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In this paper, an orthogonal basis for the space of linear splines based on linear B-splines is introduced. Some properties and modifications of this basis are investigated, then operational matrices of integration in collocation points are obtained using stable formula. Theoretical considerations are discussed. Applications to the numerical solutions for some linear and nonlinear inverse problems are given, in which the approximations are obtained using the first and second integrals of orthogonal linear splines that lead to an efficient solution procedure. For solving linear problems, the proposed method is combined with the Tikhonov regularization. Also, the trust-region-dogleg method is used for nonlinear equations. Numerical results validate the effectiveness of the orthogonal linear spline basis for linear and nonlinear inverse problems.

Some properties of orthogonal linear splines and their applications to inverse problems