Bernoulli Galerkin Matrix Method and Its Convergence Analysis for Solving System of Volterra–Fredholm Integro-Differential Equations

The principle aim of this paper is to find the numerical solution for system of Volterra–Fredholm integro-differential equations by using the Bernoulli polynomials and the Galerkin method. Through this scheme, the main problem will be transformed to a system of algebraic equations which its solutions are depend on the unknown Bernoulli coefficients. This method gives an analytic solution for systems with polynomial function solution. Better accuracy will be obtained by increasing the number of Bernoulli polynomials. Also, a mathematical proof for its convergence is provided. Moreover, some examples are presented and their numerical results are compared to the results of the Bessel collocation method to show the validity and applicability of this algorithm.