Abstract
In this study, a computational method for solving linear FredholmIntegro-Differential Equation (FIDE) of the first order under the mixed conditions using the Bernstein polynomials. First, we present some properties of these polynomials and the method is explained. These properties are then used to convert the integro-differential equation to a system of linear algebraic equations with unknown Bernstein coefficients. Using Galerkin method, we give an approximate solution. This method seems very attractive and simple to use. Illustrative examples show the efficiency and validity of the method we discuss the results using error analysis, the results are discussed.
Highlights
Integro-differential equations, which are composed by integral and differential equations, are a well-known mathematical tool and an important branch of modern mathematics
FredholmIntegroDifferential Equations (FIDE) are encountered in several areas such as biology, economics, engineering and many others, so as usual, there is no specific analytic method to solve this equations, several numerical methods are presented to approximate the solution of FIDEs
Various numerical methods take an important place in solving FIDE numerically, such as Legendre polynomials, which have been used for high-order linear FIDE (Yalçinbaş et al, 2009), rationalized Haar functions and Walsh series, differential transform method (Golubov et al, 1991) and many other known methods in the literature
Summary
Integro-differential equations, which are composed by integral and differential equations, are a well-known mathematical tool and an important branch of modern mathematics. Various numerical methods take an important place in solving FIDE numerically, such as Legendre polynomials, which have been used for high-order linear FIDE (Yalçinbaş et al, 2009), rationalized Haar functions and Walsh series, differential transform method (Golubov et al, 1991) and many other known methods in the literature. Among these methods, the polynomials of Bernstein that have been widely used to solve both linear and non linear integro-differential equations. Where, C and B are (m + 1)×1 matrices and are given by:
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