Sparse multiscale representation of Galerkin method for solving linear-mixed Volterra-Fredholm integral equations

This paper presents an efficient method for solving the linear-mixed Volterra-Fredholm equations using multiscale transformation. For this purpose, by changing the variables, the Fredholm-Volterra equation is discretized using wavelet Galerkin method. This equation reduces to a set of linear algebraic equations by using the wavelet transform matrix and the operational matrix of integration. To reach the sparse coefficients matrix for having a reduction in the computational cost, thresholding is used. This sparse system solves by generalized minimal residual (GMRES) method. If the appropriate threshold selects, the number of nonzero coefficients reduces while the error will not be less than a certain amount. The convergence analysis has been investigated. The validity and applicability of the technique are illustrated by a series of numerical tests.