  In this paper, we study a class of fractional $1$-Laplacian diffusion
equations with variable orders, proposed as a model for multiplicative noise
removal. The existence and uniqueness of the weak solution are proven. To
overcome the difficulties in the approximation process,we place particular
emphasis on studying the density propertiesof the variable-order fractional
Sobolev spaces. Numerical experiments demonstrate that our model exhibits
favorable performance across the entire image.
