A numerical method for solving variable‐order fractional diffusion equations using fractional‐order Taylor wavelets

This paper aims to provide a new numerical method for solving variable‐order fractional diffusion equations. The method is constructed using fractional‐order Taylor wavelets. By using the regularized beta function, a formula for computing the exact value of Riemann‐Liouville fractional integral operator of the fractional‐order Taylor wavelets is given. The Taylor wavelets properties and the formula are used in combination with a spectral collocation method to reduce the given diffusion equation to a system of algebraic equations. The method is easy to implement, and gives very accurate solutions. Several examples are given to show the applicability and the effectiveness of the method.