This paper considers a non-polynomial method for solving time-fractional diffusion equations on unbounded domains without discretization. First, the Caputo fractional derivative is transformed into the Riemann–Liouville derivative. The Galerkin approximation is used for the aforesaid equation in the variational form. For both time and space directions, specific types of functions are assumed separately and formed based on Laguerre polynomials. Based on fractional derivatives of this group of functions and the relations between them, the variational form equation is transformed into a linear sparse system of equations, in which the coefficient matrix is calculated explicitly and without assuming the specific computational cost. Unlike the use of classical polynomial functions, which leads to a full system and does not give an approximation for singular solutions, implementing fractional basis functions has been an efficient approach toward these drawbacks. Numerical results highlight the accuracy and efficiency of the proposed method.

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