A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions

Abstract

In this paper, a novel second-order method based on a change of variable and the symmetrical and repeated quadrature formula is presented for numerical solving second kind Volterra integral equations with non-smooth solutions. Applying the discrete Grönwall inequality with weak singularity, the convergence order $\mathcal{O}\left(N^{-2}\right)$ in $L^{\infty }$ norm is proved, where N refers to the number of time steps. Numerical results are conducted to verify the efficiency and accuracy of the method.

1. Introduction

In this paper, we consider numerical solutions of the following second kind of Volterra integral equations (VIEs) [1],

$$y\left(t\right)=g\left(t\right)+{\int }_0^t\frac{\kappa (t,s)y\left(s\right)}{{(t-s)}^{\beta }}ds, t\in [0,T], 0<\beta <1,$$

(1)

where $\kappa \in C^m(\Omega )$, $\Omega =(t,s):0\le s\le t\le T$ and $g\left(t\right)$ is a given function and we assume that $g\left(t^{\frac{1}{1-\beta }}\right)$ satisfies sufficiently smooth. The second kind VIEs have been widely used in many areas, such as science, mathematical physics, and engineering [2,3,4]. However, in most cases, the integration part can not be solved analytically by taking a weakly singular kernel into account.

In the past several years, there appears to be an increasing interest in finding numerical methods for solving the general weak singular VIEs of the second kind, such as product integration methods [5], fractional multistep methods [6], collocation methods [7,8,9,10,11,12] and so on [13,14,15,16,17,18,19]. For the sake of the comprehensive analysis and the difficulty caused by the weak regularity of the kernel, more and more investigators pay attention to the collocation methods. As pointed out in [2,20,21,22], the usual collection methods enjoy high accuracy for problems with high regularity restriction on the solutions. We shall know the convergence order will be $\mathcal{O}\left(h^{1-\beta }\right)$ for non-smooth solutions with uniform meshes whatever the degree of polynomials one chooses. In order to fill this gap, one of the most popular methods used by researchers is applying graded meshes [23,24,25,26,27,28]. In fact, as mentioned by Tang [26], graded meshes did not well in computing because enough small interval in initial time is needed. They introduced a new method with suitable transportation to solve this problem which can obtain high orders and avoid the difficulty of computation. For more detailed results, we refer readers to [29,30,31,32,33]. To our best knowledge, only fixed parameter $\beta$ is considered when they numerically solved Equation (1) using variable transformations in previous works. Then, giving a simple and straightaway approach to solve the second kind VIEs is the main purpose to do this research.

In this paper, we investigate the analysis and computation of the trapezoidal rule for the second kind of Volterra integral equations. Firstly, a common change of variable $s=t^{1-\beta }$ is used to gain a new equation whose exact solution is smooth even at the initial time. Then, the trapezoidal rule could be applied to estimate the integral part. The convergence results are conducted and proved with weakly singular discrete Grönwall inequality which was first proposed in [34] and has been widely used in the analysis of numerical schemes [9,10,26,35]. Dixon [35] applied the discrete Grönwall inequality and collocation methods to solve VIEs. The error estimate $\mathcal{O}\left(N^{2-\alpha }\right)$ is established at $t=T$. However, the previous results are obtained by applying collocation methods. The proof of present results is much more technical making use of variable changes and the non-locality of the problem.

The rest of this article is organized as follows. Numerical schemes using the product trapezoidal rule are structured under variables change in Section 2. We give a rigorous convergence analysis of the proposed method in Section 3. In Section 4, numerical tests are conducted to justify our theoretical results. Finally, we conclude this literature in Section 5.

2. The Product Trapezoidal Rule and Main Results

In this section, we present the trapezoidal scheme for solving problems (1). In order to guarantee the solution gained by the variable change is smooth even at the initial time, we assume that $g(t^{\frac{1}{1-\beta }})$ satisfies sufficiently smooth. The solution of Equation (1) can be expressed as follows, which is the same as [35],

$$y\left(t\right)=g\left(t\right)+\sum _{n=1}^{\infty }{\varphi }_n(t;\beta )t^{n(1-\beta )},t\in [0,T],$$

(2)

where ${\varphi }_n\in C^m[0,T]$ for any n. Based on the expansion (2), we introduce the change of variable

$$t^{1-\beta }=s.$$

Let $\alpha =1-\beta , h=\frac{T^{\alpha }}{N}$ and denote $s_i=ih, i=0,1,2, \dots ,N$ as a uniform mesh on $[0,T^{\alpha }]$, where N is a positive integer. For ease of exposition, we suppose that $u\left(s\right):=y\left(s^{\frac{1}{\alpha }}\right), f\left(s\right):=g\left(s^{\frac{1}{\alpha }}\right)$. Then, we can rewrite Equation (1) as follows

$$\begin{array}{ccc}u\left(s\right)& =& {\int }_0^{s^{\frac{1}{\alpha }}}\frac{\kappa (s^{\frac{1}{\alpha }},z)y\left(z\right)}{{(s^{\frac{1}{\alpha }}-z)}^{1-\alpha }}dz+f\left(s\right)\hfill \\ & =& \frac{1}{\alpha }{\int }_0^s\frac{\kappa (s^{\frac{1}{\alpha }},r^{\frac{1}{\alpha }})y\left(r^{\frac{1}{\alpha }}\right)}{{(s^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}{r}^{\frac{1}{\alpha }-1}dr+f\left(s\right)\hfill \\ & =& \frac{1}{\alpha }{\int }_0^s\frac{\kappa (s^{\frac{1}{\alpha }},r^{\frac{1}{\alpha }})u\left(r\right)}{{(s^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}{r}^{\frac{1}{\alpha }-1}dr+f\left(s\right),s\in [0,T^{\alpha }].\hfill \end{array}$$

(3)

Taking $s=s_i, u_i=u\left(s_i\right), i=0,1,2,\dots ,N$ and applying interpolation approximation at each interval $[s_j,s_{j+1}], j=0,1,2,\dots ,i-1$, we arrive at

$$\begin{array}{ccc}{u}_i& =& \frac{1}{\alpha }{\int }_0^{s_i}\frac{\kappa (s_i^{\frac{1}{\alpha }},r^{\frac{1}{\alpha }})u\left(r\right)}{{(s_i^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}{r}^{\frac{1}{\alpha }-1}dr+f\left(s_i\right)\hfill \\ & =& \frac{1}{\alpha }\sum _{j=0}^{i-1}{\int }_{s_j}^{s_{j+1}}\frac{\kappa (s_i^{\frac{1}{\alpha }},r^{\frac{1}{\alpha }})u\left(r\right)}{{(s_i^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}{r}^{\frac{1}{\alpha }-1}dr+f\left(s_i\right)\hfill \\ & \approx & \frac{1}{\alpha }\sum _{j=0}^{i-1}{\int }_{s_j}^{s_{j+1}}\frac{r^{\frac{1}{\alpha }-1}}{{(s_i^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}\left[\frac{r-s_{j+1}}{-h}{u}_j\kappa (s_i^{\frac{1}{\alpha }},s_j^{\frac{1}{\alpha }})+\frac{r-s_j}{h}{u}_{j+1}\kappa (s_i^{\frac{1}{\alpha }},s_{j+1}^{\frac{1}{\alpha }})\right]dr\hfill \\ & & +f\left(s_i\right).\hfill \end{array}$$

(4)

Omitting the truncation error and denoting $U_i$ as the approximation to $u\left(s_i\right)(0\le i\le N)$, we can obtain the product trapezoidal rule for Equation (3),

$$\begin{array}{ccc}{U}_0& =& f\left(s_0\right),\hfill \\ U_i& =& \sum _{j=0}^i{w}_{i,j}\kappa (s_i^{\frac{1}{\alpha }},s_j^{\frac{1}{\alpha }})U_j+f_i, 1\le i\le N,\hfill \end{array}$$

(5)

where

$$\begin{array}{ccc}{w}_{i,0}& =& \frac{1}{\alpha h}{\int }_{s_0}^{s_1}\frac{r^{\frac{1}{\alpha }-1}(s_1-r)}{{(s_i^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}dr, 1\le i\le N,\hfill \\ w_{i,j}& =& \frac{1}{\alpha h}{\int }_{s_j}^{s_{j+1}}\frac{r^{\frac{1}{\alpha }-1}(s_{j+1}-r)}{{(s_i^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}dr+\frac{1}{\alpha h}{\int }_{s_{j-1}}^{s_j}\frac{r^{\frac{1}{\alpha }-1}(r-s_{j-1})}{{(s_i^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}dr,\hfill \\ & & 1\le j\le i-1, 2\le i\le N,\hfill \\ w_{i,i}& =& \frac{1}{\alpha h}{\int }_{s_{i-1}}^{s_i}\frac{r^{\frac{1}{\alpha }-1}(r-s_{i-1})}{{(s_i^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}dr, 1\le i\le N.\hfill \end{array}$$

Considering the formate of coefficient $w_{i,j}$ is more complex, we give the following proposition for exact expansion.

Lemma 1. 

There exists a positive real number $M_1$ such that

$$0\le w_{i,j}\le \frac{M_1}{\alpha }{j}^{\frac{1}{\alpha }-1}h{(i^{\frac{1}{\alpha }}-j^{\frac{1}{\alpha }})}^{\alpha -1}, 1\le j<i\le N.$$

(6)

Moreover

$$0\le w_{i,i}\le \frac{{[i^{\frac{1}{\alpha }}-{(i-1)}^{\frac{1}{\alpha }}]}^{\alpha }h}{\alpha }, 1\le i\le N.$$

(7)

Proof. 

Here, we give the rigorous calculated progress of $w_{i,j}$ and $w_{i,i}$. Recall the mean-value theorem, there exist $\xi \in (s_j,s_{j+1})$ and $\eta \in (s_{j-1},s_j)$,

$$\begin{array}{cc}\hfill w_{i,j}& =\frac{{\xi }^{\frac{1}{\alpha }-1}(s_{j+1}-\xi )}{\alpha {(s_i^{\frac{1}{\alpha }}-{\xi }^{\frac{1}{\alpha }})}^{1-\alpha }}+\frac{{\eta }^{\frac{1}{\alpha }-1}(\eta -s_{j-1})}{\alpha {(s_i^{\frac{1}{\alpha }}-{\eta }^{\frac{1}{\alpha }})}^{1-\alpha }}\hfill \\ \hfill & \le \frac{h}{\alpha }\frac{s_{j+1}^{\frac{1}{\alpha }-1}}{{(s_i^{\frac{1}{\alpha }}-s_j^{\frac{1}{\alpha }})}^{1-\alpha }}+\frac{h}{\alpha }\frac{s_j^{\frac{1}{\alpha }-1}}{{(s_i^{\frac{1}{\alpha }}-s_j^{\frac{1}{\alpha }})}^{1-\alpha }}\hfill \\ \hfill & =\frac{h}{\alpha }{(j+1)}^{\frac{1}{\alpha }-1}{(i^{\frac{1}{\alpha }}-j^{\frac{1}{\alpha }})}^{\alpha -1}+\frac{h}{\alpha }{j}^{\frac{1}{\alpha }-1}{(i^{\frac{1}{\alpha }}-j^{\frac{1}{\alpha }})}^{\alpha -1}\hfill \\ \hfill & =\frac{h}{\alpha }{j}^{\frac{1}{\alpha }-1}{(i^{\frac{1}{\alpha }}-j^{\frac{1}{\alpha }})}^{\alpha -1}[1+{\left(\frac{j+1}{j}\right)}^{\frac{1}{\alpha }-1}]\hfill \\ \hfill & \le M_{1}h{j}^{\frac{1}{\alpha }-1}{(i^{\frac{1}{\alpha }}-j^{\frac{1}{\alpha }})}^{\alpha -1}.\hfill \end{array}$$

Similarly, the estimate of $w_{i,i}$ gives that

$$\begin{array}{ccc}\hfill w_{i,i}& =& \frac{1}{\alpha h}{\int }_{s_{i-1}}^{s_i}\frac{r^{\frac{1}{\alpha }}}{{(s_i^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}dr-\frac{1}{\alpha h}{\int }_{s_{i-1}}^{s_i}\frac{r^{\frac{1}{\alpha }-1}{s}_{i-1}}{{(s_i^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}dr\hfill \\ & =& \frac{1}{h}{\int }_{s_{i-1}^{\frac{1}{\alpha }}}^{s_i^{\frac{1}{\alpha }}}{z}^{\alpha }{(s_i^{\frac{1}{\alpha }}-z)}^{\alpha -1}dz-\frac{1}{h}{\int }_{s_{i-1}^{\frac{1}{\alpha }}}^{s_i^{\frac{1}{\alpha }}}{s}_{i-1}{(s_i^{\frac{1}{\alpha }}-z)}^{\alpha -1}dz\hfill \\ & =& \frac{s_i^2}{h}{\int }_{s_{i-1}^{\frac{1}{\alpha }}/s_i^{\frac{1}{\alpha }}}^1{v}^{\alpha }{(1-v)}^{\alpha -1}dv-\frac{s_{i-1}{s}_i}{h}{\int }_{s_{i-1}^{\frac{1}{\alpha }}/s_i^{\frac{1}{\alpha }}}^1{(1-v)}^{\alpha -1}dv\hfill \\ & \le & \left(\frac{s_i^2-s_{i-1}{s}_i}{h}\right){\int }_{s_{i-1}^{\frac{1}{\alpha }}/s_i^{\frac{1}{\alpha }}}^1{(1-v)}^{\alpha -1}dv\hfill \\ & =& \frac{{[i^{\frac{1}{\alpha }}-{(i-1)}^{\frac{1}{\alpha }}]}^{\alpha }h}{\alpha },\hfill \end{array}$$

which completes the proof. □

The weak singularity discrete Grönwall inequality is very important in the classical result; thus, we display it as follows.

Lemma 2 

([34]). Suppose $\left\{x_i\right\}$ and $\left\{{\psi }_i\right\}$ are two sequences of non-negative real numbers and $\left\{{\psi }_i\right\}$ increases monotonously, where $0\le i\le N$. Given the parameters $\sigma ,M\ge 0, 0<\mu <1,$ and $\lambda =\sigma +1$, if

$$x_i\le {\psi }_i+M{h}^{\sigma +1-\lambda \mu }\sum _{j=0}^{i-1}\frac{j^{\sigma }{x}_j}{{(i^{\lambda }-j^{\lambda })}^{\mu }}, 0\le i\le N,$$

then

$$x_i\le {\psi }_i{E}_{1-\mu }\left(\frac{M\Gamma (1-\mu )}{\lambda }{\left(ih\right)}^{\lambda (1-\mu )}\right), 0\le i\le N,$$

where $E_{\alpha }\left(z\right)={\sum }_{k=0}^{\infty }\frac{z^k}{\Gamma (1+k\alpha )}$ is the Mittag-Leffler function.

We present our main results in the following theorem and leave its proof in the next section.

Theorem 1. 

Suppose that the exact solution of Equation (3) $u\in C^2[0,T]$, $\kappa \in C^2[0,T]$ and $f\in C^2[0,T]$, then the scheme defined in (5) has a unique solution $U_i$ satisfying

$$\parallel u_i-U_i{\parallel }_{L^{\infty }}\le C^{*}{h}^2,i=1,2,\dots ,N,$$

(8)

where $C^{*}$ is a positive constant independent of h.

Remark 1. 

Since that $u\left(s_n\right)=y(s_n^{\frac{1}{\alpha }})$, the convergence results also can be imposed as

$$\parallel y_i-y(s_i^{\frac{1}{\alpha }}){\parallel }_{L^{\infty }}\le C^{*}{h}^2, i=1,2,\dots ,N,$$

where $y_i$ is the numerical approximation to $y(s_i^{\frac{1}{\alpha }})$.

Remark 2. 

The conclusion in the theorem can be extended by using collocation methods to approximate $u\left(r\right)$. In order to satisfy the effectiveness of the change thoughts, we only consider the case that the product trapezoidal rule.

3. Convergence of the Product Trapezoidal Rule

In this section, we will focus on the proof of Theorem 1. Considering $s=s_i$ in Equation (3) yields

$$u_i=\sum _{j=0}^i{w}_{i,j}\kappa (s_i^{\frac{1}{\alpha }},s_j^{\frac{1}{\alpha }})u_j+T_i, 0\le i\le N,$$

(9)

where $T_0=0$ and for $1\le i\le N$,

$$\begin{array}{cc}\hfill T_i=& \sum _{j=0}^{i-1}{\int }_{s_j}^{s_{j+1}}\left[u\left(r\right)\kappa (s_i^{\frac{1}{\alpha }},r^{\frac{1}{\alpha }})-\left(\frac{r-s_{j+1}}{-h}{u}_j\kappa (s_i^{\frac{1}{\alpha }},s_j^{\frac{1}{\alpha }})+\frac{r-s_j}{h}{u}_{j+1}\kappa (s_i^{\frac{1}{\alpha }},s_{j+1}^{\frac{1}{\alpha }})\right)\right]\hfill \\ \hfill & \frac{r^{\frac{1}{\alpha }-1}}{\alpha {(s_i^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}dr.\hfill \end{array}$$

Substituting (5) into (9) and taking $e_i:=u_i-U_i$, we can obtain

$$e_i=\sum _{j=0}^i{w}_{i,j}\kappa (s_i^{\frac{1}{\alpha }},s_j^{\frac{1}{\alpha }})e_j+T_i, 1\le i\le N,$$

(10)

which further implies that for any $1\le i\le N$,

$$\begin{array}{ccc}\hfill |e_i|& \le & \sum _{j=0}^i{w}_{i,j}|e_j\left|\right|\kappa (s_i^{\frac{1}{\alpha }},s_j^{\frac{1}{\alpha }})|+|T_i|\hfill \\ & \le & \sum _{j=0}^{i-1}{w}_{i,j}|e_j\left|\right|\kappa (s_i^{\frac{1}{\alpha }},s_j^{\frac{1}{\alpha }})|+w_{i,i}|e_i\left|\right|\kappa (s_i^{\frac{1}{\alpha }},s_i^{\frac{1}{\alpha }})|+|T_i|.\hfill \end{array}$$

Combining inequality (6), (7) with the assumption $M_1{max}_{0\le s\le t\le T}\left|\kappa (t,s)\right|h<1$, it yields

$$|e_i|\le C_2|T_i|+C_2\sum _{j=1}^{i-1}\frac{M_1}{\alpha }{j}^{\frac{1}{\alpha }-1}h{(i^{\frac{1}{\alpha }}-j^{\frac{1}{\alpha }})}^{\alpha -1}\left|e_j\right|,$$

(11)

where $C_2$ is a positive constant independent on h and i.

Applying the error in Lagrange interpolation and the assumptions of the exact solution and $\kappa (t,s)$, there exists a positive constant $M_2$, for any $s\in [s_j,s_{j+1}],$

$$u\left(s\right)\kappa (s_i^{\frac{1}{\alpha }},s^{\frac{1}{\alpha }})-\left[\frac{s_{j+1}-s}{h}{u}_j\kappa (s_i^{\frac{1}{\alpha }},s_j^{\frac{1}{\alpha }})+\frac{s-s_j}{h}{u}_{j+1}\kappa (s_i^{\frac{1}{\alpha }},s_{j+1}^{\frac{1}{\alpha }})\right]\le M_2{h}^2.$$

(12)

Together with Equation (10), we can further obtain

$$\begin{array}{ccc}|T_i|& \le & \frac{M_1{M}_2{h}^2}{\alpha }\sum _{j=0}^{i-1}{\int }_{s_j}^{s_{j+1}}\frac{r^{\frac{1}{\alpha }-1}}{{(s_i^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}dr=\frac{M_1{M}_2{h}^2}{\alpha }{\int }_0^{s_i}\frac{r^{\frac{1}{\alpha }-1}}{{(s_i^{\frac{1}{\alpha }}-r^{\frac{1}{\alpha }})}^{1-\alpha }}dr\hfill \\ & =& M_1{M}_2{h}^2{\int }_0^{s_i^{\frac{1}{\alpha }}}\frac{1}{{(s_i^{\frac{1}{\alpha }}-t)}^{1-\alpha }}dt\hfill \\ & =& M_1{M}_2{h}^2{\int }_0^{s_i^{\frac{1}{\alpha }}}{\left[s_i^{\frac{1}{\alpha }}(1-\frac{t}{s_i^{1/\alpha }})\right]}^{\alpha -1}dt\hfill \\ & =& \frac{M_1{M}_2{h}^2{s}_i^{\frac{1}{\alpha }-1}}{\alpha }.\hfill \end{array}$$

(13)

Then substituting above inequality (13) into (11), we can obtain

$$|e_i|\le C_3{h}^2+C_3\sum _{j=1}^{i-1}{j}^{\frac{1}{\alpha }-1}h{(i^{\frac{1}{\alpha }}-j^{\frac{1}{\alpha }})}^{\alpha -1}\left|e_j\right|,$$

(14)

where $C_3=max\{\frac{C_2{M}_1{M}_2{s}_i^{\frac{1}{\alpha }-1}}{\alpha },\frac{C_2{M}_1}{\alpha }\}.$

Applying Lemma 2, Equation (14) gives that

$$|e_i|\le C^{*}{h}^2,$$

(15)

where $C^{*}=C_3{E}_{1-\alpha }(C_3\alpha \Gamma (1-\alpha )\left(ih\right))$, which completes the proof of Theorem 1.

4. Numerical Example

In this section, some examples are given to verify our theoretical results. Here we take $\kappa =-1$ in Equation (1). Introduce the following notation

$$errors\left(\tau \right)=\underset{1\le n\le N}{max}\left|ye\left(t_n\right)-y^n\right|,err\left(t_N\right)=ye\left(t_N\right)-y^N$$

and the convergence order

$$orders:=\frac{log\left(errors\left({\tau }_1\right)\right)/log\left(errors\left({\tau }_2\right)\right)}{log({\tau }_1/{\tau }_2)},$$

where ${\tau }_1$ and ${\tau }_2$ mean the time steps.

Example 1. 

Consider the following Volterra integral equation of the second kind with a weakly singular kernel

$$y\left(t\right)=-{\int }_0^t{(t-s)}^{-\beta }y\left(s\right)ds+g\left(t\right),0\le t\le 1,$$

and $g\left(t\right)$ satisfies that the exact solution is $y=t+t^{1-\beta },0<\beta <1.$

To examine the effectiveness of our numerical methods, we take $N=100, 200, 300, 400$ for different $\beta =0.2, 0.4, 0.6$ and $\tau =1/N$. Table 1 and Table 2 show that the maximum errors and orders present that the convergence rate of our scheme is 2. Moreover, we consider the classical product trapezoidal rule without variable exchange to solve the second VIE with nonsmooth solution and the results found in Table 1 and Table 2 give that the order is $2(1-\beta )$ which is less than 2. This finding coincides with our theoretical results. In order to make our method more useful, we also test the errors and orders at $t=1$. The results found in Table 2 and Table 3 state that the scheme without variable change observes $2-\beta$ convergence orders.

Table 1. The errors and orders with $L^{\infty }$-norm.

		$\mathit{\beta }$ = 0.2		$\mathit{\beta }$ = 0.4		$\mathit{\beta }$ = 0.6
	$\mathit{N}$	Errors	Orders	Errors	Orders	Errors	Orders
Our Scheme	100	$3.91\times {10}^{-6}$	∗	$6.95\times {10}^{-6}$	∗	$2.11\times {10}^{-5}$	∗
	200	$9.85\times {10}^{-7}$	$1.99$	$1.74\times {10}^{-6}$	$2.00$	$5.34\times {10}^{-6}$	$1.97$
	300	$4.39\times {10}^{-7}$	$1.99$	$7.75\times {10}^{-7}$	$2.00$	$2.39\times {10}^{-6}$	$1.98$
	400	$2.47\times {10}^{-7}$	$1.99$	$4.36\times {10}^{-7}$	$2.00$	$1.35\times {10}^{-6}$	$1.98$
Original Scheme	100	$3.97\times {10}^{-5}$	∗	$6.20\times {10}^{-4}$	∗	$6.41\times {10}^{-3}$	∗
	200	$1.32\times {10}^{-5}$	$1.59$	$2.76\times {10}^{-4}$	$1.17$	$3.89\times {10}^{-3}$	$0.72$
	300	$6.92\times {10}^{-6}$	$1.59$	$1.71\times {10}^{-4}$	$1.18$	$2.89\times {10}^{-3}$	$0.73$
	400	$4.37\times {10}^{-6}$	$1.59$	$1.22\times {10}^{-4}$	$1.18$	$2.33\times {10}^{-3}$	$0.74$

* means here is no order.

Table 2. The errors and orders for $\beta =0.8$.

	N	Errors		$\mathit{err}\left({\mathit{t}}_{\mathit{N}}\right)$
Our Scheme	1000	$1.07\times {10}^{-6}$	∗	$1.07\times {10}^{-6}$	∗
	2000	$2.78\times {10}^{-7}$	$1.95$	$2.78\times {10}^{-7}$	$1.95$
	3000	$1.26\times {10}^{-7}$	$1.96$	$1.26\times {10}^{-7}$	$1.96$
	4000	$7.14\times {10}^{-8}$	$1.96$	$7.14\times {10}^{-8}$	$1.96$
Original Scheme	1000	$1.80\times {10}^{-2}$	∗	$4.40\times {10}^{-6}$	∗
	2000	$1.46\times {10}^{-2}$	$0.30$	$1.80\times {10}^{-6}$	$1.29$
	3000	$1.29\times {10}^{-2}$	$0.31$	$1.07\times {10}^{-7}$	$1.28$
	4000	$1.18\times {10}^{-2}$	$0.31$	$7.40\times {10}^{-6}$	$1.28$

* means here is no order.

Table 3. The errors and orders at $t=1$.

		$\mathit{\beta }$ = 0.2		$\mathit{\beta }$ = 0.4		$\mathit{\beta }$ = 0.6
	$\mathit{N}$	$\mathit{err}\left({\mathit{t}}_{\mathit{N}}\right)$	Orders	$\mathit{err}\left({\mathit{t}}_{\mathit{N}}\right)$	Orders	$\mathit{err}\left({\mathit{t}}_{\mathit{N}}\right)$	Orders
Our Scheme	100	$2.94\times {10}^{-6}$	∗	$6.95\times {10}^{-6}$	∗	$2.11\times {10}^{-5}$	∗
	200	$7.36\times {10}^{-7}$	$2.00$	$1.74\times {10}^{-6}$	$2.00$	$5.34\times {10}^{-6}$	$1.97$
	300	$3.28\times {10}^{-7}$	$2.00$	$7.75\times {10}^{-7}$	$2.00$	$2.39\times {10}^{-6}$	$1.98$
	400	$1.84\times {10}^{-7}$	$2.00$	$4.36\times {10}^{-7}$	$2.00$	$1.35\times {10}^{-6}$	$1.98$
Original Scheme	100	$6.96\times {10}^{-6}$	∗	$1.82\times {10}^{-5}$	∗	$3.85\times {10}^{-5}$	∗
	200	$2.02\times {10}^{-6}$	$1.78$	$5.88\times {10}^{-6}$	$1.63$	$1.39\times {10}^{-5}$	$1.47$
	300	$9.80\times {10}^{-7}$	$1.79$	$3.04\times {10}^{-6}$	$1.62$	$7.70\times {10}^{-6}$	$1.46$
	400	$5.86\times {10}^{-7}$	$1.79$	$1.91\times {10}^{-6}$	$1.62$	$5.08\times {10}^{-6}$	$1.45$

* means here is no order.

Example 2. 

We consider the problem (1) with the following right hand function

$$g(t,u\left(t\right))=u\left(t\right)+t^{3+2\alpha }B(\alpha ,4+\alpha )$$

where $B(p,q)$ means the Beta function. The corresponding exact function is $u\left(t\right)=t^{3+\alpha }.$

Similarly, we choose $N=100,200,300,400$ for different $\alpha =0.2,0.4,0.6,0.8.$ Table 4 and Table 5 give the errors with maximum norm and from which we can obtain the order is of 2.

Table 4. The errors and orders of $\alpha =0.2,0.4$.

	$\mathit{\alpha }$ = 0.2		$\mathit{\alpha }$ = 0.4
$\mathit{N}$	Errors	Orders	Errors	Orders
100	$9.04\times {10}^{-4}$	∗	$2.74\times {10}^{-4}$	∗
200	$2.48\times {10}^{-4}$	1.86	$7.07\times {10}^{-5}$	1.95
300	$1.15\times {10}^{-4}$	1.89	$3.19\times {10}^{-5}$	1.96
400	$6.68\times {10}^{-5}$	1.90	$1.81\times {10}^{-5}$	1.97

* means here is no order.

Table 5. The errors and orders of $\alpha =0.6,0.8$.

	$\mathit{\alpha }$ = 0.6		$\mathit{\alpha }$ = 0.8
$\mathit{N}$	Errors	Orders	Errors	Orders
100	$1.06\times {10}^{-4}$	∗	$4.93\times {10}^{-5}$	∗
200	$2.68\times {10}^{-5}$	1.99	$1.24\times {10}^{-5}$	2.00
300	$1.20\times {10}^{-5}$	1.99	$5.50\times {10}^{-6}$	2.00
400	$6.75\times {10}^{-6}$	1.99	$3.92\times {10}^{-6}$	2.00

* means here is no order.

Example 3. 

We consider the problem

$$y\left(t\right)=-{\int }_0^t{(t-s)}^{-\beta }y\left(s\right)ds+g\left(t\right),0\le t\le 1,$$

with $y\left(0\right)=1$ and right hand function $g\left(t\right)=1$ without knowing the analytical solution.

We give the following notation for more readability.

Define:

$$Err:=\underset{1\le n_2\le N}{max}\left|y^{n_2}-y_{ref}^{n_1}\right|$$

where $y_{ref}$ means the reference solution and $y_{ref}^{n_1}$ approximates to $ye\left(t_{n1}\right)$, $n_1=\frac{N_0}{N}{n}_2$, $n_2=1,2,\dots ,N$.

We use proposed methods to solve the initial problem 3 and the reference solutions are computed with $N_0=2000$. The optimal error estimate is observed by taking $N=50,100,200,400$. Table 6 and Table 7 declare that the convergence order is 2 for different $\beta =0.2,0.4,0.6,0.8$, which verifies the efficiency of our scheme.

Table 6. The errors and orders of $\beta =0.2,0.4$.

	$\mathit{\beta }$ = 0.2		$\mathit{\beta }$ = 0.4
$\mathit{N}$	Err	Orders	Err	Orders
50	$1.68\times {10}^{-5}$	∗	$2.56\times {10}^{-5}$	∗
100	$4.20\times {10}^{-6}$	2.00	$6.42\times {10}^{-6}$	1.99
200	$1.04\times {10}^{-6}$	2.01	$1.60\times {10}^{-6}$	2.00
400	$2.53\times {10}^{-7}$	2.04	$3.90\times {10}^{-7}$	2.04

* means here is no order.

Table 7. The errors and orders of $\beta =0.6,0.8$.

	$\mathit{\beta }$ = 0.6		$\mathit{\beta }$ = 0.8
$\mathit{N}$	Err	Orders	Err	Orders
50	$4.42\times {10}^{-5}$	∗	$8.67\times {10}^{-5}$	∗
100	$1.14\times {10}^{-5}$	1.95	$2.44\times {10}^{-5}$	1.83
200	$2.92\times {10}^{-6}$	1.97	$6.69\times {10}^{-6}$	1.88
400	$7.22\times {10}^{-7}$	2.02	$1.76\times {10}^{-6}$	1.93

* means here is no order.

5. Conclusions

This paper mainly presents the scheme produced by applying the trapezoidal rule to the changed equation for any $0<\beta <1$, which can be used in more general occasions. More rigorous analysis can be obtained with the help of the discrete Grönwall equality and a typical example is tested to verify our theoretical result.