Solving Fredholm Integral Equations with Bernstein Multi-Scaling Functions‎

In this article‎, ‎the efficient numerical methods for finding solution of the linear and nonlinear Fredholm integral equations of the second kind on base of Bernstein multi scaling functions are being presented‎. ‎In the beginning the properties of these functions‎, ‎which are a combination of block-pulse functions on ‎, ‎and Bernstein polynomials with the dual operational matrix are presented‎. ‎Then these properties are used for the purpose of conversion of the mentioned integral equation to a matrix equation that are compatible to a algebraic equations system‎. ‎The imperative of the Bernstein multi scaling functions are‎, ‎for the proper quantitative value of and have a high accuracy and specifically the relative errors of the numerical solutions will be minimum‎. ‎The presented methods from the standpoint of computation are very simple and attractive and the numerical examples which were presented at the end shows the efficiency and accuracy of these methods‎.