Abstract
The function-valued Padé-type approximation (2DFPTA) is used to solve two-dimensional Fredholm integral equation of the second kind. In order to compute 2DFPTA, a triangle recursive algorithm based on Sylvester identity is proposed. The advantage of this algorithm is that, in the process of calculating 2DFPTA to avoid the calculation of the determinant, it can start from the initial value, from low to high order, and gradually proceeds. Compared with the original two methods, the numerical examples show that the algorithm is effective.
1. Introduction
Consider a two-dimensional Fredholm integral equation of the second kind (2DF-II):where is a bounded domain, , and are given functions defined on , and is a real parameter called a characteristic value. It is known that a sequence of approximations leading to an infinite power series can be yielded by successive substitution, which is an iterative procedure, from this kind of integral equation. Set to be analytic and meromorphic in a neighborhood of the origin . Let its Neumann series be given byin which are continuous on and , Let and the inner product and the norm be defined by
Two-dimensional Fredholm integral equation of the second kind (2DF-II) is a useful tool to model many problems arising in fracture mechanics, transportation [1], 2D electromagnetic scattering [2], and computer graphics manipulations [3]. For domain , some works treat the square case [4–10] and some are based on discrete Galerkin method [5], Monte Carlo methods [6], or piecewise approximating polynomials [7], Nytröm methods based on cubature rules obtained as the tensor product of two-variate Gaussian rules [8, 10], and meshless method with complex factors [11]. Rational approximation has extensive application in engineering, technology and calculation [12–18]. As efficient rational approximating techniques, a generalized inverse function-valued Padé approximants had been used in solving integral equation [13, 14], and a function-valued Padé-type approximation method was used to solve one and two-dimensional Fredholm integral equation of the second kind [15–18].
In the paper, the function-valued Padé-type approximation (2DFPTA) is used to solve 2DF-II. In order to compute 2DFPTA, a recursive algorithm based on Sylvester identity is proposed. The remainder of this paper is organized as follows. In Section 2, we give the definition of 2DFPTA [18]. In Section 3, we apply Sylvester identity to propose a recursive algorithm to compute 2DFPTA. In Section 4, we compare our algorithm with the two algorithms in [4, 18].
2. Function-Valued Padé-Type Approximants and Its Convergence Analysis
In this section, we will give the definition of the function-valued Padé-type approximation [15–18]. Let be a linear functional on the polynomial space , and define it bywhere for . We can obtain from linear functional in (6) that Letwhere is a scalar polynomial of degree and assume .
Define the polynomial with function-valued coefficient byNote that acts on and is a function-valued polynomial in of degree .
Set
The polynomial is called the generating polynomial of the Padé-type approximation . It can be arbitrarily chosen and we have degrees of freedom.
It is known from (10) that implies . In view of (8), (9), and (11), is of the following form:in which . Using (10) and (12), we get
Definition 1. Let and be given by (12) and (10), respectively, and then the rational functionis defined as a two-dimensional function-valued Padé-type approximation (2DFPTA) with generating polynomial and is denoted by .
According to the relations (9) and (11), we can from [19] obtain the following error formula with the linear functional form:Note that the error formula (15) is disadvantageous to estimate error numerically, and it implies from (13) that the Padé-type approximation property isIn order to compute conveniently, we choose as the maximum absolute value of the coefficient of on in (13). We can also observe from (15) and (16) that if the coefficients are given, the order of the approximation is
3. A Triangle Recursive Algorithm of
As the definition given in Section 2, we find that the key to calculating 2DFPTA is to compute its the generating polynomial . If the generator polynomial is determined, we can compute the 2DFPTA of type according to (10) and (12). In the section, we will apply Sylvester’s identity [20] to propose a three-term recurrence formula for computing and establish a complete recursive algorithm for calculating 2DFPTA.
Lemma 2 (Sylvester identity [20]). Letand then it holds thatwhere andSet , the generating polynomial
Lemma 3 (see [15]). Let the determinant ; then the generating polynomial exists and is unique and can be expressed aswhere
Definition 4. For any th degree polynomial , by the higher-order linear functional , we define the vector inner product with order as follows:where
To discuss conveniently, we introduce the following notations:Letwhere if andFrom (20) and (25), we conclude that
To compute , we give the following theorem. According to Definition 4 and the th degree polynomial , we gain the vector inner product with order as follows:where
Theorem 5 (three-term recurrence formula). Let for all Then it holds thatwhere the initial value is
Proof. Set and Substitute them into the formula (18) and we conclude thatDue to , we have By (25) we get and substitute it into the above formula; then that is,According to the higher-order linear functional and nature of the determinant, we haveandDividing (34) by (35), we deduce thatFrom (33) and (36), we obtain that
Example 6. Let us choose an example of to illustrate the process to generate By the three-term recurrence relation (16), the calculation program of can be arranged according to the following bottom-down triangle: When computing we mainly use the left polynomial , the polynomial in the top left corner in the table, and their relation (29). It is not difficult to find that, in every column, the number of polynomial is in decline when k is bigger and bigger.
It is noticed that can also be arranged by another pattern as follows:When computing we mainly use the above polynomial , the polynomial in the top-right corner in the table, and their relation (29). Similarly, the number of polynomials is in decline when is bigger and bigger in every row. In this way, we can recursively get
From the three-term recurrence relation (29), we now build a complete recursive algorithm for calculating 2DFPTA. The advantage of this algorithm is that in the process of calculating 2DFPTA to avoid the calculation of the determinant, it can start from the initial value, from low to high order, and is gradually calculated. And its specific computational steps are as follows.
Algorithm 7 (Sylvester-type triangle algorithm). (1)Set the initial value as compute by (4).(2)For compute by (29) in turn (3)Compute (4) is written as and the denominator polynomial of function-valued Padé-type approximation is(5)From (12), numerator polynomial of is(6)In view of Definition 1, there exists such that
Example 8. Consider the second Fredholm integral equation
Its kernel is .
In the following we use the triangle algorithm to yield of the integral equation.
Solution. The first few terms of the power series for are given by Here , and we set
Step 1. Set the initial value as , , , computing
Step 2. Compute by (29)
Step 3. Obtain the generating polynomial of function-value Padé-type approximant by (27)
Step 4. From (10) get the denominator of function-value Padé-type approximant
Step 5. Get the numerator of function-value Padé-type approximant by (12) where
Step 6. Obtain function-valued Padé-type approximant
4. Numerical Results
In this section, we will consider the square domain case of . We apply Algorithm 7 for 2DFPTA to approximate the corresponding 2DF-II [11]. Compared with the methods in [4, 18], numerical experiments show that our method improves the calculation accuracy. Furthermore, our method can estimate the corresponding characteristic values of from the real roots of .
Example 9. Consider the same Fredholm integral equation with Example 8 [4, 18].
When , the exact solution of (43) is By Algorithm 7, we get the following calculation results:where when the maximum absolute error of caused by Algorithm 7 is shown in Figure 1. From Figure 1, we can observe that the error is and in the process we just only employ the first items of Neumann series (4), while in [4] the error was when the first 5 items of Chebyshev series corresponding to this equation were used and in [18] the corresponding error was .
By Algorithm 7, we also get thatwhere their eigenvalue has estimates; see Table 1.
In the computational process of , we get two possible characteristic values and . After further computing we can deduce that and both include the two real roots. Therefore we consider the two numbers as characteristic values of this equation.
In Table 2, is from Algorithm 7 in the paper and is from the method in [18], where We find that from the numerical results has better approximation effect compared with in [18].
Example 10. Consider the following [4, 18]:When , the exact solution of (56) is By Algorithm 7, the calculation of Example 10 is shown as follows:where when the maximum absolute error of caused by Algorithm 7 is shown in Figure 2. From Figure 2, we can observe that the error is and in the process we just only employ the first items of Neumann series (4). While in [4] the error was when the first 5 items of Chebyshev series corresponding to this equation were used and in [18] the corresponding error was .
From in (58), we get two possible characteristic values . By Algorithm 7, we also get thatNote that both and include the two real roots. Therefore, we consider the two real roots as the characteristic values of (56). For estimates of the characteristic values of in Example 10, see Table 3.
In Table 4, is from Algorithm 7 in the paper and is from the method in [18], where From the numerical results in Table 4, has better approximation effect compared with in [18].
Data Availability
The [tables] data used to support the findings of this study are included within the article. The [figures] data used to support the findings of this study are included within the article. The [software codes] data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is funded by the National Natural Science Foundation of China under Grant no. 11371243, Key Disciplines of Shanghai Municipality under Grant no. S30104, the Fostering Master’s Degree Empowerment Point Project of Hefei University under Grant no. 2018xs03, and the Key Natural Science Research Project of University of Anhui Province, Education Department of Anhui Province, under Grant nos. KJ2017A547, KJ2016A592, and KJ2018A565.