A new iterative method for solving linear Fredholm integral equations using the least squares method

In this paper, a new iterative method is proposed for solving linear integral equations. This method is based on the LSQR method, an algorithm for sparse linear equations and sparse least squares, reducing the solution of linear integral equations to the solution of a bidiagonal linear system of algebraic equations. A simple recurrence formula is presented for generating the sequence of approximate solutions. Some theoretical properties and error analysis of the new method are discussed. Although the new method can be used for solving the ill-posed first kind integral equations independently, combining of the new method with the method of regularization is presented to solve this kind of integral equations. Also the perturbing effect of the first kind integral equations is analyzed. Some properties and convergence theorem are proposed. Finally, some numerical examples are presented to show the efficiency of the new method.