Numerical solution of fractional partial differential equations with variable coefficients using generalized fractional-order Legendre functions

In this paper, a general formulation for the generalized fractional-order Legendre functions (GFLFs) is constructed to obtain the numerical solution of fractional partial differential equations with variable coefficients. The special feature of the proposed approach is that we define generalized fractional order Legendre functions over [0,h][0,h] based on fractional-order Legendre functions. We use these functions to approximate the unknown function on the interval [0,h]×[0,l][0,h]×[0,l]. In addition, the GFLFs fractional differential operational and product matrices are driven. These matrices combine with Tau method to transform the problem to solve systems of linear algebraic equations. By solving the linear algebraic equations, we can obtain the numerical solution. The error analysis shows that the algorithm is convergent. The method is tested on examples. The results show that the GFLFs yields better results.