}(x)+t_1^{{\displaystyle \frac{2}{n-\alpha (x)}}}{\displaystyle \frac{1}{2!}}{f}^{(n+2)}(x)-t_1^{{\displaystyle \frac{3}{n-\alpha (x)}}}{\displaystyle \frac{1}{3!}}{f}^{(n+3)}(x)+\dots =\hfill \\ & \sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{t}_1^{\frac{k}{n-\alpha (x)}}}{k!}}{f}^{(k+n)}(x),\hfill \end{array}$$

(16)

and

$$\begin{array}{cc}\hfill f^{(n)}(x+t_2^{{\displaystyle \frac{1}{n-\alpha (x)}}})=& f^{(n)}(x)+t_2^{{\displaystyle \frac{1}{n-\alpha (x)}}}{f}^{(n+1)}(x)+t_2^{{\displaystyle \frac{2}{n-\alpha (x)}}}{\displaystyle \frac{1}{2!}}{f}^{(n+2)}(x)+t_2^{{\displaystyle \frac{3}{n-\alpha (x)}}}{\displaystyle \frac{1}{3!}}{f}^{(n+3)}(x)+\dots =\hfill \\ & \sum _{k=0}^{\infty }{\displaystyle \frac{t_2^{\frac{k}{n-\alpha (x)}}}{k!}}{f}^{(k+n)}(x).\hfill \end{array}$$

(17)

Substituting Equation (16) into Equation (14) and Equation (17) into Equation (15) and then integrating leads to the expression of the Caputo fractional derivatives as an infinite sum of higher-integer-order derivatives as follows, for the left

$$\begin{array}{cc}\hfill {}_{a(x)}{}^{C}D_x^{\alpha }f(x)=& {\displaystyle \frac{1}{\Gamma (n+1-\alpha (x))}}{\int }_0^{{(x-a(x))}^{n-\alpha (x)}}{f}^{(n)}(x-t_1^{{\displaystyle \frac{1}{n-\alpha (x)}}})d{t}_1=\hfill \\ \hfill & {\displaystyle \frac{1}{\Gamma (n+1-\alpha (x))}}{\int }_0^{{(x-a(x))}^{n-\alpha (x)}}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{t}_1^{\frac{k}{n-\alpha (x)}}}{k!}}{f}^{(k+n)}(x)d{t}_1=\hfill \\ \hfill & {\displaystyle \frac{1}{\Gamma (n+1-\alpha (x))}}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k}{k!}}{\displaystyle \frac{n-\alpha (x)}{k+n-\alpha (x)}}{(x-a(x))}^{k+n-\alpha (x)}{f}^{(k+n)}(x),\hfill \end{array}$$

(18)

and right

$$\begin{array}{cc}\hfill {}_x{}^C{D}_{b(x)}^{\alpha (x)}f(x)=& {\displaystyle \frac{{(-1)}^n}{\Gamma (n+1-\alpha (x))}}{\int }_0^{{(b(x)-x)}^{n-\alpha (x)}}{f}^{(n)}(x+t_2^{{\displaystyle \frac{1}{n-\alpha (x)}}})d{t}_1=\hfill \\ \hfill & {\displaystyle \frac{{(-1)}^n}{\Gamma (n+1-\alpha (x))}}{\int }_0^{{(b(x)-x)}^{n-\alpha (x)}}\sum _{k=0}^{\infty }{\displaystyle \frac{t_2^{\frac{k}{n-\alpha (x)}}}{k!}}{f}^{(k+n)}(x)d{t}_2=\hfill \\ \hfill & {\displaystyle \frac{1}{\Gamma (n+1-\alpha (x))}}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^n}{k!}}{\displaystyle \frac{n-\alpha (x)}{k+n-\alpha (x)}}{(b(x)-x)}^{k+n-\alpha (x)}{f}^{(k+n)}(x)\hfill \end{array}$$

(19)

Caputo fractional derivatives, respectively. However, since we consider an approximation of a fractional derivative, we take a finite sum of higher-integer-order derivatives, for left

$$\begin{array}{c}\hfill {}_{a(x)}{}^{C}D_x^{\alpha (x)}f(x)\approx {\displaystyle \frac{1}{\Gamma (n+1-\alpha (x))}}\sum _{k=0}^{m-1}{\displaystyle \frac{{(-1)}^k}{k!}}{\displaystyle \frac{n-\alpha (x)}{k+n-\alpha (x)}}{(x-a(x))}^{k+n-\alpha (x)}{f}^{(k+n)}(x),\end{array}$$

(20)

and right

$$\begin{array}{c}\hfill {}_x{}^C{D}_{b(x)}^{\alpha (x)}f(x)\approx {\displaystyle \frac{1}{\Gamma (n+1-\alpha (x))}}\sum _{k=0}^{m-1}{\displaystyle \frac{{(-1)}^n}{k!}}{\displaystyle \frac{n-\alpha (x)}{k+n-\alpha (x)}}{(b(x)-x)}^{k+n-\alpha (x)}{f}^{(k+n)}(x)\end{array}$$

(21)

Caputo fractional derivatives, respectively.

4. Application Examples for Fractional Derivatives

In the following, we present two examples, including in one of them a detailed discussion about the effect of the parameter m on the resulting accuracy of the solution. We consider various combinations with constant/variable terminal $a(x)$ of the fractional derivative and constant/variable fractional order $\alpha (x)$. It is clear that computational time increases for a larger value of parameter m; however, this is case-dependent (both from the considered problem and hardware perspectives), and is not scaled linearly.

Example 1. 

The left Caputo fractional derivative of the function $f(x)=sin(x)$ is defined as

$$\left\{\begin{array}{c}{\displaystyle {}_{a(x)}{}^{C}D_x^{\alpha (x)}sin(x)=\frac{1}{\Gamma (1-\alpha (x))}{\int }_{a(x)}^x{(x-\tau )}^{-\alpha (x)}cos(\tau )d\tau ,}\hfill \\ x\in \langle 0,3\pi \rangle \mathit{and}\alpha (x)\in (0,1\rangle ,\hfill \end{array}\right.$$

(22)

and after changing the variable (Equation (14)),

$$\left\{\begin{array}{c}{\displaystyle {}_{a(x)}{}^{C}D_x^{\alpha (x)}sin(x)=\frac{1}{\Gamma (1-\alpha (x))}{\int }_0^{{(x-a)}^{1-\alpha (x)}}cos(x-t_1^{\frac{1}{1-\alpha (x)}})d{t}_1,}\hfill \\ x\in \langle a(x),3\pi \rangle \mathit{and}\alpha (x)\in (0,1\rangle .\hfill \end{array}\right.$$

(23)

Next, using Equation (20), its approximation is defined as

$$\left\{\begin{array}{c}{\displaystyle {}_{a(x)}{}^{C}D_x^{\alpha (x)}sin(x)=\frac{1}{\Gamma (2-\alpha (x))}\sum _{k=0}^{m-1}\frac{{(-1)}^k}{k!}\frac{1-\alpha (x)}{k+1-\alpha (x)}{(x-a(x))}^{k+1-\alpha (x)}{sin}^{(k+1)}(x),}\hfill \\ x\in \langle a(x),3\pi \rangle \mathit{and}\alpha (x)\in (0,1\rangle .\hfill \end{array}\right.$$

(24)

First, we consider a constant terminal of a fractional derivative equal to zero ($a=0$), and a constant order of fractional derivative $\alpha =\{1.0,0.8,0.6,0.4,0.2\}$. The exact solution of Equation (22) is known only for that case, and it is as follows (taken from [48]):

$${}_0{}^{C}D_x^{\alpha }sin(x)=x^{1-\alpha }{E}_{2,2-\alpha }(-x^2),$$

(25)

where $E_{\alpha ,\beta }$ is the Mittag–Leffler function given by

$$E_{\alpha ,\beta }(x)=\sum _{\alpha ,\beta }^{\infty }{\displaystyle \frac{x^k}{\Gamma (\alpha k+\beta )}}.$$

(26)

Figure 2 shows the comparison of the exact solution of Equation (25) and the results of the approximation according to Equation (24) for $m=31$.

Furthermore, Figure 3 shows the influence of the number of terms $m=\{1,5,11,15,21,25,31\}$ taken in the expansion of Equation (24)₁ on the accuracy of the approximation, for $\alpha =0.8$ and $\alpha =0.2$. Additional,

Table 1. Example 1: The approximation errors Equation (27) on different x coordinates $x=\{\pi /4,\pi /2,\pi ,2\pi ,3\pi \}$, as well as the Euclidean norm of the error vector $\parallel err(x)\parallel$ for x belonging to the domain Equation (24)₂, for $a=0$ and $\alpha =0.8$.

m	$\mathit{err}(\mathit{\pi }/4)$	$\mathit{err}(\mathit{\pi }/2)$	$\mathit{err}(\mathit{\pi })$	$\mathit{err}(2\mathit{\pi })$	$\mathit{err}(3\mathit{\pi })$	$\parallel \mathit{err}(\mathit{x})\parallel$
1	7.24 × 10−2	2.67 × 10−1	4.01 × 10−1	−6.28 × 10−1	7.52 × 10−1	4.44
5	6.19 × 10−5	3.50 × 10−3	5.18 × 10−2	−2.67	2.06 × 101	6.79 × 101
11	−2.16 × 10−11	−7.56 × 10−8	−4.14 × 10−5	1.71 × 10−1	−1.97 × 101	5.69 × 101
15	−3.33 × 10−16	−1.04 × 10−11	−7.07 × 10−8	4.90 × 10−3	−3.07	8.14
21	−1.11 × 10−16	−5.00 × 10−16	9.35 × 10−13	−4.29 × 10−6	3.23 × 10−2	7.64 × 10−2
25	−1.11 × 10−16	−5.00 × 10−16	4.44 × 10−16	−1.61 × 10−8	6.23 × 10−4	1.38 × 10−3
31	−1.11 × 10−16	−5.00 × 10−16	2.22 × 10−16	1.25 × 10−12	−5.63 × 10−7	1.15 × 10−6
35	−1.11 × 10−16	−5.00 × 10−16	2.22 × 10−16	2.11 × 10−15	−2.83 × 10−9	5.53 × 10−9
41	−1.11 × 10−16	−5.00 × 10−16	2.22 × 10−16	3.33 × 10−16	3.86 × 10−13	7.53 × 10−13

and

Table 2. Example 1: The approximation errors of Equation (27) on different x coordinates $x=\{\pi /4,\pi /2,\pi ,2\pi ,3\pi \}$, as well as the Euclidean norm of the error vector $\parallel err(x)\parallel$ for x belonging to the domain Equation (24)₂, for $a=0$ and $\alpha =0.2$.

m	$\mathit{err}(\mathit{\pi }/4)$	$\mathit{err}(\mathit{\pi }/2)$	$\mathit{err}(\mathit{\pi })$	$\mathit{err}(2\mathit{\pi })$	$\mathit{err}(3\mathit{\pi })$	$\parallel \mathit{err}(\mathit{x})\parallel$
1	1.55 × 10−1	8.82 × 10−1	2.34	−4.38	6.14	2.82 × 101
5	1.89 × 10−4	1.62 × 10−2	3.69 × 10−1	−2.86 × 101	2.78 × 102	8.77 × 102
11	−6.98 × 10−11	−3.71 × 10−7	−3.09 × 10−4	1.93	−2.84 × 102	7.95 × 102
15	−5.55 × 10−16	−5.18 × 10−11	−5.34 × 10−7	5.61 × 10−2	−4.49 × 101	1.16 × 102
21	−1.11 × 10−16	−1.33 × 10−15	7.13 × 10−12	−4.96 × 10−5	4.76 × 10−1	1.10
25	−1.11 × 10−16	−1.33 × 10−15	1.94 × 10−15	−1.87 × 10−7	9.23 × 10−3	2.02 × 10−2
31	−1.11 × 10−16	−1.33 × 10−15	1.67 × 10−16	1.46 × 10−11	−8.37 × 10−6	1.69 × 10−5
35	−1.11 × 10−16	−1.33 × 10−15	1.67 × 10−16	−2.21 × 10−14	−4.22 × 10−8	8.15 × 10−8
41	−1.11 × 10−16	−1.33 × 10−15	1.67 × 10−16	−3.63 × 10−14	8.20 × 10−12	1.50 × 10−11

present (for $\alpha =0.8$ and $\alpha =0.2$, respectively) the approximation errors

$$err(x)=s_e(x)-s_a(x)$$

(27)

on different coordinates $x=\{\pi /4,\pi /2,\pi ,2\pi ,3\pi \}$, as well as the Euclidean norm of the error vector $\parallel err(x)\parallel$ for x belonging to the domain Equation (24)₂, where $s_e(x)$ is the exact solution of Equation (25) and $s_a$ is an approximated solution of Equation (24)₁.

The following can be noticed:

• The Taylor series-based approximation of the fractional derivative (or integral) becomes more accurate as m increases.
• When the x-coordinate is close to the fractional derivative interval limit ($a=0$ in the example presented), a faster convergence is observed compared to the x-coordinate far from this point. Therefore, for the x-coordinate close to the point a (or b for a right-hand fractional operator), a low-order Taylor expansion can provide sufficient accuracy. On the other hand, for x far from point a (or b), higher-order terms become more significant and a higher-order Taylor series is required to maintain sufficient accuracy.
• Faster convergence is observed for a fractional derivative order α closer to the value n ($\alpha =0.8$ compared to $\alpha =0.2$ in the example presented).
• Obviously, with an integer-order derivative (or integral) $\alpha =n$ (and $\alpha =1$ in this example), $m=1$ provides an exact solution, and there is no need for $m>1$, since the higher-order terms will be equal to 0 and will not affect the obtained solution.

It should be added that for $a\ne 0$, an analytical solution of Equation (22) is undetermined, and numerical methods are necessary. Thus, Figure 4 shows an example of the solution for different values of the terminal of the derivative $a=\{-0.3\pi ,-0.2\pi ,-0.1\pi ,0,0.1\pi ,0.2\pi ,0.3\pi \}$ (and constant order $\alpha =0.2$) applying the method introduced in this paper. To demonstrate that the proposed method works well in such applications, the solutions received are compared with those obtained by the Gauss–Jacobi quadrature method in [22].

Next, Figure 5 shows a comparison of the results for constant ($a=0$) and variable ($a(x)=x-2$) terminals of a fractional derivative, while the order of fractional derivative α is constant. Meanwhile, Figure 6 shows the results extended by also adopting α as the following functions:

$$\left\{\begin{array}{c}\alpha (x)=1-{\displaystyle \frac{0.3}{\pi }}x,\hfill \\ \alpha (x)=-{\displaystyle \frac{0.4x(x-3\pi )}{{\pi }^2}}+0.1.\hfill \end{array}\right.$$

(28)

Since it is assumed that $x\in \langle 0,3\pi \rangle$, then $0.1\le \alpha (x)\le 1.0$ for all x. The $\alpha (x)$ function is plotted in Figure 6. In the adopted variable derivative terminal, instead of a fixed starting point (as in the case of $a(x)=a=\mathit{const}$), the size of the horizon from the point x is fixed ($a(x)=x-2$).

Example 2. 

The left and right fractional Caputo derivatives of the functions $f_1(x)=x^2$ and $f_2(x)={(1-x)}^2$ with variable fractional order $\alpha (x)$ are defined as:

$$\left\{\begin{array}{c}{\displaystyle {}_0{}^{C}D_x^{\alpha (x)}{f}_1(x)={}_0{}^{C}D_x^{\alpha (x)}{x}^2=\frac{1}{\Gamma (1-\alpha (x))}{\int }_0^{x}2x{(x-\tau )}^{-\alpha (x)}d\tau ,}\hfill \\ {\displaystyle {}_x{}^C{D}_1^{\alpha (x)}{f}_2(x)={}_x{}^C{D}_1^{\alpha (x)}{(1-x)}^2=\frac{1}{\Gamma (1-\alpha (x))}{\int }_x^{1}2(1-x){(x-\tau )}^{-\alpha (x)}d\tau ,}\hfill \\ x\in \langle 0,1\rangle \mathit{and}\alpha ={\displaystyle \frac{5x+1}{10}}.\hfill \end{array}\right.$$

(29)

Since it is assumed that $x\in \langle 0,1\rangle$, then $0.1\le \alpha (x)\le 0.6$ for all x. Using Equations (20) and (21), the approximation of Equation (29) is defined as:

$$\left\{\begin{array}{c}{\displaystyle {}_0{}^{C}D_x^{\alpha (x)}{x}^2=\frac{1}{\Gamma (2-\alpha (x))}\sum _{k=0}^{m-1}\frac{{(-1)}^k}{k!}\frac{1-\alpha (x)}{k+1-\alpha (x)}{x}^{k+1-\alpha (x)}{(x^2)}^{(k+1)},}\hfill \\ {\displaystyle {}_x{}^C{D}_1^{\alpha (x)}{(1-x)}^2=\frac{1}{\Gamma (2-\alpha (x))}\sum _{k=0}^{m-1}\frac{-1}{k!}\frac{1-\alpha (x)}{k+1-\alpha (x)}{(1-x)}^{k+1-\alpha (x)}{({(1-x)}^2)}^{(k+1)},}\hfill \\ x\in \langle 0,1\rangle \mathit{and}\alpha ={\displaystyle \frac{5x+1}{10}}.\hfill \end{array}\right.$$

(30)

The exact solution is (taken from [49]):

$$\left\{\begin{array}{c}{\displaystyle {}_0{}^C{D}_x^{\alpha (x)}{x}^2=\frac{2}{\Gamma (3-\alpha (x))}{x}^{2-\alpha (x)},}\hfill \\ {\displaystyle {}_x{}^C{D}_1^{\alpha (x)}{(1-x)}^2=\frac{2}{\Gamma (3-\alpha (x))}{(1-x)}^{2-\alpha (x)}.}\hfill \end{array}\right.$$

(31)

Figure 7 shows the comparison of the exact solution of Equation (31) for a variable $\alpha (x)$, as well as the constant $\alpha =\{0.1,0.6\}$ and results of the approximation according to Equation (30) for $m=2$.

5. Fractional Boundary/Initial-Value Problems

In the following, we present two examples of the application of the proposed approximation method of fractional operators in the fractional boundary/initial value problems.

Example 3. 

Consider the fractional boundary value problem,

$$\left\{\begin{array}{c}{}_0{}^C{D}_{x}y(x)+cy(x)=x+{\displaystyle \frac{c{x}^{\alpha +1}}{\Gamma (\alpha +2)}},\hfill \\ x\in \langle 0,1\rangle ,\hfill \\ y(0)=0;y(1)={\displaystyle \frac{1}{\Gamma (\alpha +2)}},\hfill \\ \alpha \in (1,2\rangle ,c\in \mathbb{R}.\hfill \end{array}\right.$$

(32)

Let us solve Equation (32) using the weighted residual method, namely the Galerkin method. Hence, the approximate solution $y_m(x)$ of Equation (32)₁ is a linear combination of the basis functions ${\varphi }_i$,

$$y(x)\approx y_m(x)=\sum _{i=1}^m{w}_i{\varphi }_i,$$

(33)

where $w_i$ are the nodal values of function $y(x)$. Basis functions are constructed such that each basic function has a value of 1 at its associated node and 0 at all other nodes. These functions are built on m Chebyshev nodes of the second kind [50,51] to minimise the problem of the Runge phenomenon [52,53], which is a problem of oscillation at the edges of an interpolation interval that occurs when interpolating with polynomials of a high degree for equally spaced interpolation points.

To simplify the derivation of the solution, and due to the fact that Chebyshev nodes are defined in the interval $\langle -1,1\rangle$, the natural coordinates ξ are used. The coordinate transformation is described as

$$x={\displaystyle \frac{x_m-x_1}{2}}\xi +{\displaystyle \frac{x_m+x_1}{2}},$$

(34)

or alternatively as

$$\xi ={\displaystyle \frac{2x}{x_m-x_1}}-{\displaystyle \frac{x_m+x_1}{x_m-x_1}},$$

(35)

where $x_m$ and $x_1$ are the coordinates at last and first node. In the example presented, we have $x_1=0$, $x_m=1$, therefore

$$x={\displaystyle \frac{1}{2}}(\xi +1),$$

(36)

and

$$\xi =2x-1.$$

(37)

Then, the left Caputo fractional derivative of the approximate solution $y_m(x)$ is

$$\begin{array}{c}\hfill {}_0{}^C{D}_x{y}_m(x)=\sum _{j=1}^m{\displaystyle \frac{w_j}{\Gamma (n+1-\alpha )}}\sum _{k=0}^{m-1}{\displaystyle \frac{{(-1)}^k}{k!}}{\displaystyle \frac{n-\alpha }{k+n-\alpha }}{x}^{k+n-\alpha }{\varphi }_j^{(k+n)}(x),\end{array}$$

(38)

and in natural coordinates (Equation (37)),

$$\begin{array}{c}\hfill {}_0{}^C{D}_x{y}_m(\xi )=\sum _{j=1}^m{\displaystyle \frac{w_j}{\Gamma (n+1-\alpha )}}\sum _{k=0}^{m-1}{\displaystyle \frac{{(-1)}^k}{k!}}{\displaystyle \frac{n-\alpha }{k+n-\alpha }}{\left({\displaystyle \frac{1}{2}}(\xi +1)\right)}^{k+n-\alpha }{2}^{k+n}{\varphi }_j^{(k+n)}(\xi ).\end{array}$$

(39)

The residual function of Equation (32)₁ is

$$\begin{array}{c}R(\xi ,w_j)={}_0{}^{C}D_x{y}_m(\xi )+c{y}_m(\xi )-{\displaystyle \frac{1}{2}}(\xi +1)-{\displaystyle \frac{c{\left(\frac{1}{2}(\xi +1)\right)}^{\alpha +1}}{\Gamma (\alpha +2)}}=\hfill \\ \hspace{1em}\sum _{j=1}^m{w}_j\left[{\displaystyle \frac{1}{\Gamma (n+1-\alpha )}}\sum _{k=0}^{m-1}{\displaystyle \frac{{(-1)}^k}{k!}}{\displaystyle \frac{n-\alpha }{k+n-\alpha }}{\left({\displaystyle \frac{1}{2}}(\xi +1)\right)}^{k+n-\alpha }{2}^{k+n}{\varphi }_j^{(k+n)}(\xi )+c{\varphi }_j(\xi )\right]+\hfill \\ \hspace{1em}-{\displaystyle \frac{1}{2}}(\xi +1)-{\displaystyle \frac{c{(\frac{1}{2}(\xi +1))}^{\alpha +1}}{\Gamma (\alpha +2)}}.\hfill \end{array}$$

(40)

Finally, the weighted residual equation (weak integral form) is

$${\int }_0^{1}R(x,w_j){\varphi }_i(x)dx={\displaystyle \frac{1}{2}}{\int }_{-1}^{1}R(\xi ,w_j){\varphi }_i(\xi )d\xi =0$$

(41)

for $j=1,2,\dots ,m$, which results in the transformation of a fractional differential problem into a system of m algebraic equations for unknown $w_i$. Utilising matrix notation, we have

$$\left\{\mathbf{K}\right\}\left\{\mathbf{w}\right\}=\left\{\mathbf{F}\right\},$$

(42)

where

$$K_{ij}={\displaystyle \frac{1}{2}}{\int }_{-1}^1\left[{\displaystyle \frac{1}{\Gamma (n+1-\alpha )}}\sum _{k=0}^{m-1}{\displaystyle \frac{{(-1)}^k}{k!}}{\displaystyle \frac{n-\alpha }{k+n-\alpha }}{\left({\displaystyle \frac{1}{2}}(\xi +1)\right)}^{k+n-\alpha }{2}^{k+n}{\varphi }_j^{(k+n)}(\xi )+c{\varphi }_j(\xi )\right]{\varphi }_i(\xi )d\xi ,$$

(43)

and

$$F_i={\displaystyle \frac{1}{2}}{\int }_{-1}^1\left({\displaystyle \frac{1}{2}}(\xi +1)+{\displaystyle \frac{c{(\frac{1}{2}(\xi +1))}^{\alpha +1}}{\Gamma (\alpha +2)}}\right){\varphi }_i(\xi )d\xi .$$

(44)

Following this procedure, one can perform numerical integration using the Gauss–Legendre quadrature method. Then, the analytical integration of a given function $g(\xi )$ over the domain space $\xi \in \langle -1,1\rangle$ is replaced by the weighted sum

$${\int }_{-1}^{1}g(\xi )d\xi =\sum _{i=1}^{n^G}g({\xi }_i^G)w_i^G,$$

(45)

where $w_i^G$ are the quadrature weights, ${\xi }_i^G$ are the location of integration points and $n^G\in \mathbb{N}$ is the number of integration points. In the study, $n^G=m$ is adopted.

The boundary conditions in Equation (32)₃ imply $w_1=1$ and $w_m={\displaystyle \frac{1}{\Gamma (\alpha +2)}}$, and they are included by modifying the matrix $\mathbf{K}$ and the vector $\mathbf{F}$, which leads to a system of equations,

$$K_{ij}{w}_i=F_i-(K_{i1}{w}_1+K_{im}{w}_m),$$

(46)

for $j=2÷m-1$, and $i=2÷m-1$. After including the boundary conditions, a system of $m-2$ equations remains to be solved to determine the values of $w_i$. We solve the linear system of equations using LU factorisation.

Appendix A contains the obtained approximate solutions for $m=5$ and $c=2.0$, and different values of fractional derivative order $\alpha =\{1.2,1.4,1.6,1.8,2.0\}$. Furthermore, the exact solution of Equation (32) (taken from [54]) is

$$y(x)={\displaystyle \frac{x^{\alpha +1}}{\Gamma (\alpha +2)}}.$$

(47)

Figure 8 shows the comparison of the exact (Equation (47)) and approximated solutions. In addition,

Table 3. Example 3: The approximation errors on different x coordinates $x=\{0.25,0.5,0.75\}$, as well as the Euclidean norm of the error vector $\parallel err(x)\parallel$ for x belonging to the domain Equation (32)₂, for $\alpha =1.8$.

m	$\mathit{err}(0.25)$	$\mathit{err}(0.5)$	$\mathit{err}(0.75)$	$\parallel \mathit{err}(\mathit{x})\parallel$
3	−2.92 × 10−6	−1.08 × 10−2	−1.57 × 10−2	9.67 × 10−2
5	−1.22 × 10−4	−6.21 × 10−5	−3.98 × 10−5	7.14 × 10−4
7	−1.14 × 10−5	−1.50 × 10−5	−5.91 × 10−6	1.06 × 10−4
9	−3.35 × 10−6	−3.22 × 10−6	−2.79 × 10−6	2.94 × 10−5
11	−1.62 × 10−6	−1.51 × 10−6	−8.13 × 10−7	1.10 × 10−5
15	−3.17 × 10−7	−3.37 × 10−7	−2.12 × 10−7	2.53 × 10−6
21	−7.95 × 10−6	−1.45 × 10−5	−1.90 × 10−5	1.38 × 10−4
25	4.93 × 10−3	8.91 × 10−3	1.15 × 10−2	8.61 × 10−2
31	−8.77 × 10−3	1.95 × 10−2	4.81 × 10−2	6.69 × 10−1

and

Table 4. Example 3: The approximation errors on different x coordinates $x=\{0.25,0.5,0.75\}$, as well as the Euclidean norm of the error vector $\parallel err(x)\parallel$ for x belonging to the domain Equation (32)₂, for $\alpha =1.2$.

m	$\mathit{err}(0.25)$	$\mathit{err}(0.5)$	$\mathit{err}(0.75)$	$\parallel \mathit{err}(\mathit{x})\parallel$
3	−1.09 × 10−2	−1.96 × 10−2	−1.76 × 10−2	1.43 × 10−1
5	−6.33 × 10−4	−2.95 × 10−4	−5.48 × 10−4	4.70 × 10−3
7	−6.33 × 10−5	−1.10 × 10−4	−4.31 × 10−5	8.09 × 10−4
9	−2.62 × 10−5	−1.77 × 10−5	−2.48 × 10−5	2.40 × 10−4
11	−1.34 × 10−5	−1.22 × 10−5	−9.22 × 10−6	9.43 × 10−5
15	−2.82 × 10−6	−2.89 × 10−6	−2.09 × 10−6	2.32 × 10−5
21	−4.59 × 10−4	−7.58 × 10−4	−9.30 × 10−4	6.99 × 10−3
25	2.60 × 10−1	4.32 × 10−1	5.31 × 10−1	3.96
31	−1.49 × 10−3	2.66 × 10−2	8.76 × 10−2	1.57

collect (for $\alpha =1.8$ and $\alpha =1.2$, respectively) the approximation errors at different coordinates $x=\{0.25,0.5,0.75\}$, as well as the Euclidean norm of the error vector $\parallel err(x)\parallel$ for x belonging to the domain in Equation (32)₂. The best approximation was obtained for $m=15$ while increasing the value of m to 21 or more, resulting in a poorer fit. When the order of the derivative α is close to n, a good alignment of the approximate solution with the exact one is obtained more quickly.

Example 4. 

Consider the fractional initial-value problem,

$$\left\{\begin{array}{c}{}_{a(x)}{}^{C}D_x^{\alpha (x)}y(x)={\displaystyle \frac{2}{\Gamma (2-\alpha (x))}}\left(x+{\displaystyle \frac{\alpha (x)-1}{2-\alpha (x)}}\right),\hfill \\ x\in \langle 0,1\rangle ,\hfill \\ y(0)=0,\hfill \end{array}\right.$$

(48)

with the following terminal $a(x)$ and fractional order $\alpha (x)$,

$$\left\{\begin{array}{c}a(x)=x-1,\hfill \\ a(x)=x-0.5,\hfill \\ a(x)=0,\hfill \end{array}\right.\mathit{and}\left\{\begin{array}{c}\alpha (x)={\displaystyle \frac{10-7x}{10}},\hfill \\ \alpha (x)=0.75.\hfill \end{array}\right.$$

(49)

To obtain the approximate solution, the same procedure as described in Example 3 was adopted. Furthermore, the exact solution of Equation (48) is known only for $a(x)=x-1$, and it is (regardless of $\alpha (x)$),

$$y(x)=x^2.$$

(50)

Figure 9 shows the comparison of the known exact solution of Equation (50) and the approximated solution for $m=5$.

6. Conclusions

The paper presents a method for approximating fractional Caputo derivatives of variable order and/or variable terminals. The procedure involves rearranging the fractional derivative as a sum of higher-integer-order derivatives. The accuracy of the presented method depends on

• the number of terms m included in the expansion—approximation becomes more accurate as m increases;
• the proximity of the x coordinate to the limit of the fractional derivative interval (points a or b)—for x close to points a or b, a low-order Taylor expansion m can provide sufficient accuracy, and for x far from points a or b, a higher-order Taylor series m is required to maintain sufficient accuracy;
• order $\alpha$ of fractional derivative—faster convergence is observed for $\alpha$ closer to value n. Additionally, with the derivative of integer order $\alpha =n$ (or integral), $m=1$ provides an exact solution, and there is no need for $m>1$, since the higher-order terms will be equal to 0 and will not affect the obtained solution.

• The examples presented demonstrate very good agreement with the exact/reference solutions. The advantage of the proposed method is its simplicity and ease of implementation. It is particularly useful when the exact solution of the fractional derivative/integral is impossible to obtain.

Furthermore, the presented method was used to solve the fractional initial/boundary value problems with the weighted residual (Galerkin) method. When the order of the derivative, denoted by $\alpha$, approaches the value of n, the approximate solution aligns more rapidly with the exact solution.