Central Part Interpolation Approach for Solving Initial Value Problems of Systems of Linear Fractional Differential Equations

Abstract

We consider an initial value problem for a system of linear fractional differential equations of Caputo type. Using an integral equation reformulation of the underlying problem, we first study the existence, uniqueness and smoothness of its exact solution. Based on the obtained results, a collocation-type method using the central part interpolation approach on the uniform grid is constructed and analyzed. Optimal convergence order of the proposed method is established and confirmed by numerical experiments.

1. Introduction

In recent years, differential equations involving fractional differential operators of fractional (i.e., of any positive real) order have received increasing attention due to their applications in the modeling of a variety of real-life processes, see, for example, [1,2,3,4]. For the fundamental theory of fractional derivatives and equations containing them we refer to [5,6,7,8], see also [9,10,11,12,13]. It follows from these works that typically, fractional differential equations pose an extra challenge compared to integer-order differential equations. Consequently, effective approximation methods for fractional differential equations have been the subject of intense investigations in the last years. For example, we point out the use of discrete convolution techniques [14,15] (see also [16]), finite-difference methods on graded meshes [17], Krylov methods [18], linear multistep methods [19], matrix-function methods [20,21], predictor–corrector schemes [22], spectral-type methods [23,24,25], and many others; we refer the reader to the review papers [26,27] for a more comprehensive discussion and comparison of various numerical methods for Caputo-type fractional differential equations.

In particular, we wish to highlight collocation-based methods, which generally enable us to obtain a stable and high-order procedure with uniform convergence on the whole interval of integration [28]. Note that, since a solution to a Caputo-type fractional differential equation will, in general, exhibit non-smooth behaviour even in the case of smooth data (see, e.g., [29,30] or Theorem 1 below), one should take this possible singular behaviour into account—numerical methods which assume smooth solutions for fractional differential equations are valid only on a small subclass of problems, as is made clear in [31,32]. In collocation methods, it is typical to overcome the non-smooth behaviour of the exact solution of the underlying fractional differential equation by applying a graded grid, where the grid points are more densely located near the point of singular behaviour [33,34,35]. However, a possible problem that may arise in the use of strongly non-uniform grids is the accumulation of round-off errors and corresponding numerical instability in the calculations. Alternative approaches include the use of non-polynomial bases [36,37,38] or the reformulation of the underlying problem using a suitable smoothing transformation [29,30].

In the present paper, our aim is to introduce and analyse a high-order collocation-based method for a class of initial value problems to a linear system of Caputo fractional differential equations. To this end, the problem is first reformulated as a system of weakly singular Volterra integral equations, and the existence, uniqueness and regularity of the exact solution of the underlying problem are studied. A smoothing transformation is then used to reduce the possible singular behaviour of the exact solution. After that, we construct and study a collocation method based on central part interpolation, using continuous piecewise polynomials on a uniform grid. In the collocation methods based on central part interpolation for a uniform grid $\left\{ih\right\}$ with the stepsize h and running integer i, we choose a parameter $m\in \mathbb{N}:=\{1,2,\dots \}$ and construct for each subinterval $[ih,(i+1\left)h\right]$ a separate Lagrange interpolation polynomial of degree $m-1$, which interpolates the solution at m grid points $\left\{\right(i+k\left)h\right\}$ (for all integers k satisfying $-m/2<k\le m/2$) surrounding the subinterval $[ih,(i+1\left)h\right]$. In other words, each subinterval $[ih,(i+1\left)h\right]$ of length h is the centre of an interpolation interval of length $(m-1)h$; for the central subinterval $[ih,(i+1\left)h\right]$, this interpolation gives a considerably more precise estimate than on the whole interval of interpolation (see Section 5 below). Note that the central part interpolation approach was introduced in [39] for weakly singular Fredholm integral equations, and in [40], it was shown that these methods have an accuracy and numerical stability advantage compared to the usual piecewise polynomial collocation schemes, where the collocation points are located inside the interval $[ih,(i+1\left)h\right]$ (including collocation using Chebyshev knots). In [41,42], this approach was applied to the numerical solution of fractional differential and integro-differential equations, and here, we extend it to systems of fractional differential equations.

The rest of the paper is organized as follows. First, we present the exact problem setting in Section 2. Then, the existence, uniqueness and regularity of the exact solution are studied in Section 3. After that, a smoothing change of variables is introduced and some of its properties are presented in Section 4. In Section 5, we provide a detailed overview of the central part interpolation approach, and in Section 6, the convergence order of the proposed collocation-type method is obtained. Finally, the theoretical results are tested by some numerical experiments in Section 7.

2. Problem Formulation and Basic Notation

Let $\alpha \in (0,1)$ and $n\in \mathbb{N}$. We consider an initial value problem for a system of fractional differential equations in the form

$$\begin{array}{cc}\hfill \left({\mathit{D}}_{\mathit{Cap}}^{\alpha }\overrightarrow{y}\right)\left(t\right)& =A\left(t\right)\overrightarrow{y}\left(t\right)+\overrightarrow{f}\left(t\right),t\in [0,1],\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \overrightarrow{y}\left(0\right)& =\overrightarrow{y_0},\hfill \end{array}$$

(2)

where

$$\begin{array}{cc}\hfill \overrightarrow{y}\left(t\right)& ={(y_1\left(t\right),\dots ,y_n\left(t\right))}^T:=\left(\begin{array}{c}{y}_1\left(t\right)\\ ⋮\\ y_n\left(t\right)\end{array}\right)\hfill \end{array}$$

is the unknown function. We assume that

$$\begin{array}{cc}\hfill A=A\left(t\right)& =\left(\begin{array}{ccc}{a}_{11}\left(t\right)& \dots & a_{1n}\left(t\right)\\ ⋮& \ddots & ⋮\\ a_{n1}\left(t\right)& \dots & a_{nn}\left(t\right)\end{array}\right),a_{ij}\in C[0,1],i,j=1,\dots ,n,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \overrightarrow{f}=\overrightarrow{f}\left(t\right)& :={(f_1\left(t\right),\dots ,f_n\left(t\right))}^T,f_i\in C[0,1],i=1,\dots ,n,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \overrightarrow{y_0}& :={(y_{01},\dots ,y_{0n})}^T,y_{0i}\in \mathbb{R}:=(-\infty ,\infty ),i=1,\dots ,n,\hfill \end{array}$$

(5)

where by $C[0,1]$ we denote the Banach space of continuous functions $y:[0,1]\to \mathbb{R}$ with the norm ${\parallel y\parallel }_{\infty }={max}_{0\le t\le 1}\left|y\left(t\right)\right|$, and

$$\begin{array}{c}\hfill \left({\mathit{D}}_{\mathit{Cap}}^{\alpha }\overrightarrow{y}\right)\left(t\right):=\left(\begin{array}{c}\left(D_{Cap}^{\alpha }{y}_1\right)\left(t\right)\\ ⋮\\ \left(D_{Cap}^{\alpha }{y}_n\right)\left(t\right)\end{array}\right)\end{array}$$

denotes the Caputo differential operator of order $\alpha$ for $\overrightarrow{y}\left(t\right),t\in [0,1]$. The Caputo fractional derivative $D_{Cap}^{\alpha }y$ of order $\alpha \in (0,1)$ for a continuous function $y\in C[0,1]$ is defined by the formula (see [8])

$$\left(D_{Cap}^{\alpha }y\right)\left(t\right)={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(J^{1-\alpha }[y-y\left(0\right)]\right)\left(t\right),0<t\le 1,$$

with $J^{\delta }$, the Riemann–Liouville integral operator of order $\delta \in [0,\infty )$, given by

$$\left(J^{\delta }y\right)\left(t\right):={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\delta \right)}}{\int }_0^t{(t-s)}^{\delta -1}y\left(s\right)\mathrm{d}s,0\le t\le 1,\delta >0;J^0:=I.$$

(6)

Here, $\mathsf{\Gamma }\left(x\right):={\int }_0^{\infty }{e}^{-s}{s}^{x-1}\mathrm{d}s(x>0)$ is the Euler gamma function, and I is the identity mapping.

It is well known (see, e.g., [43]) that $J^{\delta }(\delta >0)$ is linear, bounded and compact as an operator from $C[0,1]$ into $C[0,1]$, and we have for any $y\in C[0,1]$ that (see, e.g., [7])

$$\begin{array}{c}\hfill J^{\delta }y\in C[0,1],\left(J^{\delta }y\right)\left(0\right)=0,\delta >0,\\ \hfill D_{Cap}^{\delta }{J}^{\eta }y=J^{\eta -\delta }y,0<\delta \le \eta .\end{array}$$

(7)

Note that if $\alpha \in (0,1)$, then a sufficient condition for existence of $D_{Cap}^{\alpha }y\in C[0,1]$ is that y is a continuously differentiable function on $[0,1]$. However, this is not a necessary condition. In [9], Vainikko gives a comprehensive description of the range $J^{\delta }C[0,b]$ of $J^{\delta }$ as an operator from $C[0,1]$ into $C[0,1]$, for any $\delta >0$. Among other things, he derives the necessary and sufficient conditions for the existence of a continuous Caputo fractional derivative $D_{Cap}^{\alpha }y\in C[0,1]$ of a function $y\in C[0,1]$.

To properly characterize the regularity of solutions of problem (1)–(2), we shall use an adaptation of a weighted space of functions introduced by Vainikko for multidimensional weakly singular integral equations in [44] (for the one-dimensional case, see [43]). In what follows, we denote by $C^q\left(\mathsf{\Omega }\right)$, $q\in \mathbb{N}$, the set of q times continuously differentiable functions on $\mathsf{\Omega }\subset \mathbb{R}$. For given $q\in \mathbb{N}$ and $\nu \in \mathbb{R}$, $\nu <1$, by $C^{q,\nu }(0,1]$ we denote the set of continuous scalar functions $y:[0,1]\to \mathbb{R}$ which are q times continuously differentiable in $(0,1]$ and such that for all $x\in (0,1]$ and $i=1,\dots ,q$ the following estimates hold:

$$|y^{\left(i\right)}\left(x\right)|\le c\left\{\begin{array}{ccc}1\hfill & \mathrm{if}\hfill & i<1-\nu ,\hfill \\ 1+|logx|\hfill & \mathrm{if}\hfill & i=1-\nu ,\hfill \\ x^{1-\nu -i}\hfill & \mathrm{if}\hfill & i>1-\nu .\hfill \end{array}\right.$$

Here, $c=c\left(y\right)$ is a positive constant. In other words, $y\in C^{q,\nu }(0,1]$ if $y\in C[0,1]\cap C^q(0,1]$ and

$${\left|y\right|}_{q,\nu }:=\sum _{i=1}^q\underset{0<x\le 1}{sup}{\omega }_{i-1+\nu }\left(x\right)\left|y^{\left(i\right)}\left(x\right)\right|<\infty ,$$

where, for $x>0$, $\lambda \in \mathbb{R}$,

$${\omega }_{\lambda }\left(x\right):=\left\{\begin{array}{ccc}1\hfill & \mathrm{if}\hfill & \lambda <0,\hfill \\ {\displaystyle \frac{1}{1+|logx|}}\hfill & \mathrm{if}\hfill & \lambda =0,\hfill \\ x^{\lambda }\hfill & \mathrm{if}\hfill & \lambda >0.\hfill \end{array}\right.$$

Equipped with the norm ${\parallel y\parallel }_{C^{q,\nu }(0,1]}:={\parallel y\parallel }_{\infty }+{\left|y\right|}_{q,\nu }$, $y\in C^{q,\nu }(0,1]$, the set $C^{q,\nu }(0,1]$ becomes a Banach space, for which the embeddings

$$\begin{array}{c}\hfill C^q[0,1]\subset C^{q,\nu }(0,1]\subset C^{m,\mu }(0,1]\subset C[0,1],q\ge m\ge 1,\nu \le \mu <1\end{array}$$

(8)

hold. As an example, a function in the form $y\left(x\right)=g_1\left(x\right)x^{\mu }+g_2\left(x\right)$, where $g_1,g_2\in C^q[0,1]$, $q\in \mathbb{N}$ and $\mu >0$, satisfies $y\in C^{q,\nu }(0,1]$ for all $\nu \ge 1-\mu$.

Similarly, for vector-valued functions $\overrightarrow{y}={(y_1,\dots ,y_n)}^T$, $n\in \mathbb{N}$, we say that $\overrightarrow{y}\in C_n^{q,\nu }(0,1]$ if $y_i\in C^{q,\nu }(0,1]$, $i=1,\dots ,n$, for given $q\in \mathbb{N}$ and $\nu \in \mathbb{R}$, $\nu <1$. Clearly, $C_n^{q,\nu }(0,1]$ is a Banach space with the norm

$$\begin{array}{c}\hfill \left|\right|\overrightarrow{y}{\left|\right|}_{C_n^{q,\nu }(0,1]}=\underset{i=1,\dots ,n}{max}\left|\right|y_i{\left|\right|}_{C^{q,\nu }(0,1]},\overrightarrow{y}={(y_1,\dots ,y_n)}^T\in C_n^{q,\nu }(0,1].\end{array}$$

Correspondingly, for $\overrightarrow{y}={(y_1,\dots ,y_n)}^T$, $n\in \mathbb{N}$, we write $\overrightarrow{y}\in C_n[0,1]$ to mean that $y_i\in C[0,1]$, $i=1,\dots ,n$. Note that $C_n[0,1]$ is a Banach space with the norm

$$\begin{array}{c}\hfill \left|\right|\overrightarrow{y}{\left|\right|}_{C_n[0,1]}=\underset{i=1,\dots ,n}{max}\left|\right|y_i{\left|\right|}_{\infty },\overrightarrow{y}={(y_1,\dots ,y_n)}^T\in C_n[0,1].\end{array}$$

Further, $\overrightarrow{y}\in C_n^q[0,1]$ means that $y_i\in C^q[0,1]$, $i=1,\dots ,n$ for some $q\in \mathbb{N}$.

Finally, for a Banach space F, let $\mathcal{L}\left(F\right)=\mathcal{L}(F,F)$ be the Banach space of linear bounded operators $B:F\to F$, with the norm

$${\parallel B\parallel }_{\mathcal{L}\left(F\right)}={sup\{\parallel Bx\parallel }_F:x\in {F,\parallel x\parallel }_F\le 1\}.$$

3. Existence, Uniqueness and Smoothness of the Solution

In what follows, we introduce an integral reformulation of (1)–(2). Note that, keeping in mind the results of [9], we are interested only in the solutions $\overrightarrow{y}\in C_n[0,1]$ to problem (1)–(2) such that ${\mathit{D}}_{\mathit{Cap}}^{\alpha }\overrightarrow{y}\in C_n[0,1]$.

Let $0<\alpha <1$ and let $\overrightarrow{y}={(y_1,\dots ,y_n)}^T\in C_n[0,1]$ be an arbitrary function such that its Caputo fractional derivative $\overrightarrow{z}={\mathit{D}}_{\mathit{Cap}}^{\alpha }\overrightarrow{y}$ belongs to $C_n[0,1]$:

$$\begin{array}{c}\hfill \overrightarrow{z}={(z_1,\dots ,z_n)}^T={(D_{Cap}^{\alpha }{y}_1,\dots ,D_{Cap}^{\alpha }{y}_n)}^T\in C_n[0,1].\end{array}$$

Then (cf [8])

$$\begin{array}{c}\hfill \overrightarrow{y}\left(t\right)=\left({\mathit{J}}^{\alpha }\overrightarrow{z}\right)\left(t\right)+\overrightarrow{c},t\in [0,1],\end{array}$$

(9)

where $\overrightarrow{c}\in {\mathbb{R}}^n$ is a constant vector with arbitrary elements, and ${\mathit{J}}^{\alpha }\overrightarrow{z}$ is the component-wise application of the Riemann–Liouville integral operator $J^{\alpha }$, i.e.,

$$\left({\mathit{J}}^{\alpha }\overrightarrow{z}\right)\left(t\right)={\left(\left(J^{\alpha }{z}_1\right)\left(t\right),\dots ,\left(J^{\alpha }{z}_n\right)\left(t\right)\right)}^T,t\in [0,1].$$

Note that from (7), it follows that a function in the form (9) satisfies the condition (2) if and only if $\overrightarrow{c}=\overrightarrow{y_0}$. Thus, an arbitrary function $\overrightarrow{y}\in C_n[0,1]$ with $\overrightarrow{z}={\mathit{D}}_{\mathit{Cap}}^{\alpha }\overrightarrow{y}\in C_n[0,1]$ satisfies the condition (2) if and only if

$$\begin{array}{c}\hfill \overrightarrow{y}\left(t\right)=\left({\mathit{J}}^{\alpha }\overrightarrow{z}\right)\left(t\right)+\overrightarrow{y_0},0\le t\le 1.\end{array}$$

(10)

Let $\overrightarrow{y}\in C_n[0,1]$ be a solution to problem (1)–(2) so that $\overrightarrow{z}={\mathit{D}}_{\mathit{Cap}}^{\alpha }\overrightarrow{y}\in C_n[0,1]$. Then, from (1), we have that $\overrightarrow{z}=A\overrightarrow{y}+\overrightarrow{f}$, and by (10), we obtain

$$\begin{array}{cc}\hfill \overrightarrow{y}& ={\mathit{J}}^{\alpha }(A\overrightarrow{y}+\overrightarrow{f})+\overrightarrow{y_0}={\mathit{J}}^{\alpha }A\overrightarrow{y}+{\mathit{J}}^{\alpha }\overrightarrow{f}+\overrightarrow{y_0}.\hfill \end{array}$$

Thus, we obtain for $\overrightarrow{y}$ an integral equation

$$\begin{array}{c}\hfill \overrightarrow{y}=T\overrightarrow{y}+\overrightarrow{g},\end{array}$$

(11)

where

$$\begin{array}{cc}\hfill \left(T\overrightarrow{y}\right)\left(t\right)& =\left({\mathit{J}}^{\alpha }A\overrightarrow{y}\right)\left(t\right),t\in [0,1],\hfill \\ \hfill \overrightarrow{g}\left(t\right)& =\left({\mathit{J}}^{\alpha }\overrightarrow{f}\right)\left(t\right)+\overrightarrow{y_0},t\in [0,1].\hfill \end{array}$$

(12)

Therefore, if $\overrightarrow{y}\in C_n[0,1]$, with ${\mathit{D}}_{\mathit{Cap}}^{\alpha }\overrightarrow{y}\in C_n[0,1]$ is a solution of (1)–(2), then $\overrightarrow{y}$ is also the solution to Equation (11), and vice versa. Thus, we can use the integral Equation (11) as the basis for constructing high-order methods for the numerical solution of (1)–(2).

Following some ideas of [43] (see the proof of Lemma 2.2 in [43]), it is possible to prove the following result.

Lemma 1.

Let $K\in C\left(Q\right)$ with $Q=\left\{(t,s):0\le s\le t\le 1\right\}$. Let $\eta \in (0,1)$ and let $q,n\in \mathbb{N}$. Then operator $\mathit{S}$ defined by

$$\begin{array}{c}\hfill \left(\mathit{S}\overrightarrow{y}\right)\left(t\right)={\int }_0^t{(t-s)}^{-\eta }K(t,s)\overrightarrow{y}\left(s\right)\mathrm{d}s,\overrightarrow{y}={(y_1,\dots ,y_n)}^T\in C_n[0,1],t\in [0,1],\end{array}$$

is compact as an operator from $C_n[0,1]$ to $C_n[0,1]$. Furthermore, $\mathit{S}$ is compact as an operator from $C_n^{q,\theta }(0,1]$ into $C_n^{q,\theta }(0,1]$, where $\eta \le \theta <1$.

Below we will also need Lemmas 2–4, the proofs of which can be found in [28,43,45], respectively.

Lemma 2.

If $v_1,v_2\in C^{q,\nu }(0,1],q\in \mathbb{N},\nu \in \mathbb{R},\nu <1$, then $v_1{v}_2\in C^{q,\nu }(0,1]$, and

$$\begin{array}{c}\hfill \left|\right|v_1{v}_2{\left|\right|}_{C^{q,\nu }(0,1]}\le c\left|\right|v_1{\left|\right|}_{C^{q,\nu }(0,1]}\left|\right|v_2{\left|\right|}_{C^{q,\nu }(0,1]},\end{array}$$

with a constant c, which is independent of $v_1$ and $v_2$.

Lemma 3.

Let $v\in C[0,1]$, $v\left(t\right)\ge 0$, and let v be non-decreasing on $[0,1]$. Moreover, assume that the continuous, non-negative function $w\in C[0,1]$ satisfies the inequality

$$\begin{array}{c}\hfill w\left(t\right)\le v\left(t\right)+M{\int }_0^t{\displaystyle \frac{{(t-s)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}}w\left(s\right)ds,t\in [0,1],\end{array}$$

for some $\alpha \in (0,1)$ and constant $M>0$. Then,

$$\begin{array}{c}\hfill w\left(t\right)\le E_{\alpha }\left(M{t}^{\alpha }\right)v\left(t\right),t\in [0,1],\end{array}$$

where $E_{\alpha }$ denotes the Mittag–Leffler function, defined by

$$\begin{array}{c}\hfill E_{\beta }\left(s\right)=\sum _{j=0}^{\infty }{\displaystyle \frac{s^j}{\mathsf{\Gamma }(1+j\beta )}},s\in \mathbb{R},\beta >0.\end{array}$$

Lemma 4.

Let $m\in \mathbb{N}$, $u\in C^m[a_1,b_1]$ and $v\in C^m[a_2,b_2]$ be real-valued functions such that u maps $[a_1,b_1]$ into $[a_2,b_2]$. Then, the derivatives of the composite function $w\left(t\right)=v\left(u\right(t\left)\right)$ can be expressed by the Faà di Bruno formula

$$\begin{array}{c}\hfill w^{\left(j\right)}\left(t\right)=\sum {\displaystyle \frac{j!}{k_1!\dots k_j!}}{v}^{\left(k\right)}\left(s\right){|}_{s=u\left(t\right)}{\left({\displaystyle \frac{u^{\prime }\left(t\right)}{1!}}\right)}^{k_1}\dots {\left({\displaystyle \frac{u^{\left(j\right)}\left(t\right)}{j!}}\right)}^{k_j},0\le j\le m,\end{array}$$

where $t\in [a_1,b_1]$, $k=k_1+\dots +k_j$, and the sum is taken over all $k_1,\dots ,k_j\in \mathbb{N}\cup \left\{0\right\}$ for which $k_1+2k_2+\dots +j{k}_j=j$.

The existence, uniqueness and smoothness of the solution of problem (1)–(2) can be characterised by the following theorem.

Theorem 1.

Let $\alpha \in (0,1)$, $n\in \mathbb{N}$, and let the assumptions (3)–(5) be fulfilled. Then, the following statements hold:

1.

Problem (1)–(2) has a unique solution $\overrightarrow{y}={(y_1,\dots ,y_n)}^T$ such that $\overrightarrow{y}\in C_n[0,1]$ and ${\mathit{D}}_{\mathit{Cap}}^{\alpha }\overrightarrow{y}\in C_n[0,1]$.

2.

Let $\overrightarrow{f}\in C_n^{q,\mu }(0,1]$ and $a_{ij}\in C^{q,\mu }(0,1],i,j=1,\dots ,n$, for $q\in \mathbb{N},\mu \in \mathbb{R},\mu <1$. Then, the solution $\overrightarrow{y}$ to problem (1)–(2) and its Caputo derivative ${\mathit{D}}_{\mathit{Cap}}^{\alpha }\overrightarrow{y}$ belong to $C_n^{q,\kappa }(0,1]$, where

$$\begin{array}{c}\hfill \kappa =max\left\{1-\alpha ,\mu \right\}.\end{array}$$

(13)

Proof. 

1. First, note that $T={\mathit{J}}^{\alpha }A$ is compact from $C_n[0,1]$ to $C_n[0,1]$, since by Lemma 1, the operator ${\mathit{J}}^{\alpha }:C_n[0,1]\to C_n[0,1]$ is compact and $A:C_n[0,1]\to C_n[0,1]$ is bounded as an operator. Moreover, note that the homogeneous equation $\overrightarrow{y}=T\overrightarrow{y}$ has in $C_n[0,1]$ only the trivial solution $\overrightarrow{y}={(0,\dots ,0)}^T$. Indeed, let $\overrightarrow{w}={(w_1,\dots ,w_n)}^T\in C_n[0,1]$ be a solution of equation $\overrightarrow{y}=T\overrightarrow{y}$. Then,

$$\begin{array}{c}\hfill \overrightarrow{w}={\mathit{J}}^{\alpha }A\overrightarrow{w}=\left(\begin{array}{c}{J}^{\alpha }\left(\sum _{j=1}^n{a}_{1j}{w}_j\right)\\ ⋮\\ J^{\alpha }\left(\sum _{j=1}^n{a}_{nj}{w}_j\right)\end{array}\right)\end{array}$$

and we have for every $t\in [0,1]$ and $i=1,\dots ,n$ that

$$\begin{array}{c}\hfill |w_i\left(t\right)|\le \widehat{a}{\int }_0^t{\displaystyle \frac{{(t-s)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}}\left(\sum _{j=1}^n\left|w_j\left(s\right)\right|\right)ds,\end{array}$$

where

$$\begin{array}{c}\hfill \widehat{a}=\underset{i,j=1,\dots ,n}{max}\underset{0\le s\le 1}{max}\left|a_{ij}\left(s\right)\right|.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \sum _{i=1}^n\left|w_i\left(t\right)\right|\le n\widehat{a}{\int }_0^t{\displaystyle \frac{{(t-s)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}}\left(\sum _{i=1}^n\left|w_i\left(s\right)\right|\right)ds,\forall t\in [0,1]\end{array}$$

and, since $0<\alpha <1$, we obtain using Lemma 3 that

$$\begin{array}{c}\hfill \sum _{i=1}^n\left|w_i\left(t\right)\right|\le E_{\alpha }\left(n\widehat{a}{t}^{\alpha }\right)·0=0,\forall t\in [0,1].\end{array}$$

This yields that $\overrightarrow{w}={(w_1,\dots ,w_n)}^T=\overrightarrow{0}$.

Finally, note that it follows from $\overrightarrow{f}\in C_n[0,1]$ and (7) that ${\mathit{J}}^{\alpha }\overrightarrow{f}\in C_n[0,1]$, implying by (12) that $\overrightarrow{g}=\overrightarrow{y_0}+{\mathit{J}}^{\alpha }\overrightarrow{f}\in C_n[0,1]$. Now, by using the Fredholm alternative theorem, we obtain that the equation $\overrightarrow{y}=T\overrightarrow{y}+\overrightarrow{g}$ possesses a unique solution $\overrightarrow{y}\in C_n[0,1]$. This, together with equality (1), yields that ${\mathit{D}}_{\mathit{Cap}}^{\alpha }\overrightarrow{\mathbf{y}}\in C_n[0,1]$.

2. Assume now that $\overrightarrow{f}\in C_n^{q,\mu }(0,1]$ and $a_{ij}\in C^{q,\mu }(0,1],i,j=1,\dots ,n$, $q\in \mathbb{N},\mu \in \mathbb{R},\mu <1$. Let us prove that the solution $\overrightarrow{y}$ to problem (1)–(2) and its Caputo derivative ${\mathit{D}}_{\mathit{Cap}}^{\alpha }\overrightarrow{y}$ belong to $C_n^{q,\kappa }(0,1]$, where $\kappa$ is defined by the Formula (13). Clearly, since $1-\alpha \le \kappa$, it follows from Lemma 1 that ${\mathit{J}}^{\alpha }$ is compact as an operator from $C_n^{q,\kappa }(0,1]$ to $C_n^{q,\kappa }(0,1]$. Since $\mu \le \kappa$, it follows from Lemma 2 that $A:C_n^{q,\kappa }(0,1]\to C_n^{q,\kappa }(0,1]$ and that A is bounded. Therefore, T defined by $T={\mathit{J}}^{\alpha }A$ is linear and compact as an operator from $C_n^{q,\kappa }(0,1]$ to $C_n^{q,\kappa }(0,1]$. Note also that $\overrightarrow{g}={\overrightarrow{y}}_0+{\mathit{J}}^{\alpha }\overrightarrow{f}$, the free term of equation $\overrightarrow{y}=T\overrightarrow{y}+\overrightarrow{g}$, belongs to $C_n^{q,\kappa }(0,1]$, since $\mu \le \kappa$ and thus $\overrightarrow{f}\in C_n^{q,\mu }(0,1]\subset C_n^{q,\kappa }(0,1]$. Because $\overrightarrow{y}=T\overrightarrow{y}$ has in $C_n[0,1]$ only the trivial solution $\overrightarrow{y}=\overrightarrow{0}$, it follows from the Fredholm alternative theorem that the solution $\overrightarrow{y}$ of equation $\overrightarrow{y}=T\overrightarrow{y}+\overrightarrow{g}$ belongs to $C_n^{q,\kappa }(0,1]$.

Finally, since $\overrightarrow{y}\in C_n^{q,\kappa }(0,1]$, $\overrightarrow{f}\in C_n^{q,\mu }(0,1]\subset C_n^{q,\kappa }(0,1]$ and $a_{ij}\in C^{q,\mu }(0,1]\subset C^{q,\kappa }(0,1],i,j=1,\dots ,n$, with the help of Equation (1) and Lemma 2, we obtain that ${\mathit{D}}_{\mathit{Cap}}^{\alpha }\overrightarrow{y}\in C_n^{q,\kappa }(0,1]$. □

4. Smoothing Transformation

It follows from the results of Theorem 1 that the regularity properties of the exact solution $\overrightarrow{y}$ of problem (1)–(2) (equivalently, of integral Equation (11)) depend on both the order $\alpha$ of the fractional Caputo derivative and on the given data (the forcing function $\overrightarrow{f}$ and the matrix entries $a_{ij}$, $i,j=1,\dots ,n$). Notably, even for smooth data $\overrightarrow{f}\in C_n^q[0,1]$, $a_{ij}\in C^q[0,1]$, $q\in \mathbb{N}$, in general it does not hold that $\overrightarrow{y}\in C_n^q[0,1]$; we can only say that $\overrightarrow{y}\in C_n^{q,1-\alpha }(0,1]$. In other words, the exact solution to problem (1)–(2) will, in general, exhibit singular behaviour even when the problem data is smooth. This property of the solution greatly complicates the construction of high-order methods for its numerical approximation.

To suppress the possible singularities of the solution $\overrightarrow{y}$ of (1)–(2), we perform in Equation (11) a change of variables, using a transformation $\phi :[0,1]\to [0,1]$ defined by the formula

$$\begin{array}{c}\hfill \phi \left(t\right)=t^p,t\in [0,1],p\in \mathbb{R},p\ge 1.\end{array}$$

(14)

Note that, clearly, for any $p\ge 1$, there exists a unique continuous inverse ${\phi }^{-1}:[0,1]\to [0,1]$ and

$$\begin{array}{c}\hfill {\phi }^{-1}\left(\tau \right)={\tau }^{{\displaystyle \frac{1}{p}}},0\le \tau \le 1.\end{array}$$

For functions in the space $C_n^{q,\nu }(0,1],q\in \mathbb{N},\nu <1,$ the transformations (14) with $p>1$ possess a smoothing property. For the exact solution $\overrightarrow{y}$ of (1)–(2), the value chosen for the (smoothing) parameter p depends on the singular behaviour of $\overrightarrow{y}$, which is expressed by Theorem 1 above. For the choice of p, we can use Lemma 5 below, the proof of which uses some ideas from [46].

Lemma 5.

Let $n,q\in \mathbb{N}$ and $\nu \in \mathbb{R}$, $\nu <1$. Let $\overrightarrow{u}\in C_n^{q,\nu }\left(0,1\right]$ and ${\overrightarrow{u}}_{\phi }\left(t\right)=\overrightarrow{u}\left(\phi \left(t\right)\right)$, $t\in [0,1]$, where φ is defined by (14) with the parameter $p\ge 1$ satisfying

$$\begin{array}{ccc}p>q\hfill & \mathrm{for}\hfill & \nu \le 0,\hfill \\ p>{\displaystyle \frac{q}{1-\nu }}\hfill & \mathrm{for}\hfill & 0<\nu <1.\hfill \end{array}$$

(15)

Then ${\overrightarrow{u}}_{\phi }\in C_n^q[0,1]$ and

$${\overrightarrow{u}}_{\phi }^{\left(j\right)}\left(0\right)=\overrightarrow{0},j=1,\dots ,q.$$

Proof. 

We prove the statement component-wise, denoting $u_{\phi }:=u_{\phi ,i}$ for any $i=1,\dots ,n$, where ${\overrightarrow{u}}_{\phi }={(u_{\phi ,1},\dots ,u_{\phi ,n})}^T$. Clearly, since $\overrightarrow{u}\in C_n^{q,\nu }(0,1]$, we have that $u_{\phi }\in C[0,1]\cap C^q(0,1]$. Thus, to prove the statement of the lemma, we have to show that

$$\begin{array}{c}\hfill u_{\phi }^{\left(j\right)}\left(0\right):=\underset{t\to 0}{lim}{u}_{\phi }^{\left(j\right)}\left(t\right)=0,j=1,\dots ,q.\end{array}$$

By (14) and Lemma 4, we obtain for all $j=1,\dots ,q$ and all $t\in (0,1]$ that

$$\begin{array}{cc}\hfill u_{\phi }^{\left(j\right)}\left(t\right)& =\sum _{k_1+2k_2+\dots +j{k}_j=j}{\displaystyle \frac{j!}{k_1!\dots k_j!}}{u}^{(k_1+\dots +k_j)}\left(s\right){|}_{s=t^p}{t}^{(p-1)k_1}\dots t^{(p-j)k_j},\hfill \end{array}$$

where $k_1,\dots ,k_j$ are non-negative integers. Recall that the assumption $z\in C^{q,\nu }(0,1]$ yields for any $l=1,\dots ,q$ and $t\in (0,1]$ that

$$|z^{\left(l\right)}\left(t^p\right)|\le c_1\left\{\begin{array}{ccc}1\hfill & \mathrm{if}\hfill & l<1-\nu ,\hfill \\ 1+|log{t}^p|\hfill & \mathrm{if}\hfill & l=1-\nu ,\hfill \\ t^{p(1-\nu -l)}\hfill & \mathrm{if}\hfill & l>1-\nu ,\hfill \end{array}\right.$$

with $c_1=c_1\left(z\right)$ being a positive constant. Thus, for $j=1,\dots ,q$ and $t\in (0,1]$, we have

$$\begin{array}{cc}\hfill |u_{\phi }^{\left(j\right)}\left(t\right)|& \le c_1\sum _{k_1+2k_2+\dots +j{k}_j=j}{\displaystyle \frac{j!}{k_1!\dots k_j!}}\hfill \\ \hfill & \times \left\{\begin{array}{ccc}1\hfill & \mathrm{if}\hfill & k_1+\dots +k_j<1-\nu \hfill \\ 1+|log{t}^p|\hfill & \mathrm{if}\hfill & k_1+\dots +k_j=1-\nu \hfill \\ t^{p(1-\nu -k_1-\dots -k_j)}\hfill & \mathrm{if}\hfill & k_1+\dots +k_j>1-\nu \hfill \end{array}\right\}{t}^{p(k_1+\dots +k_j)-j}\hfill \\ \hfill & =c_1\sum _{k_1+2k_2+\dots +j{k}_j=j}{\displaystyle \frac{j!}{k_1!\dots k_j!}}\hfill \\ \hfill & \times \left\{\begin{array}{ccc}{t}^{p(k_1+\dots +k_j)-j}\hfill & \mathrm{if}\hfill & k_1+\dots +k_j<1-\nu \hfill \\ (1+p|logt\left|\right)t^{p(k_1+\dots +k_j)-j}\hfill & \mathrm{if}\hfill & k_1+\dots +k_j=1-\nu \hfill \\ t^{p(1-\nu )-j}\hfill & \mathrm{if}\hfill & k_1+\dots +k_j>1-\nu \hfill \end{array}\right\}.\hfill \end{array}$$

For $\nu >0$, we have $k_1+\dots +k_j>1-\nu$ and thus $|u_{\phi }^{\left(j\right)}\left(t\right)|\le c_2{t}^{p(1-\nu )-j}$, where $c_2$ is a positive constant independent of $t\in (0,1]$. For $\nu =0$, there is one combination of $k_1,\dots ,k_j$ such that $k_1+2k_2+\dots +j{k}_j=j$ and $k_1+\dots +k_j=1-\nu$, namely $k_1=\dots =k_{j-1}=0,k_j=1$, giving us $|u_{\phi }^{\left(j\right)}\left(t\right)|\le c_3(1+p|logt\left|\right)t^{p-j}$, where $c_3$ is a positive constant independent of $t\in (0,1]$. For $\nu <0$, the smallest exponent $p(k_1+\dots +k_j)$ again corresponds to $k_1=\dots =k_{j-1}=0,k_j=1$, giving us $|u_{\phi }^{\left(j\right)}\left(t\right)|\le c_4{t}^{p-j}$, where $c_4$ is a positive constant independent of $t\in (0,1]$. As a summary

$$\begin{array}{cc}\hfill |u_{\phi }^{\left(j\right)}\left(t\right)|& \le c\left\{\begin{array}{ccc}{t}^{p-j}\hfill & \mathrm{if}\hfill & \nu <0\hfill \\ (1+p|logt\left|\right)t^{p-j}\hfill & \mathrm{if}\hfill & \nu =0\hfill \\ t^{p(1-\nu )-j}\hfill & \mathrm{if}\hfill & \nu >0\hfill \end{array}\right\},t\in (0,1],j=1,\dots ,q,\hfill \end{array}$$

where c is a positive constant which is independent of t and j. This, together with (15), yields $u_{\phi }^{\left(j\right)}\left(0\right):={lim}_{t\to 0}{u}_{\phi }^{\left(j\right)}\left(t\right)=0$ for $j=1,\dots ,q$. □

Introducing now in the integral Equation (11), the change of variables $t=\phi \left(\tau \right)$, $\tau \in [0,1]$, with $\phi$ determined by (14) for some $p\ge 1$, we obtain an integral equation

$$\begin{array}{c}\hfill \overrightarrow{y_{\phi }}\left(\tau \right)=\left(T_{\phi }\overrightarrow{y_{\phi }}\right)\left(\tau \right)+\overrightarrow{g_{\phi }}\left(\tau \right),\tau \in [0,1],\end{array}$$

(16)

where

$$\begin{array}{c}\hfill \overrightarrow{y_{\phi }}\left(\tau \right):=\overrightarrow{y}\left({\tau }^p\right),\tau \in [0,1]\end{array}$$

(17)

is the unknown function and

$$\begin{array}{cc}\hfill \left(T_{\phi }{\overrightarrow{y}}_{\phi }\right)\left(\tau \right)& =\left({\mathit{J}}_{\mathit{p}}^{\alpha }{A}_{\phi }{\overrightarrow{y}}_{\phi }\right)\left(\tau \right),A_{\phi }\left(\tau \right):=A\left({\tau }^p\right),\tau \in [0,1],\hfill \\ \hfill \overrightarrow{g_{\phi }}\left(\tau \right)& =\overrightarrow{g}\left({\tau }^p\right)=\left({\mathit{J}}_{\mathit{p}}^{\alpha }\overrightarrow{f}\right)\left(\tau \right)+\overrightarrow{y_0},\tau \in [0,1],\hfill \end{array}$$

with

$$\begin{array}{c}\hfill \left({\mathit{J}}_{\mathit{p}}^{\alpha }\overrightarrow{u}\right)\left(\tau \right):=\left(\begin{array}{c}\hfill \left(J_p^{\alpha }{u}_1\right)\left(\tau \right)\\ \hfill ⋮\\ \hfill \left(J_p^{\alpha }{u}_n\right)\left(\tau \right)\end{array}\right),\overrightarrow{u}={(u_1,\dots ,u_n)}^T\in C_n[0,1],\end{array}$$

$$\begin{array}{c}\hfill \left(J_p^{\alpha }{u}_i\right)\left(\tau \right):={\displaystyle \frac{p}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^{\tau }{({\tau }^p-{\sigma }^p)}^{\alpha -1}{\sigma }^{p-1}{u}_i\left(\sigma \right)d\sigma ,u_i\in C[0,1],i=1,\dots ,n.\end{array}$$

We observe that ${\overrightarrow{g}}_{\phi }\in C_n[0,1]$ and $T_{\phi }={\mathit{J}}_{\mathit{p}}^{\alpha }{A}_{\phi }$ is compact as an operator from $C_n[0,1]$ to $C_n[0,1]$. Moreover, the homogeneous equation ${\overrightarrow{y}}_{\phi }=T_{\phi }{\overrightarrow{y}}_{\phi }$ corresponding to Equation (16) has in $C_n[0,1]$ only the trivial solution ${\overrightarrow{y}}_{\phi }=\overrightarrow{0}$. Therefore, Equation (16) has in $C_n[0,1]$ a unique solution ${\overrightarrow{y}}_{\phi }\in C_n[0,1]$; the solutions of (16) and (11) are related by

$$\begin{array}{c}\hfill \overrightarrow{y_{\phi }}\left(\tau \right)=\overrightarrow{y}\left({\tau }^p\right),0\le \tau \le 1;\overrightarrow{y}\left(t\right)=\overrightarrow{y_{\phi }}(t^{{\displaystyle \frac{1}{p}}}),0\le t\le 1.\end{array}$$

(18)

Note that, if the assumptions (ii) of Theorem 1 are satisfied, then, for a high enough value of parameter $p\ge 1$ (see (13) and Lemma 5), we have for the exact solution ${\overrightarrow{y}}_{\phi }$ of integral Equation (16) that ${\overrightarrow{y}}_{\phi }\in C_n^q[0,1]$. This smoothness property of the exact solution allows us to construct a piecewise polynomial application of the central part interpolation scheme, which is introduced in the next section.

5. Central Part Interpolation by Piecewise Polynomials

We start by describing the central part interpolation by polynomials. Given an interval $\left[a,b\right]$$(a<b)$ and $m\in \mathbb{N}$, $m\ge 2$, we introduce a uniform grid consisting of m points

$$t_i=a+\left(i-{\displaystyle \frac{1}{2}}\right)h,i=1,\dots ,m,h={\displaystyle \frac{b-a}{m}}.$$

(19)

We denote by ${\mathcal{P}}_{m-1}$ the set of polynomials of degree not exceeding $m-1$ and by ${\mathsf{\Pi }}_m$ the Lagrange interpolation projection operator, which assigns to any $g\in C\left[a,b\right]$ the polynomial ${\mathsf{\Pi }}_{m}g\in {\mathcal{P}}_{m-1}$ interpolating g at points (19):

$$\left({\mathsf{\Pi }}_{m}g\right)\left(t\right)=\sum _{j=1}^{m}g\left(t_j\right)\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^m{\displaystyle \frac{t-t_k}{t_j-t_k}},a\le t\le b,m\ge 2.$$

The following result is known [40].

Lemma 6.

In the case of interpolation knots (19) with $m\in \mathbb{N}$, $m\ge 2$, for $g\in C^m\left[a,b\right]$ the non-improvable estimate

$$\underset{a\le t\le b}{max}\left|g\left(t\right)-\left({\mathsf{\Pi }}_{m}g\right)\left(t\right)\right|\le {\theta }_m{h}^m\underset{a\le t\le b}{max}\left|g^{\left(m\right)}\left(t\right)\right|,$$

(20)

holds, with

$${\theta }_m={\displaystyle \frac{\left(2m\right)!}{2^{2m}{(m!)}^2}}\cong {\left(\pi m\right)}^{-{\displaystyle \frac{1}{2}}},$$

where ${\theta }_m\cong {ϵ}_m$ means that ${\theta }_m/{ϵ}_m\to 1$ as $m\to \infty$.

For $m=2k$, $k\ge 1$, we have the non-improvable estimate

$$\underset{t_k\le t\le t_{k+1}}{max}\left|g\left(t\right)-\left({\mathsf{\Pi }}_{m}g\right)\left(t\right)\right|\le {\vartheta }_m{h}^m\underset{a\le t\le b}{max}\left|g^{\left(m\right)}\left(t\right)\right|,$$

(21)

with

$${\vartheta }_m=2^{-2m}{\displaystyle \frac{m!}{{((m/2)!)}^2}}\cong \sqrt{2/\pi }{m}^{-{\displaystyle \frac{1}{2}}}{2}^{-m},$$

(22)

whereas for $m=2k+1$, $k\ge 1$, we have the non-improvable estimate

$$\underset{t_k\le t\le t_{k+2}}{max}\left|g\left(t\right)-\left({\mathsf{\Pi }}_{m}g\right)\left(t\right)\right|\le {\vartheta }_m{h}^m\underset{a\le t\le b}{max}\left|g^{\left(m\right)}\left(t\right)\right|,$$

(23)

with

$${\vartheta }_m={\displaystyle \frac{2\sqrt{3}}{9}}{\displaystyle \frac{{\left(k!\right)}^2}{\left(2k+1\right)!}}\cong {\displaystyle \frac{2\sqrt{6\pi }}{9}}{m}^{-{\displaystyle \frac{1}{2}}}{2}^{-m}.$$

(24)

When we compare the estimates (20)–(24), we observe that in the underlying central parts of $\left[a,b\right]$, the estimate of the error $g-{\mathsf{\Pi }}_{m}g$ is approximately $2^m$ times more precise than on the whole interval $[a,b]$. To properly leverage the improved error estimate attained in the central parts of the approximating polynomials, in what follows, we extend this central part approach to a piecewise polynomial method, restricting ourselves to the interval $[0,1]$.

To this end, let $N\in \mathbb{N}$ and introduce the following uniform grid:

$$\left\{jh:j\in \mathbb{Z}\right\},\mathbb{Z}:=\{\dots ,-1,0,1,\dots \},h={\displaystyle \frac{1}{N}}.$$

Let $m\in \mathbb{N},m\ge 2$, and let g be an arbitrary continuous function on $[-\delta ,1+\delta ]$ for a $\delta \ge {\displaystyle \frac{hm}{2}}$. We define a piecewise polynomial interpolant ${\mathsf{\Pi }}_{h,m}g\in C\left[0,1\right]$ as follows: on every subinterval $\left[jh,\left(j+1\right)h\right]$$(j=0,\dots ,N-1)$ the function ${\mathsf{\Pi }}_{h,m}g$ is defined independently from other subintervals as a polynomial ${\mathsf{\Pi }}_{h,m}^{\left[j\right]}g\in {\mathcal{P}}_{m-1}$ of degree less or equal to $m-1$ by the conditions

$${\mathsf{\Pi }}_{h,m}^{\left[j\right]}g\left(lh\right)=g\left(lh\right),l=j-{\displaystyle \frac{m}{2}}+1,\dots ,j+{\displaystyle \frac{m}{2}}\mathrm{if}m\mathrm{is}\mathrm{even},$$

$${\mathsf{\Pi }}_{h,m}^{\left[j\right]}g\left(lh\right)=g\left(lh\right),l=j-{\displaystyle \frac{m-1}{2}},\dots ,j+{\displaystyle \frac{m-1}{2}}\mathrm{if}m\mathrm{is}\mathrm{odd}.$$

These interpolation conditions can be written in a more compact form:

$${\mathsf{\Pi }}_{h,m}^{\left[j\right]}g\left(lh\right)=g\left(lh\right),\mathrm{for}l\in \mathbb{Z}\mathrm{such}\mathrm{that}l-j\in Z_m,$$

(25)

where

$$Z_m=\left\{k\in \mathbb{Z}:-{\displaystyle \frac{m}{2}}<k\le {\displaystyle \frac{m}{2}}\right\}.$$

For an interior knot $jh$$(j=1,\dots ,N-1)$ interpolation conditions (25) contain the condition $\left({\mathsf{\Pi }}_{h,m}^{\left[j-1\right]}g\right)\left(jh\right)=g\left(jh\right)$, as well as the condition $\left({\mathsf{\Pi }}_{h,m}^{\left[j\right]}g\right)\left(jh\right)$$=g\left(jh\right)$. Note that while the one-sided derivatives of the interpolant at the interior knots may be different, ${\mathsf{\Pi }}_{h,m}g$ is uniquely defined at the knots $jh$, and thus, ${\mathsf{\Pi }}_{h,m}g$ is continuous on $\left[0,1\right]$. Note also that the interpolant ${\mathsf{\Pi }}_{h,m}g$ is closely related to the central part interpolation of g on the uniform grid. On $\left[jh,\left(j+1\right)h\right]$, the interpolant ${\mathsf{\Pi }}_{h,m}g={\mathsf{\Pi }}_{h,m}^{\left[j\right]}g$ coincides with the polynomial interpolant ${\mathsf{\Pi }}_{m}g$ constructed for g on the interval $\left[a_j,b_j\right]$, where

$$a_j=\left(j-{\displaystyle \frac{m-1}{2}}\right)h,b_j=\left(j+{\displaystyle \frac{m+1}{2}}\right)h\left(\mathrm{in}\mathrm{the}\mathrm{case}\mathrm{of}\mathrm{even}m\right),$$

$$a_j=\left(j-{\displaystyle \frac{m}{2}}\right)h,b_j=\left(j+{\displaystyle \frac{m}{2}}\right)h\left(\mathrm{in}\mathrm{the}\mathrm{case}\mathrm{of}\mathrm{odd}m\right).$$

Moreover, $\left[jh,\left(j+1\right)h\right]$ is contained in the central part of $\left[a_j,b_j\right]$ on which the interpolation error can be estimated by (21) in the case of even m and by (23) in the case of odd m.

We now give an explicit form for the interpolants ${\mathsf{\Pi }}_{h,m}^{\left[j\right]}g$ on each subinterval $[jh,(j+1\left)h\right]$, $j=0,\dots ,N-1$. To this end, we introduce the Lagrange fundamental polynomials $L_{k,m}\in {\mathcal{P}}_{m-1}$, $k\in Z_m$:

$$L_{k,m}\left(t\right)=\prod _{l\in Z_m\setminus \left\{k\right\}}{\displaystyle \frac{t-l}{k-l}},k\in Z_m,t\in [jh,(j+1)h].$$

(26)

We see that

$$\left({\mathsf{\Pi }}_{h,m}^{\left[j\right]}g\right)\left(t\right)=\sum _{k\in Z_m}g\left(\left(j+k\right)h\right)L_{k,m}\left(nt-j\right),t\in [jh,(j+1)h],$$

(27)

where $j=0,\dots ,N-1$. This is clear, since ${\mathsf{\Pi }}_{h,m}^{\left[j\right]}g$ defined by (27) is, in fact, a polynomial of degree less than or equal to $m-1$ satisfying the interpolation conditions (25): for all $l\in \mathbb{Z}$ such that $l-j\in Z_m$, it holds that

$$\left({\mathsf{\Pi }}_{h,m}^{\left[j\right]}g\right)\left(lh\right)=\sum _{k\in Z_m}g\left(\left(j+k\right)h\right)L_{k,m}\left(l-j\right)=g\left(\left(j+\left(l-j\right)\right)h\right)=g\left(lh\right).$$

Note that, for $m=2$, the function ${\mathsf{\Pi }}_{h,m}g$ is the usual piecewise linear interpolant of g. However, for $m\ge 3$, the interpolant ${\mathsf{\Pi }}_{h,m}g$ requires values of g outside of $\left[0,1\right]$. Therefore, for a function $g\in C[0,1]$, ${\mathsf{\Pi }}_{h,m}g$ obtains a sense after an extension of g onto $\left[-\delta ,1+\delta \right]$, for a $\delta \ge {\displaystyle \frac{m}{2}}h$. Recall, for the moment (see Lemma 5), that we are interested in functions $g\in C^m\left[0,1\right]$ satisfying the boundary conditions

$$g^{\left(j\right)}\left(0\right)=0,j=1,\dots ,m.$$

Thus, to extend such a function g onto $[-\delta ,0]$, it suffices to set $g\left(t\right)=g\left(0\right)$ for $-\delta \le t\le 0$. To extend g to $[1,1+\delta ]$, however, we demand that $\delta <{\displaystyle \frac{1}{m}}$ and $\delta \ge {\displaystyle \frac{m}{2}}h$ (equivalently, that $h<2/m^2$), and use the reflection formula (see e.g., [47])

$$\begin{array}{c}\hfill g\left(t\right)=\sum _{j=0}^m{c}_{j}g\left(1-j(t-1)\right),1\le t\le 1+\delta ,\end{array}$$

where the coefficients $c_j$ are chosen so that the $C^m$-smooth joining happens at $t=1$, i.e.,

$$\underset{t\to 1-}{lim}{g}^{\left(k\right)}\left(t\right)=\underset{t\to 1+}{lim}{\left(\sum _{j=0}^m{c}_{j}g\left(1-j(t-1)\right)\right)}^{\left(k\right)},k=0,\dots ,m.$$

From these conditions, we obtain the system of equations

$$\begin{array}{c}\hfill \sum _{j=0}^m{(-j)}^k{c}_j=1,k=0,1,\dots ,m,\end{array}$$

(28)

which is satisfied for

$$\begin{array}{c}\hfill c_j={(-1)}^j{\displaystyle \frac{(m+1)!}{(m-j)!(j+1)!}},j=0,\dots ,m.\end{array}$$

(29)

Altogether, we see that the extension operator $E_{\delta }:C\left[0,1\right]\to C\left[-\delta ,1+\delta \right]$, defined by

$$\left(E_{\delta }g\right)\left(t\right)=\left\{\begin{array}{cc}g\left(0\right)& \mathrm{for}-\delta \le t\le 0\\ g\left(t\right)& \mathrm{for}0\le t\le 1\\ \sum _{j=0}^m{c}_{j}g\left(1-j(t-1)\right)& \mathrm{for}1\le t\le 1+\delta \end{array}\right\},$$

with the coefficients $\left\{c_j\right\}$ determined by Formula (29), maintains the smoothness of g on $[-\delta ,1+\delta ].$ Hence, the operator $P_{h,m}:C\left[0,1\right]\to C\left[0,1\right]$, $m\ge 2$, given by

$$P_{h,m}={\mathsf{\Pi }}_{h,m}{E}_{\delta },$$

(30)

is well defined, and $P_{h,m}^2=P_{h,m}$, i.e., $P_{h,m}$ is a projector in $C\left[0,1\right]$.

For $w_h\in \mathcal{R}\left(P_{h,m}\right)$ (the range of $P_{h,m}$), we have $w_h=P_{h,m}{w}_h={\mathsf{\Pi }}_{h,m}{E}_{\delta }{w}_h$ and due to (27), we get for $t\in [jh,(j+1\left)h\right]$ ($j=0,\dots ,N-1$) that

$$w_h\left(t\right)=\sum _{k\in Z_m}\left(E_{\delta }{w}_h\right)\left((j+k)h\right)L_{k,m}(nt-j),$$

(31)

where

$$\left(E_{\delta }{w}_h\right)\left(ih\right)=\left\{\begin{array}{ccc}{w}_h\left(0\right)& \mathrm{for}& i<0\\ w_h\left(ih\right)& \mathrm{for}& i=0,\dots ,n\\ \sum _{j=0}^m{c}_j{w}_h\left(1-j(i-n)h\right)& \mathrm{for}& i>n\end{array}\right\}.$$

Thus, $w_h\in \mathcal{R}\left(P_{h,m}\right)$ is uniquely determined on $\left[0,1\right]$ by its knot values $w_h\left(ih\right)$, $i=0,\dots ,N$. We conclude that $dim\mathcal{R}\left(P_{h,m}\right)=N+1$. It is also clear that for a $w_h\in \mathcal{R}\left(P_{h,m}\right)$ we have $w_h=0$ if and only if $w_h\left(ih\right)=0$, $i=0,\dots ,N$.

Finally, we define the interpolation operator $P_{h,m}^n:C_n[0,1]\to C_n[0,1]$ for a vector-valued function $\overrightarrow{u}\in C_n[0,1]$ by the component-wise application of $P_{h,m}$:

$$P_{h,m}^n\overrightarrow{u}:={(P_{h,m}{u}_1,\dots ,P_{h,m}{u}_n)}^T,\overrightarrow{u}\in C_n[0,1].$$

(32)

It is clear from the properties of $P_{h,m}$ that $P_{h,m}^n$ is also a projector operator in $C_n[0,1]$ and, using the same ideas as in [40], we can prove the following result.

Lemma 7.

Let $n,m\in \mathbb{N},m\ge 2,h={\displaystyle \frac{1}{N}}<2/m^2,N\in \mathbb{N}$. Let the operator $P_{h,m}^n$ be defined by the Formula (32). Then, we have for any $\overrightarrow{u}\in C_n\left[0,1\right]$ that

$$\parallel \overrightarrow{u}-P_{h,m}^n\overrightarrow{u}{\parallel }_{C_n[0,1]}\to 0\mathrm{as}N\to \infty .$$

Moreover, for $\overrightarrow{u}\in C_n^m\left[0,1\right]$, ${\overrightarrow{u}}^{\left(j\right)}\left(0\right)=\overrightarrow{0}$, $j=1,\dots ,m$, we have

$$\parallel \overrightarrow{u}-P_{h,m}^n\overrightarrow{u}{\parallel }_{C_n[0,1]}\le c{\vartheta }_m{h}^m\underset{0\le t\le 1}{max}\left|{\overrightarrow{u}}^{\left(m\right)}\left(t\right)\right|,$$

(33)

where c is a constant independent of h, with ${\vartheta }_m$ defined by (22) for even m and by (24) for odd m.

6. Collocation Method Based on the Central Part Interpolation

We now return to our system of integral Equation (16). Fix $N,m\in \mathbb{N}$, $m\ge 2$, so that $h:={\displaystyle \frac{1}{N}}<{\displaystyle \frac{2}{m^2}}$. Using the interpolation projector $P_{h,m}^n:C_n[0,1]\to C_n[0,1]$ defined by (32), we approximate Equation (16) by equation

$${\overrightarrow{y}}_{\phi ,h}=P_{h,m}^n{T}_{\phi }{\overrightarrow{y}}_{\phi ,h}+P_{h,m}^n{\overrightarrow{g}}_{\phi }.$$

(34)

This is the operator form of our method for (16). The solution ${\overrightarrow{y}}_{\phi ,h}$ of Equation (34) belongs to $\mathcal{R}\left(P_{h,m}^n\right)$, the range of operator $P_{h,m}^n$. Clearly, the knot values

$$\begin{array}{c}\hfill {\overrightarrow{y}}_{\phi ,h}\left(kh\right)={\left(y_{\phi ,h,1}\left(kh\right),\dots ,y_{\phi ,h,n}\left(kh\right)\right)}^T(k=0,\dots ,N)\end{array}$$

and

$$\begin{array}{c}\hfill {\overrightarrow{g}}_{\phi }\left(kh\right)={\left(g_{\phi ,1}\left(kh\right),\dots ,g_{\phi ,n}\left(kh\right)\right)}^T(k=0,\dots ,N)\end{array}$$

determine ${\overrightarrow{y}}_{\phi ,h}$ and ${\overrightarrow{g}}_{\phi }$ uniquely. Since $\left(P_{h,m}^n\overrightarrow{u}\right)\left(kh\right)=\overrightarrow{u}\left(kh\right)$, $k=0,\dots ,N$, for $\overrightarrow{u}\in C_n[0,1]$, Equation (34) is equivalent to a system of linear algebraic equations with respect to $y_{\phi ,h,i}\left(kh\right)$, $k=0,\dots ,N$, $i=1,\dots ,n$:

$$\begin{array}{cc}\hfill y_{\phi ,h,i}\left(kh\right)& =\sum _{j=1}^n{\displaystyle \frac{p}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^{kh}\left({\left(kh\right)}^p-{\sigma }^p\right){\sigma }^{p-1}{a}_{i,j}\left({\sigma }^p\right)y_{\phi ,h,j}\left(\sigma \right)d\sigma +g_{\phi ,i}\left(kh\right).\hfill \end{array}$$

(35)

After solving (35), we obtain the values $y_{\phi ,h,i}\left(kh\right)$, $k=0,\dots ,N$ for every $i=1,\dots ,n$. Using the representation (31), we can write $y_{\phi ,h,i}\left(\tau \right)$, $\tau \in [lh,(l+1\left)h\right]$, $l=0,\dots ,N-1$, in the form:

$$\begin{array}{cc}\hfill y_{\phi ,h,i}\left(\tau \right)& =\sum _{\widehat{k}\in Z_m}{L}_{\widehat{k},m}(N\tau -l)\hfill \\ \hfill & \times \left\{\begin{array}{cc}{y}_{\phi ,h,i}\left(0\right),\hfill & l+\widehat{k}<0,\hfill \\ y_{\phi ,h,i}\left((l+\widehat{k})h\right),\hfill & 0\le l+\widehat{k}\le N,\hfill \\ \sum _{j=0}^m{c}_j{y}_{\phi ,h,i}\left(Nh-j(l+\widehat{k}-N)h\right),\hfill & l+\widehat{k}>N,\hfill \end{array}\right.\hfill \end{array}$$

(36)

where $i=1,\dots ,n$, $L_{\widehat{k},m}(\widehat{k}\in Z_m)$ is defined by (26), and $c_0,\dots ,c_m$ are given by (29).

The approximation ${\overrightarrow{y}}_h$ for $\overrightarrow{y}$, the exact solution of problem (1)–(2) (and of (11)), is defined by the formula (see (18))

$${\overrightarrow{y}}_h(t):={\overrightarrow{y}}_{\phi ,h}(t^{{\displaystyle \frac{1}{p}}}),0\le t\le 1,h={\displaystyle \frac{1}{N}},$$

(37)

where ${\overrightarrow{y}}_{\phi ,h}={(y_{\phi ,h,1},\dots ,y_{\phi ,h,n})}^T$ is given by (36).

Theorem 2.

(i) Let $\alpha \in (0,1)$, $q,n\in \mathbb{N}$, and let the assumptions (3)–(5) be fulfilled. Let the smoothing transformation $\phi :[0,1]\to [0,1]$ be defined by (14) for any $p\ge 1$, and let $m\in \mathbb{N}$, $m\ge 2$. Then, there exists $N_0\in \mathbb{N}$ such that, for $N\ge N_0>{\displaystyle \frac{m^2}{2}}$, the collocation Equation (34) has a unique solution ${\overrightarrow{y}}_{\phi ,h}\in C_n[0,1]$ such that

$$\parallel \overrightarrow{y}-{\overrightarrow{y}}_h{\parallel }_{\infty }={\parallel {\overrightarrow{y}}_{\phi }-{\overrightarrow{y}}_{\phi ,h}\parallel }_{\infty }\to 0\mathit{for}N\to \infty \left(h={\displaystyle \frac{1}{N}}\to 0\right),$$

(38)

where ${\overrightarrow{y}}_{\phi }$ is the solution of (16), ${\overrightarrow{y}}_h$ is defined by the Formula (37), and $\overrightarrow{y}$ is the solution of (1) and (2).

(ii) In addition to (i), let the assumptions (ii) of Theorem 1 be fulfilled for $q=m$, $m\ge 2$. Let the smoothing transformation $\phi :[0,1]\to [0,1]$ be defined by (14), with the smoothing parameter p satisfying (15), where $q=m$ and κ is given by (13). Then, the following error estimate holds:

$$\parallel \overrightarrow{y}-{\overrightarrow{y}}_h{\parallel }_{\infty }\le c{\vartheta }_m{h}^m{\parallel {\overrightarrow{y}}_{\phi }^{\left(m\right)}\parallel }_{\infty },h={\displaystyle \frac{1}{N}},N\ge N_0.$$

(39)

Here, $c=c\left(m\right)$ is a positive constant independent of h (of N), and ${\vartheta }_m$ is defined by (22) for even m and by (24) for odd m.

Proof. 

Since $T_{\phi }\in \mathcal{L}\left(C_n[0,1]\right)$ is compact and since $\left\{\overrightarrow{y}\in C_n[0,1]:\overrightarrow{y}=T_{\phi }\overrightarrow{y}\right\}=\overrightarrow{0}$, the bounded inverse ${(I-T_{\phi })}^{-1}:C_n[0,1]\to C_n[0,1]$ exists (here, I is the identity mapping in $C_n[0,1]$). Denote

$$\begin{array}{c}\hfill \eta =\parallel {(I-T_{\phi })}^{-1}{\parallel }_{\mathcal{L}\left(C_n[0,1]\right)}.\end{array}$$

The compactness of $T_{\phi }\in \mathcal{L}\left(C_n[0,1]\right)$ together with the pointwise convergence of $P_{h,m}^n$ to I in $C_n[0,1]$ (see Lemma 7) implies the following norm convergence:

$${ϵ}_h:={\parallel P_{h,m}^n{T}_{\phi }-T_{\phi }\parallel }_{\mathcal{L}\left(C_n[0,1]\right)}\to 0\mathrm{as}N\to \infty \left(\mathrm{as}h={\displaystyle \frac{1}{N}}\to 0\right).$$

Hence, there is an $N_0\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \eta {ϵ}_h<1\mathrm{for}N\ge N_0\left(\mathrm{for}h\le {\displaystyle \frac{1}{N_0}}\right).\end{array}$$

With the help of this, we obtain for sufficiently large N that the operators $I-P_{h,m}^n{T}_{\phi }$ are invertible in $C_n[0,1]$ for $N\ge N_0$, and

$$\begin{array}{c}\hfill {\eta }_h:={\parallel {(I-P_{h,m}^n{T}_{\phi })}^{-1}\parallel }_{\mathcal{L}\left(C_n[0,1]\right)}\le {\displaystyle \frac{\eta }{1-\eta {ϵ}_h}}\to \eta \mathrm{as}N\to \infty .\end{array}$$

From this, it follows the unique solvability of Equation (16) for $N\ge N_0$:

$$\begin{array}{c}\hfill {\overrightarrow{y}}_{\phi ,h}={\left(I-P_{h,m}^n{T}_{\phi }\right)}^{-1}{P}_{h,m}^n{\overrightarrow{g}}_{\phi },N\ge N_0.\end{array}$$

Further, let ${\overrightarrow{y}}_{\phi }$ and ${\overrightarrow{y}}_{\phi ,h}$ be the solutions of (16) and (34), respectively. Then, we have for $N\ge N_0$ that

$$\begin{array}{cc}\hfill (I-P_{h,m}^n{T}_{\phi })({\overrightarrow{y}}_{\phi }-{\overrightarrow{y}}_{\phi ,h})& ={\overrightarrow{y}}_{\phi }-P_{h,m}^n{T}_{\phi }{\overrightarrow{y}}_{\phi }-P_{h,m}^n{\overrightarrow{g}}_{\phi }={\overrightarrow{y}}_{\phi }-P_{h,m}^n{\overrightarrow{y}}_{\phi },\hfill \\ \hfill {\overrightarrow{y}}_{\phi }-{\overrightarrow{y}}_{\phi ,h}& ={(I-P_{h,m}^n{T}_{\phi })}^{-1}({\overrightarrow{y}}_{\phi }-P_{h,m}^n{\overrightarrow{y}}_{\phi }).\hfill \end{array}$$

Thus, we can write

$$\parallel \overrightarrow{y}-{\overrightarrow{y}}_h{\parallel }_{\infty }=\parallel {\overrightarrow{y}}_{\phi }-{\overrightarrow{y}}_{\phi ,h}{\parallel }_{\infty }\le c\left(h\right){\parallel {\overrightarrow{y}}_{\phi }-P_{h,m}^n{\overrightarrow{y}}_{\phi }\parallel }_{\infty },h={\displaystyle \frac{1}{N}},N\ge N_0,$$

(40)

where the constant $c\left(h\right)={\displaystyle \frac{\eta }{1-{ϵ}_h\eta }}\to \eta$ for $h\to 0$. This, together with Lemma 7, implies the convergence (38).

In the case of (ii), it follows from Theorem 1 that the solution of $\overrightarrow{y}$ of (11) (of (1) and (2)) belongs to $C_n^{m,\kappa }(0,1]$. Therefore, due to Lemma 5, we have for ${\overrightarrow{y}}_{\phi }\left(t\right)=\overrightarrow{y}\left(\phi \left(t\right)\right)$ that ${\overrightarrow{y}}_{\phi }\in C_n^m[0,1]$ and ${\overrightarrow{y}}_{\phi }^{\left(j\right)}\left(0\right)=\overrightarrow{0}$, $j=1,\dots ,m$; due to Lemma 7 (see (33)), we obtain that $\parallel {\overrightarrow{y}}_{\phi }-P_{h,m}^n{\overrightarrow{y}}_{\phi }{\parallel }_{\infty }\le c_1{\vartheta }_m{h}^m{\parallel {\overrightarrow{y}}_{\phi }^{\left(m\right)}\parallel }_{\infty }$, where $c_1$ does not depend on h. This, together with (40), yields (39). □

7. Numerical Experiments

Example 1.

Consider the following problem (see [37], Example 2):

$$\begin{array}{cc}\hfill \left({\mathit{D}}_{\mathit{Cap}}^{\mathbf{0}.\mathbf{4}}\overrightarrow{y}\right)\left(t\right)& =A\left(t\right)\overrightarrow{y}\left(t\right)+\overrightarrow{f}\left(t\right),t\in [0,1],\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \overrightarrow{y}\left(0\right)& =\overrightarrow{y_0},\hfill \end{array}$$

(42)

where

$$\begin{array}{c}\hfill A\left(t\right)=\left(\begin{array}{cc}0& 1\\ -1& -1\end{array}\right),t\in [0,1],\end{array}$$

(43)

with $\overrightarrow{f}\left(t\right)={(f_1\left(t\right),f_2\left(t\right))}^T$, $t\in [0,1],$ comprised of

$$\begin{array}{cc}\hfill f_1\left(t\right)& =0,\hfill \\ \hfill f_2\left(t\right)& =t^{1.4}+{\displaystyle \frac{\pi csc\left(1.4\pi \right)}{\mathsf{\Gamma }(-1.4)\mathsf{\Gamma }\left(1.6\right)}}{t}^{0.6}+{\displaystyle \frac{\pi csc\left(0.4\pi \right)}{\mathsf{\Gamma }(-1.4)}}t\hfill \end{array}$$

and $y_1\left(0\right)=0,y_2\left(0\right)=0$. The solution of this problem is $\overrightarrow{y}\left(t\right)={(y_1\left(t\right),y_2\left(t\right))}^T,t\in [0,1]$, where

$$\begin{array}{cc}\hfill y_1\left(t\right)& =t^{1.4},\hfill \\ \hfill y_2\left(t\right)& ={\displaystyle \frac{0.56\pi csc\left(0.4\pi \right)}{\mathsf{\Gamma }\left(0.6\right)}}t.\hfill \end{array}$$

We see that (41)–(42) is a special case of problem (1)–(2), where $\alpha =0.4$ and $n=2$. It is clear that the entries of the matrix (43) belong to $C^q[0,1]\subset C^{q,\nu }(0,1]$ for every $q\in \mathbb{N}$, $\nu <1$ and $\overrightarrow{f}\in C_2^{q,\mu }[0,1]$, where $\mu =0.4$.

In order to find a numerical solution to problem (41)–(42) for $m\ge 2$, we assemble and solve the system of linear algebraic Equations (35). The parameter p in the smoothing transformation (14) is chosen to be greater than ${\displaystyle \frac{m}{1-\nu }}$, where

$$\nu =max\{1-\alpha ,\mu \}=0.6.$$

Thus, we have to take $p>{\displaystyle \frac{5m}{2}}$ to achieve the expected convergence order $O\left(h^m\right)$ given by Theorem 2.

In

Table 1. Numerical results for problem (41)–(42).

	$\mathit{m}=3$, $\mathit{p}=7.5$		$\mathit{m}=4$, $\mathit{p}=10$		$\mathit{m}=5$, $\mathit{p}=12.5$
$\mathit{N}$	${\mathbf{\epsilon }}_{\mathit{N}}$	${\mathbf{\epsilon }}_{\mathit{N}/\mathbf{2}}/{\mathbf{\epsilon }}_{\mathit{N}}$	${\mathbf{\epsilon }}_{\mathit{N}}$	${\mathbf{\epsilon }}_{\mathit{N}/\mathbf{2}}/{\mathbf{\epsilon }}_{\mathit{N}}$	${\mathbf{\epsilon }}_{\mathit{N}}$	${\mathbf{\epsilon }}_{\mathit{N}/\mathbf{2}}/{\mathbf{\epsilon }}_{\mathit{N}}$
8	$4.72·{10}^{-2}$		$2.21·{10}^{-2}$		$3.74·{10}^{-2}$
16	$8.57·{10}^{-3}$	$5.51$	$3.25·{10}^{-3}$	$6.80$	$2.14·{10}^{-3}$	$17.49$
32	$1.28·{10}^{-3}$	$6.67$	$3.18·{10}^{-4}$	$10.24$	$1.37·{10}^{-4}$	$15.61$
64	$1.75·{10}^{-4}$	$7.32$	$2.46·{10}^{-5}$	$12.93$	$6.07·{10}^{-6}$	$22.59$
128	$2.28·{10}^{-5}$	$7.66$	$1.70·{10}^{-6}$	$14.45$	$2.24·{10}^{-7}$	$27.04$
256	$2.92·{10}^{-6}$	$7.83$	$1.11·{10}^{-7}$	$15.24$	$7.61·{10}^{-9}$	$29.49$
512	$3.68·{10}^{-7}$	$7.92$	$7.14·{10}^{-9}$	$15.64$	$2.47·{10}^{-10}$	$30.73$
		8		16		32

, the errors

$$\begin{array}{c}\hfill {\epsilon }_N:=\underset{j=0,1,\dots ,10N}{max}\underset{i=1,2}{max}\left\{y_i\left({\displaystyle \frac{j}{10N}}\right)-y_{h,i}\left({\displaystyle \frac{j}{10N}}\right)\right\}(N=2^k,k=3,\dots ,9)\end{array}$$

(44)

are presented. Here, $\overrightarrow{y}={\left(y_1,y_2\right)}^T$ is the exact solution of problem (41) and ${\overrightarrow{y}}_h={\left(y_{h,1},y_{h,2}\right)}^T$, the approximate solution to (41), is obtained by (37). Additionally, the quotients ${\epsilon }_{N/2}/{\epsilon }_N$ for different values of m, N and p are presented. Due to Theorem 2, the expected limit value of ${\epsilon }_{N/2}/{\epsilon }_N$ is $2^m$. These values are given in the last row of Table 1. As we can see, the error ratios approach the expected limit $2^m$ as N increases.

Example 2.

Consider the following problem:

$$\begin{array}{cc}\hfill \left({\mathit{D}}_{\mathit{Cap}}^{\mathbf{0}.\mathbf{6}}\overrightarrow{y}\right)\left(t\right)& =A\left(t\right)\overrightarrow{y}\left(t\right)+\overrightarrow{f}\left(t\right),t\in [0,1],\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill \overrightarrow{y}\left(0\right)& =\overrightarrow{y_0},\hfill \end{array}$$

(46)

where

$$\begin{array}{c}\hfill A\left(t\right)=\left(\begin{array}{cc}{t}^{0.5}& 0\\ 2t& -t^{0.6}\end{array}\right),t\in [0,1],\end{array}$$

(47)

with $\overrightarrow{f}\left(t\right)={(f_1\left(t\right),f_2\left(t\right))}^T,t\in [0,1],$ comprised of

$$\begin{array}{cc}\hfill f_1\left(t\right)& =\mathsf{\Gamma }\left(1.6\right)-t^{1.1}-t^{0.5},\hfill \\ \hfill f_2\left(t\right)& =\left({\displaystyle \frac{\mathsf{\Gamma }\left(2.2\right)}{\mathsf{\Gamma }\left(1.6\right)}}-2\right)t^{0.6}-2t^{1.6}-2t+t^{1.8},\hfill \end{array}$$

and $y_1\left(0\right)=1,y_2\left(0\right)=-2$. The solution of this problem is $\overrightarrow{y}\left(t\right)={(y_1\left(t\right),y_2\left(t\right))}^T$, $t\in [0,1]$, where

$$\begin{array}{cc}\hfill y_1\left(t\right)& =t^{0.6}+1,\hfill \\ \hfill y_2\left(t\right)& =t^{1.2}-2.\hfill \end{array}$$

We see that this is a special case of problem (1)–(2), where $\alpha =0.6$ and $n=2$. It is clear that the entries of the matrix (47) belong to $C^{q,\mu }(0,1]$ and $\overrightarrow{f}\in C_2^{q,\mu }[0,1]$ for all $q\in \mathbb{N}$, where $\mu =0.5$.

In order to find a numerical solution to problem (45)–(46) for $m\ge 2$, we assemble and solve the system of the linear algebraic Equation (35). The parameter p in the smoothing transformation (14) is chosen to be greater than ${\displaystyle \frac{m}{1-\nu }}$, where

$$\nu =max\{1-\alpha ,\mu \}=0.5.$$

Thus, we have to take $p>2m$ to achieve the expected convergence order $O\left(h^m\right)$ given by Theorem 2.

In

Table 2. Numerical results for problem (45)–(46).

	$\mathit{m}=3$, $\mathit{p}=6$		$\mathit{m}=4$, $\mathit{p}=8$		$\mathit{m}=5$, $\mathit{p}=10$
$\mathit{N}$	${\mathbf{\epsilon }}_{\mathit{N}}$	${\mathbf{\epsilon }}_{\mathit{N}/\mathbf{2}}/{\mathbf{\epsilon }}_{\mathit{N}}$	${\mathbf{\epsilon }}_{\mathit{N}}$	${\mathbf{\epsilon }}_{\mathit{N}/\mathbf{2}}/{\mathbf{\epsilon }}_{\mathit{N}}$	${\mathbf{\epsilon }}_{\mathit{N}}$	${\mathbf{\epsilon }}_{\mathit{N}/\mathbf{2}}/{\mathbf{\epsilon }}_{\mathit{N}}$
8	$1.63·{10}^{-2}$		$7.30·{10}^{-3}$		$6.71·{10}^{-3}$
16	$2.46·{10}^{-3}$	$6.63$	$7.64·{10}^{-4}$	$9.55$	$4.04·{10}^{-4}$	$16.57$
32	$3.37·{10}^{-4}$	$7.31$	$5.91·{10}^{-5}$	$12.91$	$1.83·{10}^{-5}$	$22.09$
64	$4.40·{10}^{-5}$	$7.65$	$4.05·{10}^{-6}$	$14.58$	$6.73·{10}^{-7}$	$27.19$
128	$5.63·{10}^{-6}$	$7.82$	$2.64·{10}^{-7}$	$15.35$	$2.26·{10}^{-8}$	$29.72$
256	$7.11·{10}^{-6}$	$7.91$	$1.68·{10}^{-8}$	$15.70$	$7.33·{10}^{-10}$	$30.91$
512	$8.93·{10}^{-7}$	$7.96$	$1.06·{10}^{-9}$	$15.86$	$2.33·{10}^{-11}$	$31.37$
		8		16		32

, the errors ${\epsilon }_N(N=2^k,k=3,\dots ,9)$ are computed similarly to Example 1. As we can see, the error ratios approach the expected limit $2^m$ as N increases.

8. Concluding Remarks

In this work, we have constructed a high-order method based on the central part interpolation approach for the numerical solution of a class of initial value problems for systems of linear Caputo-type fractional differential equations, extending the method from the scalar problems studied in [41,42]. Taking into account the possible non-smooth behaviour of the exact solution of the underlying problem, we have carried out a rigorous convergence analysis and obtained the optimal convergence order of the proposed schemes. Finally, we have tested our theoretical results with some numerical experiments, which confirmed the validity of our method. In future studies, we hope to extend the proposed approach to the study of non-commensurate systems of linear and nonlinear fractional differential equations.