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The Scientific World Journal
Volume 2014 (2014), Article ID 413623, 8 pages
Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials
1Department of Mathematics and Statistics, Mississippi State University, MS 39762, USA
2Department of Mathematics and Statistics, Faculty of Science, Prince of Songkla University, Songkhla 90112, Thailand
Received 19 August 2013; Accepted 10 October 2013; Published 12 January 2014
Academic Editors: S. Amat, F. J. Garcia-Pacheco, H. Iiduka, and F. Minhós
Copyright © 2014 S. Mashayekhi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.
There is considerable literature that discusses approximating the solution of linear and nonlinear Hammerstein integral equations [1–6]. For Fredholm-Hammerstein integral equations, the classical method of successive approximations was introduced in . A variation of the Nystrom method was presented in , and a collocation-type method was developed in . In , Brunner applied a collocation-type method to nonlinear Volterra-Hammerstein integral equations and integrodifferential equations and discussed its connection with the iterated collocation method. Han  introduced and discussed the asymptotic error expansion of a collocation-type method for Volterra-Hammerstein integral equations. The existence of solutions to nonlinear Hammerstein integral equations was discussed in . Further, Reihani and Abadi  and Hsiao  applied rationalized Haar functions and hybrid of block-pulse functions and Legendre polynomials, respectively, for solving Fredholm and Volterra integral equations of the second kind. Several authors consider the nonlinear mixed Volterra-Fredholm integral equations of the form where and are constants and and the kernels and are given functions assumed to have th derivatives on the interval , . For the case and , where and are nonnegative integers, Yalçinbaş , Bildik and Inc , and Hashemizadeh et al.  used Taylor series, modified decomposition method, and hybrid of block-pulse functions and Legendre polynomials, respectively, to find the solution. For the case and , where and are given continuous functions which are nonlinear with respect to , Yousefi and Razzaghi  applied Legendre wavelets to obtain the solution, and for the general case, where and are given continuous functions which are nonlinear with respect to and , Ordokhani  and Marzban et al.  applied the rationalized Haar functions and hybrid of block-pulse functions and Lagrange polynomials, respectively, to get the solution.
The available sets of orthogonal functions can be divided into three classes. The first class includes sets of piecewise constant basis functions (PCBF’s) (e.g., block-pulse, Haar, and Walsh). The second class consists of sets of orthogonal polynomials (e.g., Chebyshev, Laguerre, and Legendre). The third class is the set of sine-cosine functions in the Fourier series.
Orthogonal functions have been used when dealing with various problems of the dynamical systems. The approach is based on converting the underlying differential equation into an integral equation through integration, approximating various signals involved in the equation by truncated orthogonal functions, and using the operational matrix of integration to eliminate the integral operations. The matrix can be uniquely determined based on the particular orthogonal functions (see  and references therein). Among orthogonal polynomials, the shifted Legendre polynomial is computationally more effective . The Bernoulli polynomials and Taylor series are not based on orthogonal functions; nevertheless, they possess the operational matrices of integration. However, since the integration of the cross product of two Taylor series vectors is given in terms of a Hilbert matrix , which is known to be ill-conditioned, the applications of Taylor series are limited.
In the present paper we introduce a new direct computational method to solve nonlinear mixed Volterra-Fredholm integral equations in (1). We approximate the solution not to the equation in its original form but rather to an equivalent equation and , . The functions and are approximated by hybrid functions with unknown coefficients. These hybrid functions, which consist of block-pulse functions and Bernoulli polynomials together with their operational matrices of integration and product, are given. These matrices are then used to evaluate the coefficients of the hybrid functions for solution of nonlinear mixed Volterra-Fredholm integral equations.
The outline of this paper is as follows. In Section 2, we introduce properties of Bernoulli polynomials and hybrid functions. In Section 3, the numerical method is used to approximate the nonlinear mixed Volterra-Fredholm integral equations, and in Section 4, we report our numerical findings and demonstrate the accuracy of the proposed numerical scheme by considering five numerical examples.
2. Hybrid Functions
2.1. Properties of Bernoulli Polynomials
The Bernoulli polynomials of order are defined in  by where , are Bernoulli numbers. These numbers are a sequence of signed rational numbers, which arise in the series expansion of trigonometric functions  and can be defined by the identity The first few Bernoulli numbers are with , .
The first few Bernoulli polynomials are According to , Bernoulli polynomials form a complete basis over the interval .
3. Hybrid of Block-Pulse Functions and Bernoulli Polynomials
Hybrid functions , , are defined on the interval as  where and are the order of block-pulse functions and Bernoulli polynomials, respectively.
3.1. Function Approximation
Let , and assume that is the set of hybrid of block-pulse functions and Bernoulli polynomials, and with being an arbitrary element in . Since is a finite dimensional vector space, has the unique best approximation out of such as , that is, Since , there exist unique coefficients such that where Further, by using (9), we can obtain where is a sparse invertible matrix .
3.2. Function of Two Variables Approximation
3.3. Operational Matrices of Integration and Product
The integration of the hybrid functions defined in (10) is given by where is the operational matrix of integration. The product of two hybrid functions with the vector is given by where is the product operational matrix. The matrices and are given in .
3.4. Approximation Errors
In this section we obtain bounds for the error of best approximation in terms of Sobolev norms. This norm is defined in the interval for by where denotes the th derivative of . The symbol which is introduced in  is defined by
Theorem 1. Suppose that with . If is the best approximation of then and, for , where depends on .
Proof. Let with and let be the best approximation of , which is constructed by using shifted Legendre polynomials , in the interval . Then  and, for , Since the best approximation is unique , we have and by using (23)–(25) we can obtain (21) and (22).
4. The Numerical Method
We approximate (1) as follows.
5. Illustrative Example
In this section, five examples are given to demonstrate the applicability and accuracy of our method. Example 1 is a nonlinear mixed Volterra-Fredholm-Hammerstein integral equation which was considered in  by using rationalized Haar functions (RHF) and also solved in  by applying Chebyshev approximation. Examples 2 and 3 are the integral equations reformulation of the nonlinear two-point boundary value problems considered in [13, 25] by using RHF and Adomian method, respectively. Although the reformulated integral equations in Examples 2 and 3 are Fredholm-Hammerstein integral equations, the method described here can be used. Examples 1–3 were also solved in  by using hybrid of block-pulse functions and Lagrange polynomials. For Examples 1–3, we compare our findings with the numerical results in  which have been shown to be superior to those of [13, 24, 25]. Examples 4 and 5 are Fredholm and Volterra integral equations of the second kind, respectively, which were considered in  by using RHF which were also solved in  by using hybrid of block-pulse functions and Legendre polynomials. For Examples 4 and 5, we compare our findings with the numerical results in  which have been shown to be superior to those of .
For approximating an arbitrary time function, the advantages of Bernoulli polynomials , where , over shifted Legendre polynomials , , where , are given in  and over Lagrange polynomials , where , are given below.
Advantages of Bernoulli Polynomials over Lagrange Polynomials. (a) The operational matrix of integration in Bernoulli polynomials has less error than for Lagrange polynomials. This is because, for in , we ignore the term from the integration of , while for in Lagrange polynomials in we ignore the terms from the integration of each of .
(b) Bernoulli polynomials have less terms than Lagrange polynomials. For example, , and have , and 5 terms, respectively, while all have 7 terms, and this difference will increase by increasing . Hence, for approximating an arbitrary function, we use less CPU time by applying Bernoulli polynomials as compared to Lagrange polynomials.
(c) The coefficients of individual terms in Bernoulli polynomials are smaller than the coefficient of individual terms in the Lagrange polynomials . Since the computational errors in the product are related to the coefficients of individual terms, the computational errors are less by using Bernoulli polynomials.
Example 1. Consider the integral equation given in  by
The exact solution is .
For this integral equation we choose and .
Let where , , and are obtained similar to (13). By substituting (35)–(37) in (34) we have where can be calculated similar to (12). We collocate (38) at We get Solving (40) we obtain in (35). Table 1 shows the absolute errors of exact and approximate solutions in some points of the interval obtained by the present method for and together with the method of . In this Table is order of Lagrange polynomials.
Example 2. Consider which is of great interest in hydrodynamics . This problem has the unique solution  where Here, is the root of the equation Equation (41) can be reformulated as the integral equation where Table 2 represents the computational results of the errors for different values of and obtained by the present method together with different values of and in . In Table 2, and denote the approximate and exact solutions, respectively.
Example 3. In this example we consider the mathematical model for an adiabatic tubular chemical reactor discussed in [27, 28], which, in the case of steady state solutions, can be stated as the ordinary differential equation
The problem can be converted into a Hammerstein integral equation of the form 
where is defined by
The existence and uniqueness of the solution for this Hammerstein integral equation with respect to the value of parameters , , and are given in . In  the Adomian method is used to solve the integral equation (48) for the particular values of the parameters , , and which guarantee the existence and uniqueness of the solution for this integral equation .
Table 3 gives a comparison between the numerical results of in some points of the interval obtained by the Adomian method given in , together with the method in  for and and by the present method for and .
Example 4. Consider the Fredholm integral equation of the second kind 
with the exact solution . Table 4 gives a comparison between the numerical results of in some points of the interval obtained by the method given in  for and and by the present method for and . In this Table is order of Legendre polynomials.
Example 5. Consider the Volterra linear integral equation of the second kind  with the exact solution . Table 5 gives a comparison between the numerical results of in some points of the interval obtained by the method given in  for and and by the present method for and .
In the present work, the hybrid of block-pulse functions with Bernoulli polynomials is used to solve nonlinear mixed Volterra-Fredholm integral equations. The problem has been reduced to a problem of solving a system of algebraic equations. The matrices and in (17) and (12) have large numbers of zero elements and are sparse matrices, hence, the present method is very attractive and reduces the computer memory. Illustrative examples are given to demonstrate the validity and applicability of the proposed method.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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