This paper presents an extension to the Cole–Hopf barycentric Gegenbauer
integral pseudospectral (PS) method (CHBGPM) presented in Elgindy
and Dahy [High-order numerical solution of viscous Burgers’ equation using
a Cole–Hopf barycentric Gegenbauer integral pseudospectral method, Math.
Methods Appl. Sci. 41 (2018), pp. 6226–6251] to solve an initial-boundary
value problem of Burgers’ typewhenthe boundary function k defined at the
right boundary of the spatial domain vanishes at a finite set of real numbers
or on a single/multiple subdomain(s) of the solution domain. We present a
new strategy that is computationally more efficient than that presented in
[12] in the former case, and can be implemented successfully in the latter
case when the method of [12] fails to work. Moreover, fully exponential convergence
rates are still preserved in both spatial and temporal directions if
the boundary function k is sufficiently smooth. Numerical comparisons with
other traditional methods in the literature are presented to confirm the efficiency
of the proposed method. A numerical study of the condition number
of the linear systems produced by the method is included.

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.

This paper focuses on presenting an accurate, stable, efficient, and fast pseudospectral
method to solve tempered fractional differential equations (TFDEs) in both spatial and temporal
dimensions. We employ the Chebyshev interpolating polynomial for g at Gauss–Lobatto (GL) points
in the range [?1, 1] and any identically shifted range. The proposed method carries with it a recast
of the TFDE into integration formulas to take advantage of the adaptation of the integral operators,
hence avoiding the ill-conditioning and reduction in the convergence rate of integer differential
operators. Via various tempered fractional differential applications, the present technique shows
many advantages; for instance, spectral accuracy, a much higher rate of running, fewer computational
hurdles and programming, calculating the tempered-derivative/integral of fractional order, and its
spectral accuracy in comparison with other competitive numerical schemes. The study includes
stability and convergence analyses and the elapsed times taken to construct the collocation matrices
and obtain the numerical solutions, as well as a numerical examination of the produced condition
number $k(A)% of the resulting linear systems. The accuracy and efficiency of the proposed method are
studied from the standpoint of the $L_2$ and $L_{\infty}$-norms error and the fast rate of spectral convergence.