- Open Access
Jacobi spectral solution for weakly singular integral algebraic equations of index-1
© Zhao and Wang; licensee Springer. 2014
Received: 21 January 2014
Accepted: 28 May 2014
Published: 5 June 2014
The Retraction Note to this article has been published in Advances in Difference Equations 2015 2015:205
This paper is concerned with obtaining the approximate solution of a class of semi-explicit Integral Algebraic equations (IAEs) of index-1 with weakly singular kernels. Some function transformations and variable transformations are used to change the equations into integral equations defined on the standard interval , so that the solution of the new system possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. A Jacobi collocation method is proposed and its convergence analysis is provided, which theoretically justifies the spectral rate of convergence. The numerical results are given to verify our theoretical analysis.
In practical applications, one frequently encounters system (1.1). A good source on applications of IAEs with weakly singular kernels is the initial (boundary) value problems for the semi-infinite strip and temperature boundary specification including two/three-phase inverse Stefan problems [1–4]. A wide ranging description of IAEs arising in applications is given in . Several numerical methods for integral algebraic equations have been proposed (see e.g. [5–10]). However, as far as we know the numerical analysis of WSIAEs is largely incomplete and this is a new topic for research. The existence and uniqueness for the solution of WSIAEs has been given in [11, 12]. The piecewise polynomial collocation method for system (1.1) and the concept of tractability index have been considered by Brunner ; he analyzed the regularity of the solutions using conditions that hold for the first and second kind Volterra integral equations. Recently, the effective numerical method based on the Chebyshev collocation scheme is designed for system (1.1) in  and its convergence analysis is provided.
The solutions of system (1.1) usually have a weak singularity at , whose derivatives are unbounded at . To overcome this difficulty, we use the idea of the authors in . Both function transformation and variable transformation are used to change the system into a new system defined on the standard interval , so that the solutions of the new system possess better regularity and the Jacobi spectral collocation method can be applied conveniently. The aim of this work is to use the Jacobi collocation method to numerically solve the WSIAEs (1.1) and then a rigorous error analysis is provided in the weighted -norm which shows the spectral rate of convergence is attained.
This paper is organized as follows. Some useful results for establishing the convergence results will be provided in Section 2. In Section 3, we carry out the Jacobi collocation approach for system (1.1). The convergence of the method in the weighted -norm as a main result of the paper is given in Section 4. Section 5 gives some numerical experiments. The final section contains conclusions and remarks.
2 Some useful results
In this section, we discuss how the weakly singular IAEs can be changed to treat the problem. Furthermore, the index concept for WSIAEs which plays a fundamental role in both the analysis and the development of numerical algorithms for IAEs is discussed.
2.1 Index definitions for WSIAEs
There are several definitions of index in literature, not all completely equivalent, such as the tractability index (see, e.g. Definition (8.1.7) from ), the left index  and differentiation index . Generally, the difficulties are arising in the theoretical and numerical analysis of IAEs relevant to the index notion.
In this paper, we use the concept of the differentiation index which measures how far the main WSIAE is apart from a regular system of VIEs. Namely, the number of analytical differentiations of system (1.1) until it can be formulated as a regular system of Volterra integral equations is called the differentiation index.
where () and . In fact, can be obtained using integration by parts to with .
Because , we have , then equation (2.2) together with the first equation of (1.1) is a system of regular Volterra integral equation.
But it should be noted that this reduction (differentiation) is not useful from a numerical point of view and such a definition may be useful for understanding the underlying mathematical structure of a WSIAEs, and hence choosing a suitable numerical method for their solutions.
2.2 Smoothness of the solution
and , are the smooth solutions of system (2.4).
The existence and uniqueness results and the smoothness behavior of solutions and of system (2.4) can be obtained from the corresponding discussions of the classical theory of Volterra integral equations with weakly singular kernels from  (see e.g. Theorems 6.1.6 and 6.1.14 for further details).
3 The Jacobi collocation method
Particularly, when , the Jacobi polynomials become Legendre polynomials, and when , the Jacobi polynomials become Chebyshev polynomials.
where is the th-order Jacobi polynomial.
So, the unknown coefficients and are obtained by solving the linear system (3.6) and finally the approximate solutions and will be computed by substituting these coefficients into (3.3).
4 Error estimate
To prove the error estimate in the weighted -norm, we first introduce some lemmas which are usually required to obtain the convergence results.
Lemma 1 ()
Let be a linear operator from to , then , , there exists a positive constant , such that , , s.t., .
Lemma 2 ()
Let , denote the error functions. The following main theorem reveals the convergence results of the presented scheme in the weighted -norm.
(), with ;
where , .
where and .
From now on, for simplicity, we denote by and try to derive the error bounds for the proposed method step by step.
Finally, the above estimates together with equation (4.16) lead to Theorem 1. □
It is noted in  that the function transformation generally makes the resulting equations and approximations more complicated. As discussed in [, p.80], we can also obtain the error estimates for the numerical solutions to the WSIAEs (1.1).
(), with ;
5 Numerical experiments
where () is the Jacobi weight function.
errors for Example 1
1.75 × 10−4
1.61 × 10−3
3.34 × 10−5
2.59 × 10−4
7.33 × 10−6
5.80 × 10−5
4.12 × 10−7
5.62 × 10−6
2.35 × 10−7
6.15 × 10−6
4.65 × 10−8
3.96 × 10−7
errors for Example 2
2.95 × 10−4
2.84 × 10−3
4.55 × 10−5
3.70 × 10−4
7.81 × 10−6
8.62 × 10−5
4.72 × 10−7
4.92 × 10−6
4.31 × 10−7
8.78 × 10−7
5.29 × 10−8
5.12 × 10−7
This work has been concerned with the theoretical and numerical analysis of integral algebraic equations of index 1 with weakly singular kernels. It is noted that the solutions of the WSIAEs are not sufficiently smooth. So, the original system was changed into a new system, by using some function transformations and variable transformations. We presented a spectral Jacobi collocation approximation for the new WSIAEs. The error estimation of the method in the weighted -norm was obtained. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence.
The authors would like to acknowledge the anonymous referees for their careful reading of the manuscript and constructive comments. This work is supported by the National Natural Science Foundation of China (11101109, 11271102), Natural Science Foundation of Hei-long-jiang Province of China (A201107) and SRF for ROCS, SEM.
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