A new numerical method for solving two-dimensional variable-order anomalous sub-diffusion equation

The novelty and innovativeness of this paper are the combination of reproducing kernel theory and spline, this leads to a new simple but effective numerical method for solving variable-order anomalous sub-diffusion equation successfully. This combination overcomes the weaknesses of piecewise polynomials that can not be used to solve differential equations directly because of lack of the smoothness. Moreover, new bases of reproducing kernel spaces are constructed. On the other hand, the existence of any e-approximate solution is proved and an effective method for obtaining the e-approximate solution is established. A numerical example is given to show the accuracy and effectiveness of theoretical results.