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:
For the study of the numerical solution of a scalar retarded potential integral equation posed on an infinite flat surface,
where
u is the unknown function, if
and
u and
a satisfy
,
for any
. Davis and Ducan [
2] obtained the following equation by taking the Fourier transform
where
denotes the Bessel function of the first kind and of order zero. When
, the kernel function
is highly oscillatory.
In [
11], Beezley and Krueger obtained an equivalent formula to the Helmholtz equation, which is expressed as
where
denotes Green’s function and
denotes the convolution
. There are situations in which this equation can be solved explicitly, such as when
or
Here,
is the first kind of the first-order modified Bessel function, and
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,
where
is a known smooth function,
is the frequency and
is an unknown function. When
, Equation (
1) is a highly oscillatory integral equation. Furthermore, the convergence rate of solving Equation (
1) improves from
for the method in the literature [
23] to
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 into N equal parts to obtain the sub-integration interval of width h, where On each interval , approximate by .
Then, Equation (2) can be written as
Solving Equation (3) yields
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
denote the linear Volterra integral operator defined by
where the kernel
is continuous on
Lemma 1 ([
31,
39]).
Let and let R denote the resolvent kernel associated with K. Then, for any Equation (1) has a unique solution , which is given by Let
be the collocation points, satisfying
, and let
denote the linear interpolation through points
and
.
At any collocation point, we have
By taking Equations (
4) and (
5), we obtain
while by approximating
with
, one obtains
In particular, when
,
. Then, from Equation (7), we get
where
means the modified moment, which is given by
and
is the Gamma function defined as
.
is a Kummer hypergeometric function that can be efficiently computed for smaller
[
40],
This power series is convergent for all
and
x if
. For larger
, it can be computed as [
40],
and if
, the above equation takes “+”. Conversely, if
, the above equation takes “−” and the remaining term satisfies
the operation of that appears above is
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
Lemma 2 ([
23,
31]).
Assume that the function is adequately smooth. Then, for where . Lemma 3 ([
23,
31]).
Assume that the function is adequately smooth. Then, derivatives of the solution of Equation (1) can be written aswhere if Remark 1 ([
23,
31]).
Since ,we obtain that is uniformly bounded in . If the interpolation condition satisfies
, then it is called Hermite interpolation. Choosing two points
and
and Hermite interpolation, we have
where
denote the basic polynomial with respect to and .
By differentiating the two sides of Equation (
1) with respect to the variable
x, we get
As Equation (15) holds at every collocation point, we have
It follows from Equations (11), (14) and (16) that
where
is an approximation of
.
From Equations (17) and (18), we get
Combining Equations (19) and (20) yields
,
where
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
is bounded as
, we write
Lemma 4 ([
23]).
Let be a smooth real-valued function on with for all and fixed values of r. Moreover, suppose and . Then, we have Lemma 5 ([
43]).
Suppose that and with , then we have,
where , denotes the smallest integer that is not less than x.
Lemma 6 ([
44]).
Suppose that , for all and , then we get,
where , denotes the smallest integer that is not less than x.
Lemma 7 ([
41,
44]).
Suppose that is Nth differentiable and , then for , we have,
where,
.
Theorem 1. Using direct linear interpolation, the error for solving Equation (1) satisfies .
Proof. For Equation (
1) at any collocation point, there is
Let the variables
, then we get
Approximating
by
, we have
It follows from Equations (24) and (25) that
It is assumed that the error at the interpolation point
is
Then, we can obtain
where
is the linear interpolation residual term. Substituting Equation (27) into Equation (26) yields
Therefore, we can express the interpolation error at the point
as
According to Lemma 4, we get that
when
. By the nature of the direct linear interpolation of the residual term, we have
Applying Lemma 6, we have
□
Theorem 2. The error estimate for solving Equation (1) using direct higher-order interpolation is .
Proof. Based on the fact that Equation (
1) holds at any collocation point, we have
Approximating
by
, we have
Equations (30) and (31) yield
When the error at the interpolation point
is
, we get
where
is the interpolation residual term. Then, Equation (32) can be written as
Substituting Equation (33) into Equation (34), we get
The interpolation error at
is thus found to be
By transforming the variables
, and noting that
, then we obtain
Applying Lemma 4, we get that
. According to the property of the direct higher interpolation residual term, we have
Then, applying Lemma 6, we have
□
Theorem 3. The error estimate obtained by solving Equation (1) using the direct hermite interpolation method is Proof. According the direct Hermite interpolation method in
Section 4, we know that
Approximating
by
, we have
It follows from Equations (37)–(40) that
Based on Hermite interpolation, the error function is expressed as follows
where
is the residual term of Hermite interpolation. Furthermore, the errors at the collocation point
satisfy
. Substituting Equation (43) into Equations (41) and (42), respectively, we have
Combining Equations (44) and (45) yields
,
Applying Lemma 4, we can get that
when
. According to the property of direct Hermite interpolation of the residual term, we have
According to Lemmas 4 and 6, we get
Therefore, we can get
.
□
Theorem 4. The error estimate obtained by solving Equation (1) using the piecewise hermite interpolation method is Proof. In the case of the piecewise Hermite interpolation collocation method, we have
Approximate
by
, and we have
It follows from Equations (47) and (48) that
where
and
. A similar argument to that used in Theorem (3) shows that
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.
denotes direct linear interpolation,
denotes direct higher order interpolation,
denotes direct Hermite interpolation and
denotes piecewise Hermite interpolation.
From
Figure 1, we can see that the oscillation of the
function depends only on
; the larger
, the higher the oscillation.
Example 1. We consider the following equation
When , then the exact solution of the equation is .
Table 1,
Table 2,
Table 3,
Table 4,
Table 5 and
Table 6 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
at the points
and
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
When , then the exact solution of the equation is .
Example 3. We consider the following equation
When , then the exact solution of the equation is .
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.