Application of fractional derivatives for obtaining new Tikhonov regularization matrices

A linear discrete ill-posed problem has a perturbed right-hand side vector and an ill-conditioned coefficient matrix. The solution to such a problem is very sensitive to perturbation. Replacement of the coefficient matrix by a nearby one that has less condition number is one of the well-known approaches for decreasing the sensitivity of the problem to perturbation. This work is intended to obtain some new Tikhonov regularization matrices based on the discretization of fractional derivatives such as Grunwald–Letnikov and Caputo. The new regularization matrices are the extension of the classic regularization ones based on first and second derivatives. One of the advantages of this work is to achieve the null spaces of new matrices explicitly which are generally used to show the uniqueness of the regularized solution. Numerical results indicate the efficiency and effectiveness of the new matrices compared to the classic ones.