- Research
- Open Access
Sinc Nyström method for a class of nonlinear Volterra integral equations of the first kind
- Yanying Ma^{1}Email author,
- Jin Huang^{1},
- Changqing Wang^{2} and
- Hu Li^{1}
https://doi.org/10.1186/s13662-016-0849-8
© Ma et al. 2016
- Received: 19 January 2016
- Accepted: 28 April 2016
- Published: 8 June 2016
Abstract
Two numerical methods are proposed to solve nonlinear Volterra integral equations of the first kind. By using variable transformations, the problem is converted into linear Volterra integral equations of the second kind. These methods are implemented by utilizing Sinc quadrature, and then the problem is reduced to linear algebraic system equations. We state error analysis for the proposed methods, which show that these methods obtain exponential convergence order. Numerical examples are presented to confirm the theoretical estimation and illustrate the effectiveness of the proposed methods.
Keywords
- nonlinear Volterra integral equation
- Sinc Nyström method
- SE transformation
- DE transformation
1 Introduction
Volterra integral equations of the first kind arise in many fields of science and engineering, for example, in diffusion problems, fluid dynamics, heat conduction problems, nonlinear dynamic systems identification, concrete problems of mechanics, et cetera. As we all know, Volterra integral equations of the first kind are ill-posed problems because their solutions are generally unstable, and slight changes can make large errors [1, 2]. So it is difficult to find exact solutions of these equations in many cases. Furthermore, since the small error may lead to an unbounded error, it is also difficult to find numerical solutions. Some works were motivated by the aforementioned discussion, and several regularization methods were introduced to conquer the ill-posedness in [3–5].
There exist several methods to solve linear Volterra integral equations of the first kind [6–9], such as block-pulse functions method [6], modified block-pulse functions method [7], and wavelet method [8]. However, nonlinear problems are still a challenge. Babolian et al. introduced the operational matrices method by using piecewise constant orthogonal functions and homotopy perturbation method for solving nonlinear Volterra integral equations of the first kind separately in [10] and [11]. The Adomian method [12] and optimal homotopy asymptotic method [13] were applied to solve nonlinear Volterra integral equations. Inderdeep and Sheo presented the Haar wavelet method for numerical solution of a class of nonlinear Volterra integral equations of the first kind in [14].
It is common to employ a collocation method or analytic method based on the use of polynomial base functions to solve Volterra integral equations. Recently, several authors introduced SE and DE Sinc quadratures for solving integral equations. Muhammad and Mori proposed a numerical method of indefinite integration based on the DE transformation together with Sinc expansion of the integrand in [15]. Haber provided two formulas and approximation error for approximating the indefinite integral over a finite interval in [16]. The Sinc Nyström method for numerical solution of one-dimensional Cauchy singular integral equations given on a smooth arc in the complex plane has been described in [17]. Muhammad et al. [18] presented a technique for linear integral equations using the Sinc collocation method based on the DE transformation. Rashidinia and Zarebnia [19] developed an analogous approach for the system of linear Fredholm integral equations by means of SE transformation. More recently, Okayama et al. [20] reported error estimates with explicit constants for the Sinc approximation, Sinc quadrature, and Sinc indefinite integration. Furthermore, the theoretical analysis of Sinc Nyström methods for linear integral and differential equations have been discussed in [21–23]. Similar numerical approaches for nonlinear Fredholm and Volterra integral equations of the second kind are also presented in [24, 25]. However, nonlinear Volterra integral equations of the first kind are still not solved. In this work, we develop SE and DE Sinc methods to solve Eq. (1) in terms of SE and DE Sinc quadrature rules; these methods have a simple structure and perfect approximate properties. The convergence rates of these methods are exponential. Therefore, the proposed methods improve the conventional polynomial convergence rate. Furthermore, the proposed schemes are stable because the discrete coefficient matrices are very well conditioned.
In this paper, the basic ideas are organized as follows. In Section 2, we present some definitions and preliminary results about the Sinc function and SE, DE Sinc quadrature for indefinite integral. In Section 3, Sinc Nyström methods for the nonlinear Volterra integral equations of the first kind are developed. In Section 4, the convergence analysis with errors are described for the current methods. Both of these two algorithms are exponentially convergent. In Section 5, numerical examples are presented to validate the effectiveness of these methods. Numerical results of the proposed methods are compared with existing methods to confirm the reliability of the proposed methods. Finally, a conclusion is given in Section 6.
2 Preliminaries
2.1 Sinc indefinite integral on the real axis
2.2 SE and DE Sinc indefinite integral
From the above we can see that the approximation of Eq. (4) is valid on \(\mathbb{R}\), whereas Eq. (1) is defined on finite interval \([a, x]\). Equation (4) can be applicable to infinite intervals using variable transformations. Here, the smoothing variable transformations with standard SE and DE transformation functions \(\phi(x)\) are utilized.
Definition 2.1
- (i)
f is analytic in \(\mathscr{D}_{d}\);
- (ii)
\(|f(z)|\leq M_{0}|Q(z)|^{\alpha}\) for all z in \(\mathscr {D}_{d}\), where \(Q(z)=(z-a)(b-z)\), and \(M_{0}\) is a constant.
Based on the Sinc approximation and SE transformation, we can implement a quadrature rule designated as the SE Sinc quadrature and present exponential convergence in the following theorem.
Theorem 2.1
(see [20])
Theorem 2.2
(see [20])
3 Sinc Nyström method for Volterra integral equations
Further, the proposed numerical methods for Eq. (1) will be fully discussed in two subsections, where we state the SE Sinc Nyström method and DE Sinc Nyström method for efficient evaluation of Volterra integral equations.
3.1 SE Sinc Nyström method
3.2 DE Sinc Nyström method
4 Convergence analysis for numerical method
Throughout this section, we provide a convergence analysis of the associated SE and DE Sinc Nyström methods. Let us first consider the SE case. Tomoaki Okayama and his coauthors have given the theoretical analysis of Sinc Nyström methods for linear Volterra equation in [22] by utilizing error estimates with explicit constants for Sinc quadrature. They display that approximate solutions have exponential convergence order.
Theorem 4.1
Proof
We refer to [22]. □
Theorem 4.2
Proof
We refer to [22]. □
According to these results, we can give an error analysis of the nonlinear Volterra integral equations of the first kind.
Theorem 4.3
Proof
Next, we take into account the error analysis of DE case for Eq. (1). The proof is similar to that in the SE case, so we only state the results.
Theorem 4.4
Theorem 4.5
Theorem 4.6
Remark 1
Here, the values of \(C_{0}\) in formulas (30) and (34) are different. In addition, the convergence rate of the approximate solution in (1) and (12) are consistent while the inverse function of \(H (u(t) )\) satisfies the Lipschitz condition. We can get the same conclusion for the DE case. Further, the convergence speed of the DE Sinc Nyström method is much faster than that of the SE Sinc Nyström method.
5 Numerical examples
Example 1
The numerical results of the SE Sinc Nyström method for Example 1
N | \(\mathbf{Error}_{\boldsymbol{v(x)}}\) | \(\mathbf{Error}_{\boldsymbol{u(x)}}\) | \(\boldsymbol{\rho_{N}}\) | Cond |
---|---|---|---|---|
4 | 9.203e − 003 | 2.442e − 003 | ∗ | 5.403 |
8 | 9.862e − 004 | 2.329e − 004 | 3.390 | 5.434 |
16 | 3.027e − 005 | 7.625e − 006 | 4.933 | 5.437 |
32 | 2.164e − 007 | 4.670e − 008 | 7.351 | 5.437 |
64 | 1.536e − 010 | 3.001e − 011 | 10.604 | 5.437 |
The numerical results of the DE Sinc Nyström method for Example 1
N | \(\mathbf{Error}_{\boldsymbol{v(x)}}\) | \(\mathbf{Error}_{\boldsymbol{u(x)}}\) | \(\boldsymbol{\rho_{N}}\) | Cond |
---|---|---|---|---|
4 | 8.238e − 002 | 1.887e − 002 | ∗ | 5.500 |
8 | 5.987e − 003 | 1.305e − 003 | 3.854 | 5.438 |
16 | 1.871e − 005 | 3.476e − 006 | 8.553 | 5.437 |
32 | 9.852e − 010 | 2.360e − 010 | 13.846 | 5.437 |
64 | 3.553e − 015 | 8.882e − 016 | 18.020 | 5.437 |
Example 2
The numerical results of the SE Sinc Nyström method for Example 2
N | \(\mathbf{Error}_{\boldsymbol{v(x)}}\) | \(\mathbf{Error}_{\boldsymbol{u(x)}}\) | \(\boldsymbol{\rho_{N}}\) | Cond |
---|---|---|---|---|
4 | 1.701e − 003 | 4.167e − 003 | ∗ | 5.403 |
8 | 1.766e − 004 | 4.658e − 004 | 3.161 | 5.434 |
16 | 9.051e − 006 | 2.365e − 005 | 4.301 | 5.437 |
32 | 1.039e − 007 | 2.602e − 007 | 6.505 | 5.437 |
64 | 1.360e − 010 | 3.175e − 010 | 9.678 | 5.437 |
The numerical results of the DE Sinc Nyström method for Example 2
N | \(\mathbf{Error}_{\boldsymbol{v(x)}}\) | \(\mathbf{Error}_{\boldsymbol{u(x)}}\) | \(\boldsymbol{\rho_{N}}\) | Cond |
---|---|---|---|---|
4 | 1.541e − 002 | 2.591e − 002 | ∗ | 5.500 |
8 | 8.811e − 004 | 2.325e − 003 | 3.478 | 5.438 |
16 | 1.308e − 005 | 3.266e − 005 | 6.154 | 5.437 |
32 | 6.183e − 010 | 1.139e − 009 | 14.817 | 5.437 |
64 | 5.551e − 016 | 8.882e − 016 | 20.291 | 5.437 |
Example 3
The numerical results of the SE Sinc Nyström method for Example 3
N | \(\mathbf{Error}_{\boldsymbol{v(x)}}\) | \(\mathbf{Error}_{\boldsymbol{u(x)}}\) | \(\boldsymbol{\rho_{N}}\) | Cond |
---|---|---|---|---|
4 | 3.565e − 003 | 5.677e − 002 | ∗ | 2.814 |
8 | 3.096e − 004 | 4.437e − 003 | 3.678 | 2.823 |
16 | 1.388e − 005 | 2.006e − 004 | 4.467 | 2.824 |
32 | 1.547e − 007 | 1.211e − 006 | 7.372 | 2.824 |
64 | 2.502e − 010 | 6.334e − 010 | 10.901 | 2.824 |
The numerical results of the DE Sinc Nyström method for Example 3
N | \(\mathbf{Error}_{\boldsymbol{v(x)}}\) | \(\mathbf{Error}_{\boldsymbol{u(x)}}\) | \(\boldsymbol{\rho_{N}}\) | Cond |
---|---|---|---|---|
4 | 3.044e − 002 | 9.255e − 002 | ∗ | 2.866 |
8 | 1.686e − 003 | 3.949e − 003 | 4.551 | 2.825 |
16 | 1.439e − 005 | 8.880e − 005 | 5.475 | 2.824 |
32 | 2.405e − 009 | 4.197e − 008 | 11.047 | 2.824 |
64 | 7.772e − 016 | 8.295e − 015 | 22.271 | 2.824 |
Example 4
The numerical results of the SE Sinc Nyström method for Example 4
N | \(\mathbf{Error}_{\boldsymbol{v(x)}}\) | \(\mathbf{Error}_{\boldsymbol{u(x)}}\) | \(\boldsymbol{\rho_{N}}\) | Cond |
---|---|---|---|---|
4 | 4.079e − 003 | 3.518e − 003 | ∗ | 2.330 |
8 | 3.576e − 004 | 2.783e − 004 | 3.660 | 2.337 |
16 | 9.208e − 006 | 1.611e − 005 | 4.111 | 2.338 |
32 | 9.411e − 008 | 2.283e − 007 | 6.140 | 2.338 |
64 | 1.498e − 010 | 3.041e − 010 | 9.552 | 2.338 |
The numerical results of the DE Sinc Nyström method for Example 4
N | \(\mathbf{Error}_{\boldsymbol{v(x)}}\) | \(\mathbf{Error}_{\boldsymbol{u(x)}}\) | \(\boldsymbol{\rho_{N}}\) | Cond |
---|---|---|---|---|
4 | 3.100e − 002 | 2.895e − 002 | ∗ | 2.339 |
8 | 1.912e − 003 | 1.362e − 004 | 4.410 | 2.338 |
16 | 1.420e − 005 | 4.278e − 005 | 4.992 | 2.338 |
32 | 8.818e − 009 | 9.879e − 0010 | 15.402 | 2.338 |
64 | 2.220e − 016 | 3.053e − 016 | 21.626 | 2.338 |
6 Conclusion
In the present study, the SE and DE Nyström method are presented by converting nonlinear Volterra integral equations of the first kind into linear Volterra integral equations of the second kind. The proposed methods are stable and avoid the ill-conditioning and nonlinear iteration problems. The condition numbers have good reliability and efficiency. Numerical results are in agreement with the theoretical analysis. It is obvious that the convergence rate of the approximate solutions are exponential when the inverse function of \(H(u(t))\) satisfies the Lipschitz condition. In future work, we will utilize the proposed methods to deal with the general nonlinear Volterra integral equations of the first kind and nonlinear Volterra integral equation systems of the first kind.
Declarations
Acknowledgements
The authors are very grateful to the referees for their detailed comments and valuable suggestions, which greatly improved the manuscript. This work was partially supported by the financial support from National Natural Science Foundation of China (Grant No. 11371079).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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