Multi-Effective Collocation Methods for Solving the Volterra Integral Equation with Highly Oscillatory Fourier Kernels

Abstract

In this paper, we focus on the numerical solution of the second kind of Volterra integral equation with a highly oscillatory Fourier kernel. Based on the calculation of the modified moments, we propose four collocation methods to solve the equations: direct linear interpolation, direct higher order interpolation, direct Hermite interpolation and piecewise Hermite interpolation. These four methods are simple to construct and can quickly compute highly oscillatory integrals involving Fourier functions. We present the corresponding error analysis and it is easy to see that, in some cases, our proposed method has a fast convergence rate in solving such equations. In some cases, our proposed methods have significant advantages over the existing methods. Some numerical experiments demonstrating the efficiency of the four methods are also presented.

1. Introduction

Volterra integral equations occur widely in all areas of mathematical physics, such as electromagnetic scattering [1], transient scattering problems [2], acoustic scattering problems [3], the subdiffusive and superdiffusive regime [4], transport dynamics [5], static and quasi-static electrodynamics [6], and virus transmission models [7,8,9]. Moreover, in 1995, Berrone [10] introduced a heat transfer model while studying the changes in the state of materials at specific temperatures:

$$u\left(x\right)=f\left(x\right)+{\int }_0^x{(x-t)}^{-\alpha }k(x,t)u\left(t\right)dt.$$

For the study of the numerical solution of a scalar retarded potential integral equation posed on an infinite flat surface,

$${\int }_L\frac{u(x^{\prime },t-\left|x^{\prime }-x\right|)}{\left|x^{\prime }-x\right|}d{x}^{\prime }=a(x,t),(x,t)\in L\times (0,T),$$

where u is the unknown function, if $L={\mathbb{R}}^2$ and u and a satisfy $u\equiv 0$, $a\equiv 0$ for any $t\le 0$. Davis and Ducan [2] obtained the following equation by taking the Fourier transform

$$2\pi {\int }_0^t{J}_0\left(\omega s\right)\widehat{u}(\omega ,t-s)ds=\widehat{a}(\omega ,t),t\in I:=[0,T],T<\infty ,$$

where $J_0(·)$ denotes the Bessel function of the first kind and of order zero. When $\omega \gg 1$, the kernel function $J_0\left(\omega s\right)$ is highly oscillatory.

In [11], Beezley and Krueger obtained an equivalent formula to the Helmholtz equation, which is expressed as

$$4R\left(t\right)+G\left(t\right)+(G\ast (2R+R\ast R\left)\right)\left(t\right)=0,t>0,$$

where $G(·)$ denotes Green’s function and $f\ast g$ denotes the convolution $(f\ast g)\left(x\right)={\int }_0^{x}f\left(t\right)g(x-t)dt$. There are situations in which this equation can be solved explicitly, such as when

$$G\left(t\right)=\gamma e^{\beta t}\iff R\left(t\right)=e^{(\beta -\gamma /2)t}\frac{I_1(\gamma t/2)}{t},$$

or

$$G\left(t\right)=\gamma t{e}^{\beta t}\iff R\left(t\right)=2e^{\beta t}\frac{J_2\left(\sqrt{\gamma }t\right)}{t}.$$

Here, $I_1$ is the first kind of the first-order modified Bessel function, and $J_2$ is the first kind of the second-order Bessel function. For more details, see [11,12].

In recent decades, lots of methods have been proposed to solve Volterra integral equations. For example, Linz [13] proposed the classical product–integral technique to solve Volterra integral equations of the first type with a highly oscillatory kernel. Anderssen and White [14] proposed the use of approximations for the kernel function k and the unknown function u, respectively, to solve Volterra integral equations of the first type. Levin [15] proposed a collocation method for approximating integrals of rapidly oscillatory functions. Levin [16] analyzed the rate of convergence of a collocation method and showed that the error decreases with increasing frequency. Uddin [17] constructed a numerical scheme for approximation of a class of Volterra integral equations of convolution type with highly oscillatory kernels. Xiang and Brunner [18] proposed three collocation methods to solve the Volterra integral equation with weakly singular highly oscillatory Bessel kernels. Xiang and Wu [19] presented a piecewise constant and linear collocation methods for approximation of the solution to a Volterra integral equation with weakly singular highly oscillatory trigonometric kernels. Wu [20] compared collocation methods on graded meshes with those on uniform meshes for a Volterra integral equation with weakly singular highly oscillatory Fourier kernels. Fang, He and Xiang [21] proposed two Hermite-type collocation methods in the study of a Volterra integral equation with a highly oscillatory Bessel kernel, and the corresponding error analysis and numerical experiments were given. Ma and Kang [22] presented a frequency-explicit convergence analysis of collocation methods for Volterra integral equations with weakly singular highly oscillatory kernels. Ma, Fang and Xiang [23] studied the rate of convergence of the direct Filon method for Volterra integral equations with Fourier kernels. Liu and Ma [24] introduced a kind of generalized multi-step collocation method for Volterra integral equations with highly oscillatory Fourier kernels, and studied two convergence analyses. Research into solving the Volterra equation with time and space variables is also quite well established. For example, Yang, Wu and Zhang [25] and Jiang, Wang, Wang and Zhang [26] proposed a space–time spectral-order Sinc collocation method to deal with the singularity of the equation. They presented a predictor–corrector compact difference scheme for a nonlinear fractional differential equation. Tian, Yang, Zhang and Xu [27] constructed an implicit robust difference method with graded meshes for the modified Burgers model with nonlocal dynamic properties. Yang, Zhang and Tang [28] presented a high-order method based on the orthogonal spline collocation (OSC) method for the solution of the fourth-order subdiffusion problem. Zhang, Yang, Tang and Xu [29] proposed an orthogonal spline collocation (OSC) method to solve the fourth-order multi-term subdiffusion equation. In considering these techniques, some authors have made great contributions to the numerical solution methods for highly oscillatory problems, such as collocation methods [30,31], Filon–Clenshaw–Curtis quadratures [32,33], the Levin method [34], fast multipole methods [35], Clenshaw–Curtis algorithms [36], Clenshaw–Curtis–Filon-type methods [37], BBFM–collocation [38] and so on.

It is well known that integral equations with highly oscillatory kernels are prominent in many fields such as astronomy, computerized tomography, electromagnetic and seismology. For the study of highly oscillatory problems, classical quadrature rules, such as the Newton–Cotes rule, the Clenshaw–Curtis rule or the Gauss rule, cannot compute such integrals. Therefore, we consider the following Volterra integral equation with a highly oscillatory Fourier kernel,

$$u\left(x\right)-{\int }_0^x{e}^{i\omega (x-t)}u\left(t\right)dt=f\left(x\right),x\in [0,T],$$

(1)

where $f\left(x\right)$ is a known smooth function, $\omega$ is the frequency and $u\left(x\right)$ is an unknown function. When $\omega \gg 1$, Equation (1) is a highly oscillatory integral equation. Furthermore, the convergence rate of solving Equation (1) improves from $O\left({\omega }^{-1}\right)$ for the method in the literature [23] to $O\left({\omega }^{-3}\right)$ for the method in this paper.

Regarding the study of highly oscillatory integral equations, Linz [13] proposed the classical product–integral technique for solving Volterra integral equations of the first kind, and we present the computational procedure of this method to solve Equation (1).

First, we divide the integration interval $[0,x_j]$ into N equal parts to obtain the sub-integration interval $[x_{i-1},x_i]$ of width h, where $x_i=ih,x_0=0,x_N=x_j.$ On each interval $[x_{i-1},x_i]$, approximate $u(x_i-h/2)$ by $u_i$.

Then, we get

$$u\left(x_j\right)-\sum _{i=1}^j{m}_i\left(x_j\right)u_i=f\left(x_j\right),$$

(2)

where

$$m_i\left(x\right)={\int }_{x_{i-1}}^{x_i}{e}^{i\omega \left(x-t\right)}dt.$$

Then, Equation (2) can be written as

$$u_j-\sum _{i=1}^{j-1}{m}_i\left(x_j\right)u_i-m_j\left(x_j\right)u_j=f\left(x_j\right).$$

(3)

Solving Equation (3) yields $u_j$

$$u_j=\frac{f\left(x_j\right)-{\sum }_{i=1}^{j-1}{m}_i\left(x_j\right)u_i}{1-m_j\left(x_j\right)}.$$

This paper is composed of the following sections: The background of the equations and the current research status are given in Section 1. Section 2 introduces the direct linear interpolation collocation method, Section 3 leads to the direct high order interpolation collocation method, Section 4 studies the direct Hermite interpolation collocation method, and Section 5 discusses the piecewise Hermite interpolation collocation method. Section 6 contains the error analysis of the method in our paper, and Section 7 contains the numerical experiments.

2. Direct Linear Interpolation Collocation Method (DL)

Volterra [39] introduced the classical theory of linear Volterra integral equations. Brunner [31] gave a resolvent representation of the solution by applying Picard’s iterative method to solve such integral equations.

Let $\mathcal{V}:C\left(I\right)\to C\left(I\right)$ denote the linear Volterra integral operator defined by

$$\left(\mathcal{V}\varphi \right)\left(t\right):={\int }_0^{t}K(x,t)\varphi \left(x\right)dx,t\in I:=[0,T],$$

where the kernel $K=K(x,t)$ is continuous on $D:=(x,t):0\le x\le t\le T.$

Lemma 1

([31,39]). Let $K\in C\left(D\right)$ and let R denote the resolvent kernel associated with K. Then, for any $f\in C\left(I\right),$ Equation (1) has a unique solution $u\in C\left(I\right)$, which is given by

$$u\left(x\right)=f\left(x\right)+{\int }_0^{x}R(x,t)f\left(t\right)dt,x\in I.$$

Let ${\left\{x_k\right\}}_{k=0}^N$ be the collocation points, satisfying $0=x_0\le x_1\le x_2\le \dots \le x_N=1$, and let

$$u_d\left(x\right)=u\left(0\right)+\frac{u\left(x_k\right)-u\left(0\right)}{x_k}x,$$

(4)

denote the linear interpolation through points $x=0$ and $x=x_k$.

At any collocation point, we have

$$u\left(x_k\right)-{\int }_0^{x_k}{e}^{i\omega (x_k-t)}u\left(t\right)dt=f\left(x_k\right).$$

(5)

By taking Equations (4) and (5), we obtain

$$u\left(x_k\right)-{\int }_0^{x_k}{e}^{i\omega (x_k-t)}{u}_d\left(t\right)dt\approx f\left(x_k\right),$$

(6)

while by approximating $u\left(x_k\right)$ with $u_k$, one obtains

$$u_k-{\int }_0^{x_k}{e}^{i\omega (x_k-t)}\left(u\left(0\right)+\frac{u_k-u\left(0\right)}{x_k}t\right)dt=f\left(x_k\right),$$

(7)

In particular, when $x_k=0$, $u\left(0\right)=f\left(0\right)$. Then, from Equation (7), we get

$$u_k=\frac{f\left(x_k\right)+f\left(0\right)M(0,\omega ,x_k)-\frac{1}{x_k}M(1,\omega ,x_k)}{1-\frac{1}{x_k}M(1,\omega ,x_k)},$$

(8)

where $M(m,\omega ,x_k)$ means the modified moment, which is given by

$$\begin{array}{cc}\hfill M(m,\omega ,x_k)& =e^{i\omega x_k}{\int }_0^{x_k}{t}^m{e}^{-i\omega t}dt\hfill \\ \hfill & ={\left(\frac{x_k}{2}\right)}^{m+1}{e}^{3i\omega x_k/2}{\int }_{-1}^1{(1+\theta )}^m{e}^{i\omega x_k\theta /2}d\theta \hfill \\ \hfill & =x_k^{m+1}{e}^{i\omega x_k}\frac{\Gamma (m+1)\Gamma \left(1\right)}{\Gamma (m+2)}{}_1\mathrm{F}_1(1+m;m+2;-i{x}_k\omega ),\hfill \end{array}$$

(9)

and $\Gamma \left(x\right)$ is the Gamma function defined as

$\Gamma \left(x\right)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}dt$.

${}_1\mathrm{F}_1(\alpha ;\beta ;x)$ is a Kummer hypergeometric function that can be efficiently computed for smaller $\left|x\right|$ [40],

${}_1\mathrm{F}_1(\alpha ;\beta ;x)=\frac{\Gamma \left(\beta \right)}{\Gamma \left(\alpha \right)}{\sum }_{n=0}^{\infty }\frac{\Gamma (\alpha +n)}{\Gamma (\beta +n)}\frac{x^n}{n!},$

This power series is convergent for all $\alpha ,\beta$ and x if $\beta \ne 0,-1,-2,\dots ,$. For larger $\left|x\right|$, it can be computed as [40],

$$\begin{array}{cc}\hfill {}_1\mathrm{F}_1(\alpha ;\beta ;x)& =\frac{\Gamma \left(\beta \right)e^{\pm i\pi \alpha }{x}^{-\alpha }}{\Gamma (\beta -\alpha )}[\sum _{j=0}^{R-1}\frac{{\alpha }_j{(1+\alpha -\beta )}_j}{j!}{(-x)}^{-j}+O\left({\left|x\right|}^{-R}\right)]\hfill \\ \hfill & +\frac{\Gamma \left(\beta \right)e^x{x}^{\alpha -\beta }}{\Gamma \left(\alpha \right)}[\sum _{j=0}^{S-1}\frac{{(\beta -\alpha )}_j{(1-\alpha )}_j}{j!}{x}^{-j}+O\left({\left|x\right|}^{-S}\right)],\hfill \end{array}$$

and if $-\frac{1}{2}\pi <argx<\frac{3}{2}\pi$, the above equation takes “+”. Conversely, if $-\frac{3}{2}\pi <argx<-\frac{1}{2}\pi$, the above equation takes “−” and the remaining term satisfies

$O\left({\left|x\right|}^{-R}\right)=\frac{{\left(\alpha \right)}_R{(1+\alpha -\beta )}_R}{R!}{(-x)}^{-R}[\frac{3}{4}+\frac{1/8+\beta /4-\alpha /2-R/4}{x}+O\left({\left|x\right|}^{-2}\right)],$

$O\left({\left|x\right|}^{-S}\right)=\frac{{(\beta -\alpha )}_S{(1-\alpha )}_S}{S!}{x}^{-S}[\frac{2}{3}-\beta +2\alpha +x-S+O\left({\left|x\right|}^{-1}\right)].$

the operation of ${\left(\alpha \right)}_j$ that appears above is ${\left(\alpha \right)}_j=\alpha (\alpha +1)(\alpha +2)\dots (\alpha +j-1),$$j\ge 1,(j\in N).$

3. Direct High-Order Interpolation Collocation Method (DO)

In this section, we will introduce the direct higher interpolation collocation method to solve Equation (1).

Let ${\left\{x_k\right\}}_{k=0}^N$ be the collocation points, satisfying $0=x_0<x_1<x_2<\dots <x_N=1$, then let

$$u_d\left(x\right)=d_1\left(x\right)u\left(0\right)+d_2\left(x\right)u\left(x_k\right),$$

(10)

denote an interpolation polynomial with the property $u_d\left(0\right)=u\left(0\right),u_d\left(x_k\right)=u\left(x_k\right),$ where

$d_1\left(x\right)=(1+\frac{2x}{x_k}){\left(\frac{x-x_k}{x_k}\right)}^2,d_2\left(x\right)=(1+2\frac{x-x_k}{-x_k}){\left(\frac{x}{x_k}\right)}^2.$

As Equation (1) holds at any of the collocation points, we have

$$u\left(x_k\right)-{\int }_0^{x_k}{e}^{i\omega (x_k-t)}u\left(t\right)dt=f\left(x_k\right).$$

(11)

It follows from Equations (10) and (11) that

$$u_k-{\int }_0^{x_k}{e}^{i\omega (x_k-t)}(d_1\left(t\right)u\left(0\right)+d_2\left(t\right)u_k)dt=f\left(x_k\right),$$

(12)

where $u_k$ denotes an approximation of $u\left(x_k\right)$. In particular, when $x=0$, there is $u\left(0\right)=f\left(0\right)$.

Solving Equation (12) yields

$$u_k=\frac{f\left(x_k\right)+f\left(0\right)[\frac{2}{x_k^3}M(3,\omega ,x_k)-\frac{3}{x_k^2}M(2,\omega ,x_k)+M(0,\omega ,x_k)]}{1-\frac{3}{x_k^2}M(2,\omega ,x_k)+\frac{2}{x_k^3}M(3,\omega ,x_k)},$$

(13)

where $M(x_k,\alpha ,\beta )$ denotes the modified moment, calculated as (9).

4. Direct Hermite Interpolation Collocation Method (DH)

In this section, we will give the direct Hermte interpolation collocation method to solve Equation (1) and give some Lemmas. For simplicity, we write Equation (1) as

$$(I-\mathcal{V})u=f$$

Lemma 2

([23,31]). Assume that the function $h=h\left(x\right)$ is adequately smooth. Then, for $j=1,2,3,\dots ,$

$$D{\mathcal{V}}^{j}h=h\left(0\right){\mathcal{V}}^{j-1}r+{\mathcal{V}}^{j}Dh,$$

where $r\left(x\right)=e^{i\omega x}$.

Lemma 3

([23,31]). Assume that the function $f=f\left(x\right)$ is adequately smooth. Then, derivatives of the solution of Equation (1) can be written as

$$Du=\sum _{j=0}^{\infty }(f\left(0\right){\mathcal{V}}^{j-1}r+{\mathcal{V}}^{j}Df).$$

where ${\mathcal{V}}^j=0$ if $j\le 0.$

Remark 1

([23,31]). Since $\left|\right|{\mathcal{V}}^j\left|\right|\le max\left\{\left|e^{i\omega g(x-t)}\right|:(x,t)\in I\times I\right\}/(j-1)!$,

$$\left|\right|Du\left|\right|\le \sum _{j=0}^{\infty }\left(f\left(0\right)\left|\right|{\mathcal{V}}^{j-1}\left|\right|\left|\right|r\left|\right|+\left|\right|{\mathcal{V}}^j\left|\right|\left|\right|Df\left|\right|\right),$$

we obtain that $\left|\right|Du\left|\right|$ is uniformly bounded in $\omega \in (0,+\infty )$.

If the interpolation condition satisfies $H\left(x_k\right)=f\left(x_k\right),H^{\prime }\left(x_k\right)=f^{\prime }\left(x_k\right)$, then it is called Hermite interpolation. Choosing two points $x=0$ and $x=x_k$ and Hermite interpolation, we have

$$u_h\left(x\right)=h_1\left(x\right)u\left(0\right)+h_2\left(x\right)u\left(x_k\right)+h_3\left(x\right)u^{\prime }\left(0\right)+h_4\left(x\right)u^{\prime }\left(x_k\right),$$

(14)

where

$h_1\left(x\right)=(1+\frac{2x}{x_k}){\left(\frac{x-x_k}{x_k}\right)}^2,h_2\left(x\right)=(1+2\frac{x-x_k}{-x_k}){\left(\frac{x}{x_k}\right)}^2,$

$h_3\left(x\right)=x{\left(\frac{x-x_k}{-x_k}\right)}^2,h_4\left(x\right)=(x-x_k){\left(\frac{x}{x_k}\right)}^2,$

denote the basic polynomial with respect to $x=0$ and $x=x_k$.

By differentiating the two sides of Equation (1) with respect to the variable x, we get

$$u^{\prime }\left(x\right)-u\left(x\right)-i\omega {\int }_0^x{e}^{i\omega (x-t)}u\left(t\right)dt=f^{\prime }\left(x\right).$$

(15)

As Equation (15) holds at every collocation point, we have

$$u^{\prime }\left(x_k\right)-u\left(x_k\right)-i\omega {\int }_0^{x_k}{e}^{i\omega (x_k-t)}u\left(t\right)dt=f^{\prime }\left(x_k\right).$$

(16)

It follows from Equations (11), (14) and (16) that

$$u_k^d-{\int }_0^{x_k}{e}^{i\omega (x_k-t)}(h_1\left(t\right)u\left(0\right)+h_2\left(t\right)u_k+h_3\left(t\right)u^{\prime }\left(0\right)+h_4\left(t\right)u_k^{\prime })dt=f\left(x_k\right)$$

(17)

$$u_k^{\prime d}-u_k^d-i\omega {\int }_0^{x_k}{e}^{i\omega (x_k-t)}(h_1\left(t\right)u\left(0\right)+h_2\left(t\right)u_k+h_3\left(t\right)u^{\prime }\left(0\right)+h_4\left(t\right)u_k^{\prime })dt=f^{\prime }\left(x_k\right)$$

(18)

where $u_k^d$ is an approximation of $u\left(x_k\right)$.

From Equations (17) and (18), we get

$$\begin{array}{cc}\hfill u_k^d& =\frac{f\left(x_k\right)+(1/x_k^{2}M(3,\omega ,x_k)-2/x_{k}M(2,\omega ,x_k)+M(1,\omega ,x_k))u^{\prime }\left(0\right)}{1-(3/x_k^{2}M(2,\omega ,x_k)-2/x_k^{3}M(3,\omega ,x_k))}\hfill \\ \hfill & +\frac{(1/x_k^{2}M(3,\omega ,x_k)-1/x_{k}M(2,\omega ,x_k))u_k^{\prime }}{1-(3/x_k^{2}M(2,\omega ,x_k)-2/x_k^{3}M(3,\omega ,x_k))}\hfill \\ \hfill & +\frac{(2/x_k^{3}M(3,\omega ,x_k)-3/x_k^{2}M(2,\omega ,x_k)+M(0,\omega ,x_k))u\left(0\right)}{1-(3/x_k^{2}M(2,\omega ,x_k)-2/x_k^{3}M(3,\omega ,x_k))},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill u_k^{\prime d}& =\frac{f^{\prime }\left(x_k\right)+i\omega (1/x_k^{2}M(3,\omega ,x_k)-2/x_{k}M(2,\omega ,x_k)+M(1,\omega ,x_k))u^{\prime }\left(0\right)}{1-i\omega (1/x_k^{2}M(3,\omega ,x_k)-1/x_{k}M(2,\omega ,x_k))}\hfill \\ \hfill & +\frac{(1+i\omega (3/x_k^{2}M(2,\omega ,x_k)-2/x_k^{3}M(3,\omega ,x_k)))u_k}{1-i\omega (1/x_k^{2}M(3,\omega ,x_k)-1/x_{k}M(2,\omega ,x_k))}\hfill \\ \hfill & +\frac{i\omega (2/x_k^{3}M(3,\omega ,x_k)-3/x_k^{2}M(2,\omega ,x_k)+M(0,\omega ,x_k))u\left(0\right)}{1-i\omega (1/x_k^{2}M(3,\omega ,x_k)-1/x_{k}M(2,\omega ,x_k))},\hfill \end{array}$$

(20)

Combining Equations (19) and (20) yields

$u_k^d=\frac{a_{2}s+b_{2}c}{a_2{b}_1+b_2{a}_1},u_k^{\prime d}=\frac{a_{1}s-b_{1}c}{a_1{b}_2+b_1{a}_2}$,

where

$$\begin{array}{cc}\hfill a_1& =1-3/x_k^{2}M(2,\omega ,x_k)+2/x_k^{3}M(3,\omega ,x_k),\hfill \\ \hfill a_2& =1/x_{k}M(2,\omega ,x_k)-1/x_k^{2}M(3,\omega ,x_k),\hfill \\ \hfill b_1& =-1-i\omega (3/x_k^{2}M(2,\omega ,x_k)-2/x_k^{3}M(3,\omega ,x_k))\hfill \\ \hfill b_2& =1-i\omega (1/x_k^{2}M(3,\omega ,x_k)-1/x_{k}M(2,\omega ,x_k))\hfill \\ \hfill c& =f\left(x_k\right)+(1/x_k^{2}M(3,\omega ,x_k)-2/x_{k}M(2,\omega ,x_k)+M(1,\omega ,x_k))u^{\prime }\left(0\right)\hfill \\ & +(2/x_k^{3}M(3,\omega ,x_k)-3/x_k^{2}M(2,\omega ,x_k)+M(0,\omega ,x_k))u\left(0\right),\hfill \\ \hfill s& =f^{\prime }\left(x_k\right)+i\omega (1/x_k^{2}M(3,\omega ,x_k)-2/x_{k}M(2,\omega ,x_k)+M(1,\omega ,x_k))u^{\prime }\left(0\right)\hfill \\ \hfill & +i\omega (2/x_k^{3}M(3,\omega ,x_k)-3/x_k^{2}M(2,\omega ,x_k)+M(0,\omega ,x_k))u\left(0\right).\hfill \end{array}$$

5. Piecewise Hermite Interpolation Collocation Method (PH)

In this section, we present the piecewise Hermite interpolation collocation method to solve Equation (1).

Based on the piecewise Hermite interpolation, we know that

$$\widehat{u}\left(x\right)={\widehat{H}}_{0k}u\left(x_{k-1}\right)+{\widehat{H}}_{1k}u\left(x_k\right)+{\widehat{H}}_{2k}{u}^{\prime }\left(x_{k-1}\right)+{\widehat{H}}_{3k}{u}^{\prime }\left(x_k\right),$$

where

$$\begin{array}{cc}\hfill {\widehat{H}}_{0k}& =(1+2\frac{x-x_{k-1}}{x_k-x_{k-1}}){\left(\frac{x-x_k}{x_k-x_{k-1}}\right)}^2=(1+2\frac{x-x_{k-1}}{h}){\left(\frac{x-x_k}{h}\right)}^2,\hfill \\ \hfill {\widehat{H}}_{1k}& =(1+2\frac{x-x_{k+1}}{x_k-x_{k+1}}){\left(\frac{x-x_k}{x_{k+1}-x_k}\right)}^2=(1+2\frac{x-x_{k+1}}{h}){\left(\frac{x-x_k}{h}\right)}^2,\hfill \\ \hfill {\widehat{H}}_{2k}& =(x-x_{k-1}){\left(\frac{x-x_k}{x_k-x_{k-1}}\right)}^2=(x-x_{k-1}){\left(\frac{x-x_k}{h}\right)}^2,\hfill \\ \hfill {\widehat{H}}_{3k}& =(x-x_k){\left(\frac{x-x_{k-1}}{x_k-x_{k-1}}\right)}^2=(x-x_k){\left(\frac{x-x_{k-1}}{h}\right)}^2,\hfill \end{array}$$

denotes the polynomial of the function through nodes $x_{k-1}$ and $x_k$. Then, we have

$$u_k-\sum _{p=1}^{k-1}{\int }_{x_{p-1}}^{x_p}{e}^{i\omega (x_k-t)}{\widehat{u}}_p\left(t\right)dt-{\int }_{x_{k-1}}^{x_k}{e}^{i\omega (x_k-t)}{\widehat{u}}_k\left(t\right)dt=f_k,$$

(21)

$$u_k^{\prime }-u_k-i\omega \sum _{p=1}^{k-1}{\int }_{x_{p-1}}^{x_p}{e}^{i\omega (x_k-t)}{\widehat{u}}_p\left(t\right)dt-i\omega {\int }_{x_{k-1}}^{x_k}{e}^{i\omega (x_k-t)}{\widehat{u}}_k\left(t\right)dt=f_k^{\prime }.$$

(22)

Thus, we can get

$$\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\left[\begin{array}{c}{u}_k\\ u_k^{\prime }\end{array}\right]=\left[\begin{array}{c}{r}_1\\ r_2\end{array}\right]$$

where

$$\begin{array}{cc}\hfill & a_{11}=1-T_{kk1},a_{12}=-T_{kk3},a_{21}=-1-T_{kk1},a_{22}=1-T_{kk3},\hfill \\ \hfill & r_1=f_j+\sum _{p=1}^{k-1}(T_{pk0}{u}_{p-1}+T_{pk1}{u}_p+T_{pk2}{u}_{p-1}^{\prime }+T_{pk3}{u}_p^{\prime })+T_{kk0}{u}_{k-1}+T_{kk2}{u}_{k-1}^{\prime },\hfill \\ \hfill & r_2=f_k^{\prime }+i\omega (\sum _{p=1}^{k-1}(T_{pk0}{u}_{p-1}+T_{pk1}{u}_p+T_{pk2}{u}_{p-1}^{\prime }+T_{pk3}{u}_p^{\prime })+T_{kk0}{u}_{k-1}+T_{kk2}{u}_{k-1}^{\prime }),\hfill \end{array}$$

and $T_{pkm}$ denotes the moment

$$T_{pkm}={\int }_{x_{p-1}}^{x_p}{\widehat{H}}_{mp}{e}^{i\omega (x_k-t)}dt,m=0,1,2,3.$$

The specific calculation formula can be written as

$$\begin{array}{cc}\hfill T_{pk0}& =(M(0,\omega ,(k-p+1)h)-M(0,\omega ,(k-p)h))(2k-2p+3){(k-p)}^2\hfill \\ \hfill & -\left(M\right(1,\omega ,(k-p+1)h)-M(1,\omega ,(k-p)h\left)\right)(k-p+1)(k-p)6/h\hfill \\ \hfill & +(M(2,\omega ,(k-p+1)h)-M(2,\omega ,(k-p)h))3(2k-2p+1)/h^2\hfill \\ \hfill & -(M(3,\omega ,(k-p+1)h)-M(3,\omega ,(k-p)h))2/h^3,\hfill \\ \hfill T_{pk1}& =(M(0,\omega ,(k-p+1)h)-M(0,\omega ,(k-p)h)){(k-p+1)}^2(-2k+2p+1)\hfill \\ \hfill & +\left(M\right(1,\omega ,(k-p+1)h)-M(1,\omega ,(k-p)h\left)\right)(k-p+1)(k-p)6/h\hfill \\ \hfill & -(M(2,\omega ,(k-p+1)h)-M(2,\omega ,(k-p)h))3(2k-2p+1)/h^2\hfill \\ \hfill & +(M(3,\omega ,(k-p+1)h)-M(3,\omega ,(k-p)h))2/h^3,\hfill \\ \hfill T_{pk2}& =(M(0,\omega ,(k-p+1)h)-M(0,\omega ,(k-p)h))(k-p+1){(k-i)}^{2}h\hfill \\ \hfill & -\left(M\right(1,\omega ,(k-p+1)h)-M(1,\omega ,(k-p)h\left)\right)(3k-3p+2)(k-p)\hfill \\ \hfill & +\left(M\right(2,\omega ,(k-p+1)h)-M(2,\omega ,(k-p)h\left)\right)(3k-3p+1)/h\hfill \\ \hfill & -(M(3,\omega ,(k-p+1)h)-M(3,\omega ,(k-p)h))/h^2,\hfill \\ \hfill T_{pk3}& =(M(0,\omega ,(k-p+1)h)-M(0,\omega ,(k-p)h)){(k-p+1)}^2(k-p)h\hfill \\ \hfill & -\left(M\right(1,\omega ,(k-p+1)h)-M(1,\omega ,(k-p)h\left)\right)(k-p+1)(3k-3p+1)\hfill \\ \hfill & +\left(M\right(2,\omega ,(k-p+1)h)-M(2,\omega ,(k-p)h\left)\right)(3k-3p+2)/h\hfill \\ \hfill & -(M(3,\omega ,(k-p+1)h)-M(3,\omega ,(k-p)h))/h^2.\hfill \end{array}$$

6. Error Analysis

In this section, the convergence analysis of our proposed collocation method will be given. Before that, we give some lemmas.

We shall use the order symbols O in the same sense as in refs. [41,42]. If $F\left(x\right)/\left(G\right(x\left)\right)$ is bounded as $x\to \infty$, we write

$$F\left(x\right)=O\left(G\right(x\left)\right),x\to \infty .$$

Lemma 4

([23]). Let $g\left(x\right)$ be a smooth real-valued function on $(a,b)$ with $g^r\left(x\right)\ge 1$ for all $x\in (a.b)$ and fixed values of r. Moreover, suppose $\varphi \in C^1(0,1)$ and ${\varphi }^{\prime }\in L[0,1]$. Then, we have

$$\left|{\int }_0^x{e}^{i\omega g\left(x\right)}\varphi \left(x\right)dx\right|=O\left({\omega }^{-1/r}\right).$$

Lemma 5

([43]). Suppose that $\alpha >-1,\beta >-1,\omega \gg 1$ and $f\left(x\right)\in C^{r+1}$ with $f\left(a\right)=f\left(b\right)=0$, then we have

${\int }_a^b{(t-a)}^{\alpha }{(b-t)}^{\beta }{e}^{i\omega t}f\left(t\right)dt=O\left({\omega }^{-2-min\left\{\alpha ,\beta \right\}}\right)$,

where $r=\left[min\left\{\alpha ,\beta \right\}\right]$, $\left[x\right]$ denotes the smallest integer that is not less than x.

Lemma 6

([44]). Suppose that $f\left(x\right)\in C^{r+1}$, for all $\alpha >-1,\beta >-1,$ and $\omega \gg 1$, then we get

${\int }_a^b{(t-a)}^{\alpha }{(b-t)}^{\beta }{e}^{i\omega t}f\left(t\right)dt=O\left({\omega }^{-1-min\left\{\alpha ,\beta \right\}}\right)$,

where $r=\left[min\left\{\alpha ,\beta \right\}\right]$, $\left[x\right]$ denotes the smallest integer that is not less than x.

Lemma 7

([41,44]). Suppose that $f\left(x\right)$ is Nth differentiable and $-1<\alpha ,\beta \le 0$, then for $a<t<b$, we have

${\int }_a^b{(t-a)}^{\alpha }{(b-t)}^{\beta }{e}^{i\omega t}f\left(t\right)dt=B_N\left(\omega \right)-A_N\left(\omega \right)+O\left({\omega }^{-N}\right)$,

where$A_N\left(\omega \right)={\sum }_{n=0}^{N-1}\frac{\Gamma (n+\alpha +1)}{n!}{e}^{i\pi (n+\alpha -1)/2}{\omega }^{-n-\alpha -1}{e}^{i\omega a}{\left[\frac{d^n}{d{t}^n}\left\{{(b-t)}^{\beta }f\left(t\right)\right\}\right]}_{t=a}$,

$B_N\left(\omega \right)={\sum }_{n=0}^{N-1}\frac{\Gamma (n+\beta +1)}{n!}{e}^{i\pi (n-\beta -1)/2}{\omega }^{-n-\beta -1}{e}^{i\omega b}{\left[\frac{d^n}{d{t}^n}\left\{{(t-a)}^{\alpha }f\left(t\right)\right\}\right]}_{t=b}$.

Theorem 1.

Using direct linear interpolation, the error for solving Equation (1) satisfies

$\left|u_k-u\left(x_k\right)\right|=O\left({\omega }^{-2}\right)$.

Proof. 

For Equation (1) at any collocation point, there is

$$u\left(x_k\right)-{\int }_0^{x_k}{e}^{i\omega (x_k-t)}u\left(t\right)dt=f\left(x_k\right).$$

(23)

Let the variables $\theta =x_k-t$, then we get

$$u\left(x_k\right)-{\int }_0^{x_k}{e}^{i\omega \theta }u(x_k-\theta )dt=f\left(x_k\right).$$

(24)

Approximating $u\left(x_k\right)$ by $u_k$, we have

$$u_k-{\int }_0^{x_k}{e}^{i\omega \theta }{u}_d(x_k-\theta )dt=f\left(x_k\right).$$

(25)

It follows from Equations (24) and (25) that

$$u\left(x_k\right)-u_k={\int }_0^{x_k}{e}^{i\omega \theta }(u(x_k-\theta )-u_d(x_k-\theta ))d\theta .$$

(26)

It is assumed that the error at the interpolation point $x=0$ is $E\left(0\right)=0.$ Then, we can obtain

$$E\left(x\right)=u\left(x\right)-u_h\left(x\right)=\frac{x}{x_k}E\left(x_k\right)+R\left(x\right),$$

(27)

where $R\left(x\right)$ is the linear interpolation residual term. Substituting Equation (27) into Equation (26) yields

$$E\left(x_k\right)={\int }_0^{x_k}{e}^{i\omega \theta }(\frac{x_k-\theta }{x_k}E\left(x_k\right)+R(x_k-\theta ))d\theta .$$

(28)

Therefore, we can express the interpolation error at the point $x=x_k$ as

$$E\left(x_k\right)=\frac{{\int }_0^{x_k}{e}^{i\omega \theta }R(x_k-\theta )d\theta }{1-{\int }_0^{x_k}{e}^{i\omega \theta }\frac{x_k-\theta }{x_k}d\theta }=\frac{I_1}{I_2}.$$

According to Lemma 4, we get that $I_2=1-{\int }_0^{x_k}{e}^{i\omega \theta }\frac{x_k-\theta }{x_k}d\theta =O\left(1\right)$ when $\omega \to \infty$. By the nature of the direct linear interpolation of the residual term, we have

$$R\left(x\right)=u\left(x\right)-u_h\left(x\right)=c_{d}x(x-x_k),$$

(29)

Applying Lemma 6, we have

$$I_1=c_d{\int }_0^{x_k}(x_k-\theta )\theta e^{i\omega \theta }d\theta =O\left({\omega }^{-2}\right).$$

Therefore, we can get

$$E\left(x_k\right)=\frac{I_1}{I_2}=O\left({\omega }^{-2}\right).$$

□

Theorem 2.

The error estimate for solving Equation (1) using direct higher-order interpolation is

$\left|u_k-u\left(x_k\right)\right|=O\left({\omega }^{-2}\right)$.

Proof. 

Based on the fact that Equation (1) holds at any collocation point, we have

$$u\left(x_k\right)-{\int }_0^{x_k}{e}^{i\omega (x_k-t)}u\left(t\right)dt=f\left(x_k\right),$$

(30)

Approximating $u\left(x_k\right)$ by $u_k$, we have

$$u_k-{\int }_0^{x_k}{e}^{i\omega (x_k-t)}{u}_h\left(t\right)dt=f\left(x_k\right),$$

(31)

Equations (30) and (31) yield

$$u\left(x_k\right)-u_k={\int }_0^{x_k}{e}^{i\omega (x_k-t)}(u\left(t\right)-u_h\left(t\right))dt.$$

(32)

When the error at the interpolation point $x=0$ is $E\left(0\right)=0$, we get

$$E\left(x\right)=u\left(x\right)-u_h\left(x\right)=d_2\left(x\right)E\left(x_k\right)+R\left(x\right),$$

(33)

where $R\left(x\right)$ is the interpolation residual term. Then, Equation (32) can be written as

$$E\left(x_k\right)={\int }_0^{x_k}{e}^{i\omega (x_k-t)}E\left(t\right)dt.$$

(34)

Substituting Equation (33) into Equation (34), we get

$$E\left(x_k\right)={\int }_0^{x_k}{e}^{i\omega (x_k-t)}(d_2\left(t\right)E\left(x_k\right)+R\left(t\right))dt.$$

(35)

The interpolation error at $x=x_k$ is thus found to be

$$E\left(x_k\right)=\frac{{\int }_0^{x_k}{e}^{i\omega (x_k-t)}R\left(t\right)dt}{1-{\int }_0^{x_k}{e}^{i\omega (x_k-t)}{d}_2\left(t\right)dt}.$$

By transforming the variables $\theta =x_k-t$, and noting that $d_2\left(x\right)=(1+2\frac{x-x_k}{-x_k}){\left(\frac{x}{x_k}\right)}^2$, then we obtain

$$E\left(x_k\right)=\frac{{\int }_0^{x_k}{e}^{i\omega \theta }R(x_k-\theta )d\theta }{1-{\int }_0^{x_k}{e}^{i\omega \theta }{d}_2(x_k-\theta )d\theta }=\frac{I_1}{I_2}.$$

Applying Lemma 4, we get that $I_2=O\left(1\right)$. According to the property of the direct higher interpolation residual term, we have

$$R\left(x\right)=u\left(x\right)-u_h\left(x\right)=c\left(x\right)x(x-x_k),$$

(36)

Then, applying Lemma 6, we have

$$I_1={\int }_0^{x_k}{e}^{i\omega \theta }(x_k-\theta )\theta c(x_k-\theta )d\theta =O\left({\omega }^{-2}\right).$$

Therefore, we can get

$$E\left(x_k\right)=\frac{I_1}{I_2}=O\left({\omega }^{-2}\right).$$

□

Theorem 3.

The error estimate obtained by solving Equation (1) using the direct hermite interpolation method is

$$\left|u_k-u\left(x_k\right)\right|=O\left({\omega }^{-3}\right).$$

Proof. 

According the direct Hermite interpolation method in Section 4, we know that

$$u\left(x\right)-{\int }_0^{x_k}{e}^{i\omega \theta }u(x-\theta )d\theta =f\left(x\right),$$

(37)

$$\begin{array}{c}\hfill u^{\prime }\left(x\right)-u\left(x\right)-i\omega {\int }_0^{x_k}{e}^{i\omega \theta }u(x-\theta )d\theta =f^{\prime }\left(x\right).\end{array}$$

(38)

Approximating $u\left(x_k\right)$ by $u_k$, we have

$$u_k-{\int }_0^{x_k}{e}^{i\omega \theta }{u}_h(x_k-\theta )d\theta =f\left(x_k\right),$$

(39)

$$\begin{array}{c}\hfill u_k^{\prime }-u_k-i\omega {\int }_0^{x_k}{e}^{i\omega \theta }{u}_h\left((x_k-\theta )\right)d\theta =f^{\prime }\left(x_k\right).\end{array}$$

(40)

It follows from Equations (37)–(40) that

$$u\left(x_k\right)-u_k={\int }_0^{x_k}{e}^{i\omega \theta }(u(x_k-\theta )-u_h(x_k-\theta ))d\theta ,$$

(41)

$$\begin{array}{c}\hfill u^{\prime }\left(x_k\right)-u_j^{\prime }-(u\left(x_k\right)-u_k)=i\omega {\int }_0^{x_k}{e}^{i\omega \theta }(u(x_k-\theta )-u_h\left((x_k-\theta )\right))d\theta .\end{array}$$

(42)

Based on Hermite interpolation, the error function is expressed as follows

$$E\left(x\right)=u\left(x\right)-u_h\left(x\right)=h_2\left(x\right)E\left(x_k\right)+h_4\left(x\right)E^{\prime }\left(x_k\right)+R\left(x\right),$$

(43)

where $R\left(x\right)$ is the residual term of Hermite interpolation. Furthermore, the errors at the collocation point $x=0$ satisfy $E\left(0\right)=E^{\prime }\left(0\right)=0$. Substituting Equation (43) into Equations (41) and (42), respectively, we have

$$\begin{array}{cc}\hfill (1-& {\int }_0^{x_k}{e}^{i\omega \theta }{h}_2(x_k-\theta )d\theta )E\left(x_k\right)-{\int }_0^{x_k}{e}^{i\omega \theta }{h}_4(x_k-\theta )d\theta E^{\prime }\left(x_k\right)\hfill \\ \hfill =& {\int }_0^{x_k}{e}^{i\omega \theta }R(x_k-\theta )d\theta ,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill (1& -i\omega {\int }_0^{x_k}{e}^{i\omega \theta }{h}_4(x_k-\theta )d\theta )E^{\prime }\left(x_k\right)-(1+i\omega {\int }_0^{x_k}{e}^{i\omega \theta }{h}_2(x_k-\theta )d\theta )E\left(x_k\right)\hfill \\ \hfill & =i\omega {\int }_0^{x_k}{e}^{i\omega \theta }R(x_k-\theta )d\theta .\hfill \end{array}$$

(45)

Combining Equations (44) and (45) yields

$E\left(x_k\right)=\frac{Q_2}{Q_1},E^{{}^{\prime }}\left(x_k\right)=\frac{Q_3}{Q_1}$,

where

$$\begin{array}{cc}\hfill Q_1& =(1-{\int }_0^{x_k}{e}^{i\omega \theta }{h}_2(x_k-\theta )d\theta )\ast (1-i\omega {\int }_0^{x_k}{e}^{i\omega \theta }{h}_4(x_k-\theta )d\theta )\hfill \\ & -{\int }_0^{x_k}{e}^{i\omega \theta }{h}_4(x_k-\theta )d\theta \ast (1+i\omega {\int }_0^{x_k}{e}^{i\omega \theta }{h}_2(x_k-\theta )d\theta ),\hfill \\ \hfill Q_2& ={\int }_0^{x_k}{e}^{i\omega \theta }{h}_4(x_k-\theta )d\theta \ast i\omega {\int }_0^{x_k}{e}^{i\omega \theta }R(x_k-\theta )d\theta \hfill \\ & +{\int }_0^{x_k}{e}^{i\omega \theta }R(x_k-\theta )d\theta \ast (1-i\omega {\int }_0^{x_k}{e}^{i\omega \theta }{h}_4(x_k-\theta )d\theta ),\hfill \\ \hfill Q_3& =(1-{\int }_0^{x_k}{e}^{i\omega \theta }{h}_2(x_k-\theta )d\theta )\ast i\omega {\int }_0^{x_k}{e}^{i\omega \theta }R(x_k-\theta )d\theta \hfill \\ \hfill & +{\int }_0^{x_k}{e}^{i\omega \theta }R(x_k-\theta )d\theta \ast (1+i\omega {\int }_0^{x_k}{e}^{i\omega \theta }{h}_2(x_k-\theta )d\theta ).\hfill \end{array}$$

Applying Lemma 4, we can get that $Q_1=O\left(1\right)$ when $\omega \to \infty$. According to the property of direct Hermite interpolation of the residual term, we have

$$R\left(x\right)-u\left(x\right)-u_h\left(x\right)=c_h{x}^2{(x-x_k)}^2,$$

(46)

According to Lemmas 4 and 6, we get

$$\begin{array}{cc}\hfill Q_2& ={\int }_0^{x_k}{e}^{i\omega \theta }{h}_4(x_k-\theta )d\theta \ast c_{h}i\omega {\int }_0^{x_k}{e}^{i\omega \theta }{\theta }^2{(\theta -x_k)}^{2}d\theta \hfill \\ & +c_h{\int }_0^{x_k}{e}^{i\omega \theta }{\theta }^2{(\theta -x_k)}^{2}d\theta \ast (1-i\omega {\int }_0^{x_k}{e}^{i\omega \theta }{h}_4(x_k-\theta )d\theta )\hfill \\ & =O\left({\omega }^{-3}\right).\hfill \end{array}$$

Therefore, we can get

$E\left(x_k\right)=O\left({\omega }^{-3}\right)$.

□

Theorem 4.

The error estimate obtained by solving Equation (1) using the piecewise hermite interpolation method is

$$\left|u_k-u\left(x_k\right)\right|=O\left({\omega }^{-3}h\right).$$

Proof. 

In the case of the piecewise Hermite interpolation collocation method, we have

$$u\left(x_k\right)-\sum _{m=1}^{k-1}{\int }_{x_{m-1}}^{x_m}{e}^{i\omega (x_m-t)}u\left(t\right)dt-{\int }_{x_{k-1}}^{x_k}{e}^{i\omega (x_k-t)}u\left(t\right)dt=f\left(x_k\right),$$

(47)

Approximate $u\left(x_k\right)$ by $u_k$, and we have

$$u_k-\sum _{m=1}^{k-1}{\int }_{x_{m-1}}^{x_m}{e}^{i\omega (x_m-t)}{\widehat{u}}_k\left(t\right)dt-{\int }_{x_{k-1}}^{x_k}{e}^{i\omega (x_k-t)}{\widehat{u}}_j\left(t\right)dt=f\left(x_k\right),$$

(48)

It follows from Equations (47) and (48) that

$${\epsilon }_k=\frac{{\sum }_{m=1}^{k-1}{\epsilon }_m{\int }_{x_{m-1}}^{x_m}{e}^{i\omega (x_m-t)}dt+{\sum }_{m=1}^k{\int }_{x_{m-1}}^{x_m}{e}^{i\omega (x_m-t)}{r}_m\left(t\right)dt}{1-{\int }_0^{x_1}{e}^{i\omega t}dt},$$

(49)

where ${\epsilon }_k=u\left(x_k\right)-u_k,k=1,2,\dots ,N$ and $r_m\left(t\right)=(u\left(t\right)-{\widehat{u}}_m){|}_t\in [x_{m-1},x_m]$. A similar argument to that used in Theorem (3) shows that

$$\frac{{\sum }_{m=1}^j{\int }_{x_{m-1}}^{x_m}{e}^{i\omega (x_m-t)}{r}_m\left(t\right)dt}{1-{\int }_0^{x_1}{e}^{i\omega t}dt}=O\left({\omega }^{-3}h\right),$$

(50)

By using the generalized discrete Gronwall inequality, the desired results can be derived [31]. □

7. Numerical Experiments

In this section, we present numerical experiments that are consistent with the convergence analysis in Section 6. $DL$ denotes direct linear interpolation, $DO$ denotes direct higher order interpolation, $DH$ denotes direct Hermite interpolation and $PH$ denotes piecewise Hermite interpolation.

From Figure 1, we can see that the oscillation of the $e^{i\omega x}$ function depends only on $\omega$; the larger $\omega$, the higher the oscillation.

Example 1.

We consider the following equation

$u\left(x\right)-{\int }_0^x{e}^{i\omega (x-t)}u\left(t\right)dt=f\left(x\right),$

When $f\left(x\right)=e^x-{\int }_0^x{e}^{i\omega (x-t)}{e}^{t}dt$, then the exact solution of the equation is $u\left(x\right)=e^x$.

Table 1. The absolute error of the piecewise Hermite interpolation method for solving Example 1 varies with the step size h when $\omega =100$.

$\mathbf{Steps}\left(\mathit{h}\right)\setminus \mathit{x}$	0.2	0.4	0.6	0.8
$0.1$	8.5811 $\times {10}^{-6}$	1.0505e $\times {10}^{-5}$	1.2809 $\times {10}^{-5}$	1.5639 $\times {10}^{-5}$
$0.05$	5.4129 $\times {10}^{-7}$	6.6218 $\times {10}^{-7}$	8.0717 $\times {10}^{-7}$	9.8618 $\times {10}^{-7}$
$0.01$	8.5811 $\times {10}^{-10}$	1.0488 $\times {10}^{-9}$	1.2766 $\times {10}^{-9}$	1.5604 $\times {10}^{-9}$
$0.005$	5.3575 $\times {10}^{-11}$	6.7978 $\times {10}^{-11}$	7.5534 $\times {10}^{-11}$	1.0026 $\times {10}^{-10}$

,

Table 2. Comparison of the step size $h=1/30$ for the product–integration technique in ref. [13] with the absolute error obtained by solving Example 1 by each of the four methods we propose, where the step size $h=0.1$ for the piecewise Hermite.

$\mathit{\omega }\setminus \mathit{x}$		0.2	0.4	0.6	0.8	Times (s)
	method in ref. [13]	9.0505 $\times {10}^{-3}$	2.4929 $\times {10}^{-2}$	5.1834 $\times {10}^{-2}$	9.5648 $\times {10}^{-2}$
	$DL$	7.1590 $\times {10}^{-4}$	4.4221 $\times {10}^{-3}$	8.4436 $\times {10}^{-3}$	5.8542 $\times {10}^{-3}$	0.07
10	$DO$	1.6541 $\times {10}^{-3}$	1.1349 $\times {10}^{-2}$	2.6547 $\times {10}^{-2}$	3.5627 $\times {10}^{-2}$	0.17
	$DH$	5.0565 $\times {10}^{-7}$	1.5197 $\times {10}^{-5}$	8.0397 $\times {10}^{-5}$	1.8113 $\times {10}^{-4}$	0.18
	$PH$	8.5811 $\times {10}^{-6}$	1.0505 $\times {10}^{-5}$	1.2809 $\times {10}^{-5}$	1.5639 $\times {10}^{-5}$	12.01
	method in ref. [13]	5.3956 $\times {10}^{-3}$	1.4430 $\times {10}^{-2}$	2.9208 $\times {10}^{-2}$	5.3042 $\times {10}^{-2}$
	$DL$	1.7381 $\times {10}^{-5}$	1.8079 $\times {10}^{-5}$	1.7382 $\times {10}^{-5}$	8.4881 $\times {10}^{-5}$	0.08
100	$DO$	1.7398 $\times {10}^{-4}$	2.4167 $\times {10}^{-4}$	2.7397 $\times {10}^{-4}$	2.3903 $\times {10}^{-4}$	0.18
	$DH$	5.7609 $\times {10}^{-9}$	3.1650 $\times {10}^{-8}$	7.9237 $\times {10}^{-8}$	1.1255 $\times {10}^{-7}$	0.19
	$PH$	8.3020 $\times {10}^{-7}$	1.0136 $\times {10}^{-6}$	1.2384 $\times {10}^{-6}$	1.5133 $\times {10}^{-6}$	12.14
	method in ref. [13]	4.4797 $\times {10}^{-4}$	1.2024 $\times {10}^{-3}$	2.4321 $\times {10}^{-3}$	4.4078 $\times {10}^{-3}$
	$DL$	1.9208 $\times {10}^{-7}$	2.4344 $\times {10}^{-7}$	8.3147 $\times {10}^{-8}$	6.5576 $\times {10}^{-7}$	0.08
1000	$DO$	1.0842 $\times {10}^{-6}$	2.1712 $\times {10}^{-6}$	2.8220 $\times {10}^{-6}$	2.8309 $\times {10}^{-6}$	0.19
	$DH$	3.5449 $\times {10}^{-12}$	2.8325 $\times {10}^{-11}$	8.1719 $\times {10}^{-11}$	1.3838 $\times {10}^{-10}$	0.19
	$PH$	1.2003 $\times {10}^{-9}$	1.4653 $\times {10}^{-9}$	1.7894 $\times {10}^{-9}$	2.1859 $\times {10}^{-9}$	12.24
	method in ref. [13]	2.5400 $\times {10}^{-5}$	4.6830 $\times {10}^{-5}$	5.7729 $\times {10}^{-5}$	1.0625 $\times {10}^{-4}$
	$DL$	1.2448 $\times {10}^{-9}$	1.8349 $\times {10}^{-9}$	8.0239 $\times {10}^{-9}$	9.0117 $\times {10}^{-9}$	0.09
10,000	$DO$	1.8448 $\times {10}^{-8}$	2.3232 $\times {10}^{-8}$	1.0113 $\times {10}^{-8}$	2.3808 $\times {10}^{-8}$	0.20
	$DH$	5.7004 $\times {10}^{-15}$	2.9783 $\times {10}^{-14}$	1.8947 $\times {10}^{-14}$	1.0765 $\times {10}^{-13}$	0.20
	$PH$	1.2245 $\times {10}^{-12}$	1.4954 $\times {10}^{-12}$	1.8288 $\times {10}^{-12}$	2.2344 $\times {10}^{-12}$	12.41
	method in ref. [13]	7.5131 $\times {10}^{-6}$	2.0218 $\times {10}^{-5}$	4.0961 $\times {10}^{-5}$	7.4200 $\times {10}^{-5}$
	$DL$	5.5605 $\times {10}^{-11}$	8.6924 $\times {10}^{-11}$	5.9600 $\times {10}^{-11}$	7.9871 $\times {10}^{-11}$	0.13
60,000	$DO$	2.6945 $\times {10}^{-10}$	5.4208 $\times {10}^{-10}$	7.6252 $\times {10}^{-10}$	8.8118 $\times {10}^{-10}$	0.24
	$DH$	2.0344 $\times {10}^{-15}$	5.2951 $\times {10}^{-16}$	4.4443 $\times {10}^{-15}$	1.2404 $\times {10}^{-14}$	0.28
	$PH$	5.0765 $\times {10}^{-15}$	7.2941 $\times {10}^{-15}$	9.8994 $\times {10}^{-15}$	9.6185 $\times {10}^{-15}$	12.57

,

Table 3. Absolute error of the four methods to solve Example 2, where the step size $h=0.1$ for piecewise Hermite interpolation.

$\mathit{\omega }\setminus \mathit{x}$		0.2	0.4	0.6	0.8
	$DL$	6.4692 $\times {10}^{-5}$	7.2174 $\times {10}^{-4}$	1.8880 $\times {10}^{-3}$	1.8278 $\times {10}^{-3}$
10	$DO$	1.3290 $\times {10}^{-3}$	8.0655 $\times {10}^{-3}$	1.6386 $\times {10}^{-2}$	1.8575 $\times {10}^{-2}$
	$DH$	4.5631 $\times {10}^{-8}$	2.4625 $\times {10}^{-8}$	1.7470 $\times {10}^{-5}$	4.6872 $\times {10}^{-5}$
	$PH$	6.8836 $\times {10}^{-6}$	6.4721 $\times {10}^{-6}$	5.8028 $\times {10}^{-6}$	4.8986 $\times {10}^{-6}$
	$DL$	1.6481 $\times {10}^{-6}$	3.8164 $\times {10}^{-6}$	6.3979 $\times {10}^{-6}$	2.1824 $\times {10}^{-5}$
100	$DO$	1.5431 $\times {10}^{-4}$	1.8583 $\times {10}^{-4}$	1.7695 $\times {10}^{-4}$	1.1951 $\times {10}^{-4}$
	$DH$	5.2188 $\times {10}^{-10}$	5.1081 $\times {10}^{-9}$	1.7083 $\times {10}^{-8}$	2.9020 $\times {10}^{-8}$
	$PH$	6.6663 $\times {10}^{-7}$	6.2640 $\times {10}^{-7}$	5.6120 $\times {10}^{-7}$	4.7362 $\times {10}^{-7}$
	$DL$	1.7580 $\times {10}^{-8}$	4.4871 $\times {10}^{-8}$	5.6898 $\times {10}^{-8}$	1.7894 $\times {10}^{-7}$
1000	$DO$	9.5127 $\times {10}^{-7}$	1.6639 $\times {10}^{-6}$	1.8252 $\times {10}^{-6}$	1.4591 $\times {10}^{-6}$
	$DH$	3.4056 $\times {10}^{-13}$	4.6007 $\times {10}^{-12}$	1.7571 $\times {10}^{-11}$	3.5168 $\times {10}^{-11}$
	$PH$	9.6336 $\times {10}^{-10}$	9.0524 $\times {10}^{-10}$	8.1094 $\times {10}^{-10}$	6.8432 $\times {10}^{-10}$
	$DL$	1.2468 $\times {10}^{-10}$	3.7801 $\times {10}^{-10}$	1.7088 $\times {10}^{-9}$	2.1301 $\times {10}^{-9}$
10,000	$DO$	1.6407 $\times {10}^{-8}$	1.7858 $\times {10}^{-8}$	4.3324 $\times {10}^{-9}$	1.1816 $\times {10}^{-8}$
	$DH$	5.1895 $\times {10}^{-16}$	4.8727 $\times {10}^{-15}$	5.1573 $\times {10}^{-15}$	2.7606 $\times {10}^{-14}$
	$PH$	9.8325 $\times {10}^{-13}$	9.2382 $\times {10}^{-13}$	8.2756 $\times {10}^{-13}$	6.9947 $\times {10}^{-13}$
	$DL$	5.0667 $\times {10}^{-12}$	1.4987 $\times {10}^{-11}$	1.9306 $\times {10}^{-11}$	3.1078 $\times {10}^{-11}$
60,000	$DO$	2.3506 $\times {10}^{-10}$	4.1277 $\times {10}^{-10}$	4.9203 $\times {10}^{-10}$	4.6249 $\times {10}^{-10}$
	$DH$	1.1938 $\times {10}^{-16}$	1.7093 $\times {10}^{-18}$	1.4874 $\times {10}^{-15}$	2.9122 $\times {10}^{-15}$
	$PH$	4.4944 $\times {10}^{-15}$	4.2769 $\times {10}^{-15}$	5.0636 $\times {10}^{-15}$	2.6914 $\times {10}^{-15}$

,

Table 4. Absolute error of the piecewise Hermite interpolation method for solving Example 2 varies with the step size h when $\omega =100$.

$\mathbf{Steps}\left(\mathit{h}\right)\setminus \mathit{x}$	0.2	0.4	0.6	0.8
$0.1$	6.8836 $\times {10}^{-6}$	6.4721 $\times {10}^{-6}$	5.8028 $\times {10}^{-6}$	4.8986 $\times {10}^{-6}$
$0.05$	4.3376 $\times {10}^{-7}$	4.0792 $\times {10}^{-7}$	3.6573 $\times {10}^{-7}$	3.0879 $\times {10}^{-7}$
$0.01$	6.8726 $\times {10}^{-10}$	6.4635 $\times {10}^{-10}$	5.7926 $\times {10}^{-10}$	4.8900 $\times {10}^{-10}$
$0.005$	4.2871 $\times {10}^{-11}$	4.0620 $\times {10}^{-11}$	3.5909 $\times {10}^{-11}$	2.9635 $\times {10}^{-11}$

,

Table 5. Absolute error of the four methods to solve Example 3, where the step size $h=0.1$ for piecewise Hermite interpolation.

$\mathit{\omega }\setminus \mathit{x}$		0.2	0.4	0.6	0.8
	$DL$	1.5048 $\times {10}^{-3}$	9.7657 $\times {10}^{-3}$	1.9603 $\times {10}^{-2}$	1.4660 $\times {10}^{-2}$
10	$DO$	2.2412 $\times {10}^{-3}$	1.6209 $\times {10}^{-2}$	3.9982 $\times {10}^{-2}$	5.6128 $\times {10}^{-2}$
	$DH$	2.0739 $\times {10}^{-6}$	6.3914 $\times {10}^{-5}$	3.4675 $\times {10}^{-4}$	8.0126 $\times {10}^{-4}$
	$PH$	1.0302 $\times {10}^{-5}$	1.4780 $\times {10}^{-5}$	2.0528 $\times {10}^{-5}$	2.8177 $\times {10}^{-5}$
	$DL$	3.6570 $\times {10}^{-5}$	4.0623 $\times {10}^{-5}$	4.4738 $\times {10}^{-5}$	2.1014 $\times {10}^{-4}$
100	$DO$	1.9558 $\times {10}^{-4}$	3.0131 $\times {10}^{-4}$	3.8122 $\times {10}^{-4}$	4.0400 $\times {10}^{-5}$
	$DH$	2.3633 $\times {10}^{-8}$	1.3319 $\times {10}^{-7}$	3.4215 $\times {10}^{-7}$	4.9957 $\times {10}^{-7}$
	$PH$	9.9416 $\times {10}^{-7}$	1.4154 $\times {10}^{-6}$	1.9780 $\times {10}^{-6}$	2.7214 $\times {10}^{-6}$
	$DL$	4.0409 $\times {10}^{-7}$	5.4286 $\times {10}^{-7}$	2.7346 $\times {10}^{-7}$	1.6420 $\times {10}^{-6}$
1000	$DO$	1.2514 $\times {10}^{-6}$	2.7275 $\times {10}^{-6}$	3.9153 $\times {10}^{-6}$	4.5590 $\times {10}^{-6}$
	$DH$	1.4547 $\times {10}^{-11}$	1.1920 $\times {10}^{-10}$	3.5286 $\times {10}^{-10}$	6.1365 $\times {10}^{-10}$
	$PH$	1.4394 $\times {10}^{-9}$	2.0484 $\times {10}^{-9}$	2.8583 $\times {10}^{-9}$	3.9296 $\times {10}^{-9}$
	$DL$	2.6219 $\times {10}^{-9}$	4.1222 $\times {10}^{-9}$	1.8699 $\times {10}^{-8}$	2.2262 $\times {10}^{-8}$
10,000	$DO$	2.0596 $\times {10}^{-8}$	2.8986 $\times {10}^{-8}$	2.0550 $\times {10}^{-8}$	4.0667 $\times {10}^{-8}$
	$DH$	2.5041 $\times {10}^{-14}$	1.2785 $\times {10}^{-13}$	8.1372 $\times {10}^{-14}$	4.9120 $\times {10}^{-13}$
	$PH$	1.4654 $\times {10}^{-12}$	2.0875 $\times {10}^{-12}$	2.9210 $\times {10}^{-12}$	4.0130 $\times {10}^{-12}$
	$DL$	1.1697 $\times {10}^{-10}$	1.9308 $\times {10}^{-10}$	1.4828 $\times {10}^{-10}$	2.2263 $\times {10}^{-10}$
60,000	$DO$	3.1514 $\times {10}^{-10}$	6.9042 $\times {10}^{-10}$	1.0630 $\times {10}^{-9}$	1.3734 $\times {10}^{-9}$
	$DH$	1.5612 $\times {10}^{-16}$	5.6320 $\times {10}^{-16}$	2.8927 $\times {10}^{-15}$	7.0474 $\times {10}^{-15}$
	$PH$	6.7350 $\times {10}^{-15}$	9.6907 $\times {10}^{-15}$	1.5380 $\times {10}^{-14}$	1.8024 $\times {10}^{-14}$

and

Table 6. Absolute error of the piecewise Hermite interpolation method for solving Example 3 varies with the step size h when $\omega =100$.

$\mathbf{Steps}\left(\mathit{h}\right)\setminus \mathit{x}$	0.2	0.4	0.6	0.8
$0.1$	9.9416 $\times {10}^{-7}$	1.4154 $\times {10}^{-6}$	1.9780 $\times {10}^{-6}$	2.7214 $\times {10}^{-6}$
$0.05$	2.2131 $\times {10}^{-6}$	3.1525 $\times {10}^{-6}$	4.3998 $\times {10}^{-6}$	6.0463 $\times {10}^{-6}$
$0.01$	9.8247 $\times {10}^{-9}$	1.4001 $\times {10}^{-8}$	1.9544 $\times {10}^{-8}$	2.6853 $\times {10}^{-8}$
$0.005$	6.3141 $\times {10}^{-10}$	8.9961 $\times {10}^{-10}$	1.2560 $\times {10}^{-9}$	1.7224 $\times {10}^{-9}$

show the errors of our proposed methods for solving Equation (1)at collocation points 0.2, 0.4,0.6 and 0.8. And the error in solving the equation with step size h for the piecewise Hermite interpolation collocation method is given in Table 1, Table 4 and Table 6. In Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8, we give the error plots for solving Equation (1) using the methods of this paper. Where in Figure 2, Figure 3, Figure 4 and Figure 5, Figure 7 and Figure 8, plots of the error with $\omega$ at the points $x=0.2,0.5$ and $0.8$ are given for our four proposed methods. In Figure 6, we present the curves of the variation in the error with x for solving Example 2 using the direct linear interpolation collocation method and the direct higher interpolation collocation method when omega = 10,000. It can be seen that the direct higher interpolation collocation method is more stable than the direct linear collocation method in solving the equation.

Example 2.

We consider the following equation

$u\left(x\right)-{\int }_0^x{e}^{i\omega (x-t)}u\left(t\right)dt=f\left(x\right),$

When $f\left(x\right)=sin\left(x\right)-{\int }_0^x{e}^{i\omega (x-t)}sin\left(t\right)dt$, then the exact solution of the equation is $u\left(x\right)=sin\left(x\right)$.

Example 3.

We consider the following equation

$u\left(x\right)-{\int }_0^x{e}^{i\omega (x-t)}u\left(t\right)dt=f\left(x\right),$

When $f\left(x\right)=x{e}^x-{\int }_0^x{e}^{i\omega (x-t)}t{e}^{t}dt$, then the exact solution of the equation is $u\left(x\right)=x{e}^x$.

From the graphs and tables of the above numerical experiments, it is easy to see that each of our four proposed collocation methods has its own merits and all of them can efficiently solve the second kind of Volterra integral equations with highly oscillatory Fourier kernels.

8. Conclusions

Volterra integral equations with highly oscillatory kernels have a wide range of applications in mathematical physics and are a hot topic for mathematicians. At present, the theoretical study of this type of integral equation is quite mature. Based on the calculation of the modified moments, solving highly oscillatory problems is usually considered to be effective. Accordingly, four collocation methods (direct linear interpolation, direct higher order interpolation, direct Hermite interpolation and piecewise Hermite interpolation) are proposed in this paper to solve the second kind of Volterra integral equation with a highly oscillatory Fourier kernel. These four methods can solve this class of integral equation efficiently and they all have the characteristic of decreasing error with increasing frequency. In comparison with the classical technique of product integration, the four methods we propose have higher convergence rates. The presented convergence analysis and numerical experiments demonstrate the effectiveness of our proposed methods.

However, in this paper, we have only discussed solving Volterra integral equations with highly oscillating Fourier kernels, and in future work, the four methods we have proposed can be used to try to solve integral equations with different highly oscillating kernels.