1. Introduction
Accurate computation of the initial value problems with oscillatory forcing term (IVPs) is one of the ambitious task in scientific computing. In this work, we focused on a particular case of the IVPs involving an ODE system
where
is a square matrix of order
n,
is a
n-vector of functions, and
is a rapidly oscillating scaler function with a frequency parameter
. Particularly, we assume
. Initial value problem (
1) is the simplest model of the more complicated problems of electronic engineering. Generally, nonlinear ODEs and differential algebraic equations (DAEs) with oscillatory forcing term are frequently appeared in this field. The analytic solution of the ODEs (
1) in terms of symmetric integral operator and is given as
where
[
1]. In this paper, we consider that the phase function
has no stationary point for
.
High-frequency transmissions are regularly encountered in the field of radio frequency. This is a result of modulation, which allows a lower-frequency information stream to be imposed on a higher-frequency carrier. The goal is to make it possible to implement audio transmission using antennae that are reasonable in size. If modulation is not used, long-distance antennas on the order of several hundred miles to several thousand miles are necessary. When it comes to radio frequency communication systems, transmissions with frequencies in the Mega Hertz range and above are typical. Because of the existence of solid-state amplifiers, mixers, and other components in radio frequency transmission systems, the nonlinearity of the system might occur [
2].
In most radio frequency systems, the linear part of (
1) appears because linear indicators, i.e., resistors, and capacitors are used in conjunction with the system. While the nonlinear portion is caused by amplifiers, mixers, or nonlinear controlled resistors and capacitors. The linear portion is caused by a combination of these components. ODEs (
1) is a straight forward model containing nonlinearity. The emergence of the phase
is due to the application of sine waves to the terminals of circuits including diodes or transistors, which causes the term to appear. It is important to note that sinusoids and amplifiers employing transistors are ubiquitous in indication systems and should not be overlooked [
2,
3,
4].
As a result of the ongoing rapid changes in the radio frequency and telecommunications industries, we must build faster simulations, faster designs, and faster product outputs to keep up with the times. It is necessary to update the existing computer aided design (CAD) tools in order to accommodate the new algorithms. Furthermore, the increasing complexity of modulation formats makes it unreasonably slow to interpret software tools, resulting in dissatisfaction with the end output. It is now imperative that rapid and precise numerical techniques be developed in order to deal with the situation.
Many well-known researchers have made significant contributions to this field, recognizing the importance of these issues and the frequency with which they are encountered in many domains of science and engineering. A number of rapid approaches, such as the Runge–Kutta method of order 4, multistep methods, such as the Adam–Bashforth method and the Adam–Moulton method [
5,
6], as well as several other advanced numerical methods, were proposed to estimate IVPs (
1). When Taylor’s series was used to reduce multistep methods to single-step methods, and then Newton-Cotes quadrature was employed to approximate the ODEs, and the result was cited as [
7].
Fast numerical procedures [
8,
9] have been derived for evaluation of rapidly oscillatory ordinary differential equations. Recently, the state-of-the-art numerical methods, i.e., asymptotic and Filon-types of methods, have been implemented to approximate highly oscillatory linear and nonlinear ODEs [
1]. The generalized Filon’s method is used to solve stochastic differential equations and extendable to evaluate ordinary differential equations with rapidly oscillating factors [
10].
In [
1], the oscillatory IVPs are transformed to highly oscillatory integrals with Fourier kernel. The integrals are evaluated numerically by some state-of-the-art methods such as the Levin collocation method [
11,
12,
13,
14,
15,
16,
17], asymptotic method [
18,
19,
20], numerical steepest decent method [
21,
22], and Filon(-type) methods [
1,
23,
24,
25,
26,
27].
In [
10], the authors presented generalized Filon quadrature to evaluate a first-order stochastic differential equation with rapidly oscillating factor. The method is applicable to solve a variety of oscillatory ODE problems. Recently, the authors [
28] constructed a multivalue mixed collocation method based on generalized Hermite polynomials for numerical solution of oscillatory ODEs system. Some theoretical results are also derived in the paper.
The motivation is made from the work [
1], in which the integral is evaluated by the asymptotic method. Levin collocation method is one of the accurate tool which can evaluate oscillatory integrals with complicated phase function, where the asymptotic method fails. Secondly, radial basis function interpolation is one of the best approximation techniques which accurately approximates irregular oscillatory function. The Levin collocation method based on multiquadric RBF is implemented in this work to solve the oscillatory-type linear ODEs. The method improves accuracy with the inverse power of
. The method is applicable to oscillatory problems with stationary free phase functions. Moreover, the integrand in (
2) is interpolated by the RBF collocation method instead of the Newton backward interpolant, following the multistep methods and evaluated the integral by quadrature methods or matlab built-in code "quadv". Finally, a nonlinear IVPs (
1) can be transformed into linear IVPs by Bernoulli’s transformation and then approximated by the proposed algorithms.
2. Numerical Procedures
In this section, we discuss two numerical techniques for solution of oscillatory and nonoscillatory ODEs. According to the first method, the integrand of (
2) is interpolated by the RBF collocation method, and then the integral is evaluated by the built-in code of MATLAB programming ‘quadv’ or uses multi-resolution quadratures to approximate the targeted integral. The method accurately approximates the nonoscillatory linear ODEs. Secondly, a generic method is implemented to approximate both nonoscillatory and highly oscillatory linear ODEs. The Levin collocation method based on multiquadric RBFs is used to approximate the oscillatory integral of the ODE solution (
2). Finally, nonlinear ODEs are transformed to linear ODEs and then approximated by the proposed procedures.
2.1. Procedure Based on RBF Interpolation
A modified form of IVP (
1) is given as
By the Euler method, the exact solution of ODE (
3) can be written as
On descritization over the intervals
, (
4) is reduced to
Now, we interpolate the integrand by the RBF collocation method. The unknown values of the integrand can be predicted by the RK4 method.
The global RBFs
are univariate smooth functions with a free shape parameter
. The accuracy and shape of the RBF interpolation depend upon the optimal value of
. For a given set of m-centers,
, an approximate function
is supposed to satisfy the following interpolation condition:
The equation confers a system of linear equations, and can be written in matrix form as
By solving linear equation’s system, (
6), the unknown coefficients,
could easily be found out. The matrix
is a square matrix called system matrix, and
and
are
m-vectors. The entries of
are given as
Thus, Equation (
5) becomes
The integral can be numerically evaluated by the multiresolution quadratures such as hybrid functions or Haar wavelets [
11,
12,
29]. It can also be evaluated by the MATLAB built-in codes such as quadgk, quadv, and quadl. In the current work, we have used “quadv” for evaluation of the targeted integral.
For optimal accuracy, we have chosen multiquadric RBF, , where are the radial distances and the free parameter is called shape parameter. An optimal value of is an open problem. In this work, we have taken a fixed value of .
Now, the desired solution of (
5) at
-th time level is given by
For each , the integral can be evaluated recursively by the RBF collocation method, and the numerical results can be obtained at different time levels.
2.2. Levin Collocation Method
This method is efficiently implementable to evaluate nonoscillatory ODEs and the ODEs with highly oscillatory forcing term
. The initial value problem (
1) can be reduced to a single ODE as
with analytical solution
where
A is assumed as a constant parameter for single ODE [
30]. In the current work, we also assume that
for any
and
. The descretized form of (
8) is given by
In order to approximate highly oscillatory integral , we focus on the Levin quadrature theory which is briefly discussed as:
According to the Levin approach, an approximate function
is supposed to be a solution of the following ordinary differential equation
Integral on the right of (
9) can be written as
Thus, the desired approximate solution of the IVP (
7) is given as
To find the approximate function
, we substitute
in the ODE (
10), and by applying interpolation condition, we obtain
where
is the multi-quadric RBF and
is the free shape parameter of the RBF interpolant.
Equation (
13) also generates a linear equation’s system. This linear system of equations could be rewritten in the form of a matrix as
or
For
, the system matrix
becomes a square matrix with entries
By solving the system of linear Equation (
14) by Gauss-elimination or LU-factorization method to obtain the unknown coefficients
, we can find, as a result, the approximate RBF solution
. The desired solution of the ODE is obtain then by the Levin Formula (
11) and is denoted by LCM.
2.3. Nonlinear Highly Oscillatory ODEs
In this section, we briefly describe a procedure that can transform Bernoulli-type nonlinear highly oscillatory ODEs to a linear form. Generally, the Bernoulli’s equation can be written as
For
or
, (
16) is a linear ODE. Otherwise, we use the following transformation
On differentiating, we obtain
Substituting the value of
from (
16) in (
17), we obtain
or
On rearranging, we obtain
The transformed ODE (
18) is linear and can be approximated by the two proposed methods.
Theorem 1 ([
12]).
Let be a stationary point of order , and let , . Then, the following holds true:- i.
The error bound of the RBF method with Levin approach to approximate the oscillatory integral is given by for m collocation points.
- ii.
For computation of by the same method at , the error bound is given byfor .
As shown in (
12), solution by the Levin collocation method of the IVP (
7) in the local domain
is given as
The error estimate of the method
is demonstrated in (
19). Thus, the desired error estimate in the local domain interval
of the new method to compute IVP (
7) is given by
For
m collocation points, the error estimate (
22) is reduced to
where
m represents the number of collocation points of the LCM. Thus, the asymptotic error estimate reach to
.
3. Numerical Assessment
Few benchmark nonoscillatory and highly oscillatory IVPs have been considered from the literature to verify accuracy of the proposed procedures. The results of the new methods are compared with asymptotic method [
1]. Accuracies are measured in terms of infinity norm
and absolute errors
. CPU time (in seconds) is also computed in some problems. The comparison of the new work is also performed with some classical methods such as RK4 and the Adam–Bashforth method in case of nonoscillatory ODEs.
Test Problem 1.Consider the following nonoscillatory ODE [5]with analytical solution The ODE (
24) is evaluated numerically by the AF4, RK4, and the proposed RBF collocation methods. The results at different time levels in terms of
and
are calculated and analyzed in
Table 1 and
Figure 1. It is shown in the table that the proposed method performs better than the other classical methods. The result of the RBF method depends upon the values of shape parameter
and collocation points
m as shown in
Figure 1 (left). We see better accuracy on increasing the collocation points
m. The CPU time (in seconds) of the proposed method is shown in
Figure 1 (right). It is obvious from the results analyzed in the table and the figure that the RBF method improves the results for increasing
m at low computational time.
Test Problem 2.Consider the following nonoscillatory ODEwith analytical solution The ODE (
25) is approximated by the RBF collocation method and the classical methods. Numerical results are analyzed in
Table 2 and
Figure 2. From
Figure 2 (left), it is shown that the proposed method improves accuracy on increasing
m at fixed free shape parameter
. The CPU time comparison (in seconds) of the new method is shown in
Figure 2 (right). This demonstrates that the new method decreases the errors on increasing the collocation points.
The results of the new method for fixed
m and
are compared with RK4 at different time levels and are shown in
Table 2. It is evidence that the proposed method is accurate and efficient even at fewer nodal points.
Test Problem 3.Consider the following highly oscillatory IVP [1] The analytical solution obtained by MAPLE 16 is given as The given ODE is highly oscillatory. The irregular oscillations of the analytical solution are shown in
Figure 3 (left) for
. The highly oscillatory IVP is hard to solve by the classical numerical methods such as RK4 and AF4 for higher values of
. The oscillatory problem is computed by the proposed Levin collocation method (LCM). Numerical results by the proposed method are compared with the asymptotic method [
1] as shown in
Table 3. The table demonstrates that the new method is more accurate and faster than the asymptotic method for higher values of
. The method is tested for lager collocation points
m on increasing values of
. The results are analyzed in
Table 4.
The method is also implemented at different time levels on increasing
m, and it obtained higher accuracies. The results are presented in
Table 5. In addition, the results of the LCM for fixed time level and increasing
m are shown in
Figure 3 (right). We see that the new method improves accuracy on increasing
m as well as
at different time levels.
Test Problem 4.Consider the following nonlinear highly oscillatory IVP [10] Using Bernoulli’s transformation, the nonlinear ODE (
26) is transformed into the linear form as
The analytical solution of (
27) is given as
The transformed linear ODE (
27) is computed by the new method LCM. Results and CPU time are analyzed in
Table 6 and
Table 7 and
Figure 4.
Table 6 demonstrates that the new method improves the numerical results for increasing
m and
. The proposed method is also examined for varying
m at different time levels. Better results are obtained and are shown in
Table 7.
The CPU time related to
m is calculated and displayed in
Figure 4 (left). We see that the new method is efficient as well. The exact solution of the oscillatory ODE is displayed in
Figure 4 (right). From the whole discussion, it is obvious that the LCM is an accurate tool for approximating the oscillatory-type linear and nonlinear IVPs.
Test Problem 5.Consider the following highly oscillatory IVP [28]with exact solution The ODE (
28) is highly oscillatory and is computed by the new method LCM. The results of the proposed method are compared with the multivalue collocation method reported in [
28]. The absolute errors are analyzed in
Table 8, which demonstrates that the proposed method is better than the multivalue collocation method [
28]. It is also shown in the table that the new method is highly accurate compared to the existing method [
28].