This paper develops two tempered fractional matrices that are computationally accurate, efficient,
and stable to treat myriad tempered fractional differential problems. The suggested approaches are
versatile in handling both spatial and temporal dimensions and treating integer- and fractionalorder
derivatives as well as non-tempered scenarios via utilizing pseudospectral techniques. We
depend on Lagrange basis functions, which are derived from the tempered Jacobi-Müntz functions
based on the left- and right-definitions of Erdélyi-Kober fractional derivatives. We aim to obtain
the pseudospectral-tempered fractional differentiation matrices in two distinct ways. The study
involves a numerical measurement of the condition number of tempered fractional differentiation
matrices and the time spent to create the collocation matrices and find the numerical solutions.