br Numerical solutions for nonlinear Hammerstein integral equations of the

Numerical solutions for nonlinear Hammerstein integral equations of the second kind
Let us consider a special case of Eq. (1) as follows: Suppose that exists and rewrite Eq. (34) as follows: where: Therefore, Eq. (36) can be rewritten in the following form:

Results and discussion
In this section, we will present some numerical examples in order to investigate the performance and efficiency of the proposed method. For this purpose, we have compared our results with those of  [3] as well as with the exact solutions. This comparison is done based on error functions presented as: and: with: where is the approximate solution obtained by our method in Section  2 and is the approximate solution presented in Section  4. Also, it is assumed that is the best polynomial approximation of degree at most for the exact solution of the integral equation determined using maple software.


Reduced order models are highly desirable for engineers in system analysis, synthesis and simulation of complicated high-order systems, since the analysis and design of such systems is not an easy task. Various methods for model reduction are reported in the literature in the time and frequency domains. Model reduction was started by Davison in 1966  [1] and followed by Chidambara, who suggested several modifications to Davison’s approach  [2–4]. After that, different approaches were proposed using purchase CX-4945 eigenvalue retention  [1,5], Routh approximation  [6], Hurwitz polynomial approximation  [7,8], the stability equation method  [9,10], moments matching [11–14], the continued fraction method  [15–17], Pade approximation  [18] and etc.
The issue of optimality in model reduction was considered by Wilson  [19,20], who suggested an optimization approach based on minimization of the integral squared impulse response error between full and reduced-order models. This attempt was continued by other researchers through other approaches  [21–24].
In 1981  [25], the controllability and observability of the states were considered in model reduction by Moore. The suggested approach suffered from steady state errors but the stability of the reduced model was assured if the original system was also stable  [26]. Furthermore, the concept of , , and were used for model reduction in  [27–30].
In recent decades, evolutionary techniques, such as Particle Swarm Optimization (PSO) and Genetic Algorithm (GA), have been used for order reduction of systems  [31–33]. In these approaches, the reduced order model parameters are achieved by minimizing a fitness function, which is often Integral Square Error (ISE), Integral Absolute Error (IAE), norm or norm  [34–36].
This paper proposes an alternative method for order reduction based on Chebyshev rational functions, using the HS algorithm. The full order system is expanded and then the first coefficients of Chebyshev rational function expansion are obtained. A desire fixed structure for the reduced order model is considered and a set of parameters are defined, whose values determine the reduced order system. These unknown parameters are determined using the harmony search algorithm by minimizing the errors between the first coefficients of Chebyshev rational function expansion of full and reduced systems. To assure stability, the Routh criterion is applied, as used in  [37], where it states optimization problems as constraints, which, subsequently, are converted to constrained optimization problems. To show the accuracy of the proposed method, three systems are reduced by the proposed method and compared with those available in the literature.
To make a proper background, Chebyshev rational functions and the harmony search are briefly explained in Sections  2 and 3, respectively. The proposed method is explained in Section  4. The ability of the proposed approach is shown in Section  5, and, finally, the paper is concluded in Section  6.