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In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

It is a long-standing problem to preserve fine scale features such as texture in the process of deblurring. In order to deal with this challenging but imperative issue, we establish a framework of nonlinear fractional diffusion equations, which performs well in deblurring images with textures. In the new model, a fractional gradient is used for regularization of the diffusion process to preserve texture features and a source term with blurring kernel is used for deblurring. This source term ensures that the model can handle various blurring kernels. The relation between the regularization parameter and the deblurring performance is investigated theoretically, which ensures a satisfactory recovery when the blur type is known. Moreover, we derive a digital fractional diffusion filter that lives on images. Experimental results and comparisons show the effectiveness of the proposed model for texturepreserving deblurring.

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