This paper presents an accurate exponential tempered fractional spectral collocation method
(TFSCM) to solve one-dimensional and time-dependent tempered fractional partial differential
equations (TFPDEs). We use a family of tempered fractional Sturm–Liouville
eigenproblems (TFSLP) as a basis and the fractional Lagrange interpolants (FLIs) that generally
satisfy the Kronecker delta (KD) function at the employed collocation points. Firstly,
we drive the corresponding tempered fractional differentiation matrices (TFDMs). Then, we
treat with various linear and nonlinear TFPDEs, among them, the space-tempered fractional
advection and diffusion problem, the time-space tempered fractional advection–diffusion
problem (TFADP), the multi-term time-space tempered fractional problems, and the timespace
tempered fractional Burgers’ equation (TFBE) to investigate the numerical capability
of the fractional collocation method. The study includes a numerical examination of the
produced condition number $\kappa (A)$ of the linear systems. The accuracy and efficiency of the
proposed method are studied from the standpoint of the $L^{\infty}$-norm error and exponential rate
of spectral convergence.