In this paper, we consider numerical methods for two-sided space variable-order fractional diffusion equations (VOFDEs) with a nonlinear source term. The implicit Euler (IE) method and a shifted Grünwald (SG) scheme are used to approximate the temporal derivative and the space variable-order (VO) fractional derivatives, respectively, which leads to an IE-SG scheme. Since the order of the VO derivatives depends on the space and the time variables, the corresponding coefficient matrices arising from the discretization of VOFDEs are dense and without the Toeplitz-like structure. In light of the off-diagonal decay property of the coefficient matrices, we consider applying the preconditioned generalized minimum residual methods with banded preconditioners to solve the discretization systems. The eigenvalue distribution and the condition number of the preconditioned matrices are studied. Numerical results show that the proposed banded preconditioners are efficient.

Recent advances have been made in a wide range of imaging and pattern recognition applications, including picture categorization and object identification systems. These systems necessitate a robust feature extraction method. This study proposes a new class of orthogonal functions known as orthogonal mountain functions (OMFs). Using these functions, a novel set of orthogonal moments and associated scaling, rotation, and translation (SRT) invariants are presented for building a color image’s feature vector components. These orthogonal moments are presented as quaternion orthogonal mountain Fourier moments (QOMFMs). To demonstrate the validity of our theoretically recommended technique, we conduct a number of image analysis and pattern recognition experiments, including a comparison of the performance of the feature vectors proposed above to preexisting orthogonal invariant moments. The result of this study experimentally proves the effectiveness and quality of our QOMFMs.