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This paper introduces a new class of fractional-order anisotropic diffusion equations for noise removal. These equations are Euler-Lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function, so the proposed equations can be seen as generalizations of second-order and fourth-order anisotropic diffusion equations. We use the discrete Fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain. It is one important aspect of the algorithm that it considers the input image as a periodic image. To overcome this problem, we use a folded algorithm by extending the image symmetrically about its borders. Finally, we list various numerical results on denoising real images. Experiments show that the proposed fractional-order anisotropic diffusion equations yield good visual effects and better signal-to-noise ratio.