Numerical Solution of the Mixed Volterra-Fredholm Integral Equations Using the Bivariate Shifted Legendre Functions

new method is proposed for the numerical solution of the linear two-dimensional mixed Volterra-Fredholm integral equations. The proposed numerical algorithm uses the operational matrices of the bivariate shifted Legendre functions. A new operational matrix of the bivariate shifted Legendre functions is introduced. The method reduces the mixed Volterra-Fredholm integral equation to the solution of a system of linear algebraic equations. Finally, the method is tested on a problem and numerical results are given. The numerical results confirm that the computational method provides high accurate results.