Abstract
In this work, shifted fractional-order Jacobi orthogonal function in the interval [0, T] is outputted of the classical Jacobi polynomial (see Definition 2.3). Also, we list and derive some facts related to the shifted fractional-order Jacobi orthogonal function. Spectral collocation techniques are addressed to solve the multidimensional distributed-order diffusion equations (MDODEs). A mixed of shifted Jacobi polynomials and shifted fractional order Jacobi orthogonal functions are used as basis functions to adapt the spatial and temporal discretizations, respectively. Based on the selected basis, a spectral collocation method is listed to approximate the MDODEs. By means of the selected basis functions, the given conditions are automatically satisfied. We conclude with the application of spectral collocation method for multi-dimensional distributed-order diffusion equations.