This paper is devoted to extend an iterative algorithm for sparse least-squares problems for solving one-and two-dimensional linear Fredholm integral equations. We consider the operator form of these equations and then develop the LSMR method for solving them in an appropriate manner. The proposed method is based on bidiagonalization process of Golub-Kahan and reducing the linear operator L to the lower bidiagonal matrix form. Convergence property of the numerical solution associated with the suggested scheme is also provided. The method is fast and very accurate in solving two-dimensional equations. Other advantage of the proposed method is its high accuracy for solving Fredholm integral equations with non-smooth solutions. Solving such problems with other numerical methods, for instance, spectral methods, are not accurate as our proposed LSMR method. Numerical results confirm this claim.

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