However, each of these methods has its inherent advantages and disadvantages. Iterative method was used to approximate the solutions of the nonlinear Fredholm integral equations (1) by transforming the integral equation into a discretized form. Literature presented a computational method for solving nonlinear Fredholm integral equations of the second kind which is based on the use of Haar wavelets. Optimal homotopy asymptotic method (OHAM) was introduced in literature as a reliable and efficient technique for finding the solutions of integral equations. Harmonic wavelet method was employed as basis functions in the collocation method towards approximate solutions of the Fredholm type integral equations. Triangular functions method (TFM) was utilized as a basis in collocation method to reduce the solutions of nonlinear Fredholm integral equations to the solutions of algebraic equations by using the optimal coefficients. The accuracy of PDFM relies on the choices of radial basis functions. Positive definite function method (PDFM) is based on interpolation by radial basis functions (RBFs) to approximate the solutions of the nonlinear Fredholm integral equations. Homotopy perturbation method (HPM) introduced by was used to solve the nonlinear integral equations (1). Discrete Adomian decomposition method (DADM) arose when the quadrature rules are used to approximate the definite integrals which can not be computed analytically. The results of algorithm depend explicitly on the selection of the starting interval which decreases the speed of convergence substantially. Literature applied a different method (DM) to solve the equation (1) by convert the problem to an optimal control problem based on introducing an artificial control function. Many different numerical and approximate techniques were introduced to obtain the solutions of nonlinear integral equations. Some deterministic techniques have been applied to obtaining solutions of the nonlinear integral equations during the past several years. Consequently, our aim here is to find the unknown function which is the solution of problem (1). However, it is difficult to solve the nonlinear kinds with them, (1)especially analytically, where is a known continuous function over the interval, is a known nonlinear function with respect to and is a kernel function which is known and continuous too, at the same time bounded on the square which is the upper bound on the square. These proposed methods of literatures – have been used to discuss the linear Fredholm integral equations. Many numerical computational methods including deterministic techniques and Monte Carlo methods are introduced in this area ( – ). The problem of solving numeric solutions for the linear Fredholm integral equations of the second kinds is one of the oldest problems in the applied mathematics literature. In most cases, the problem of finding numeric solutions for the nonlinear Fredholm integral equations is more difficult than the problem of the linear Fredholm integral equations. Therefore, numeric solutions of integral equations have been a subject of great interest of many researchers. Integral equations are usually difficult to solve analytically. It is well known that linear and nonlinear integral equations arise in many scientific fields such as the population dynamics, spread of epidemics, thermal control engineering, and semi-conductor devices. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Ĭompeting interests: The authors have declared that no competing interests exist. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.įunding: The paper is supported by National Natural Science Foundation of China (11161031), Specialized Research Fund for the Doctoral Program of Higher Education (20131514110005) and Natural Science Foundation of Inner Mongolia (2013MS0108). Received: NovemAccepted: JPublished: July 29, 2014Ĭopyright: © 2014 Hong et al. PLoS ONE 9(7):Įditor: Guido Germano, University College London, United Kingdom Citation: Hong Z, Yan Z, Yan J (2014) Random Search Algorithm for Solving the Nonlinear Fredholm Integral Equations of the Second Kind.
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