We present a novel high-order numerical method to solve viscous Burgers' equation with smooth initial and boundary data. The proposed method combines Cole-Hopf transformation with well-conditioned integral reformulations to reduce the problem into either a single easy-to-solve integral equation with no constraints or an integral equation provided by a single integral boundary condition. Fully exponential convergence rates are established in both spatial and temporal directions by embracing a full Gegenbauer collocation scheme based on Gegenbauer-Gauss mesh grids using apt Gegenbauer parameter values and the latest technology of barycentric Gegenbauer differentiation and integration matrices. The global collocation matrices of the reduced algebraic linear systems were derived allowing for direct linear system solvers to be used. Rigorous error and convergence analyses are presented in addition to two easy-to-implement pseudocodes of the proposed computational algorithms. We further show three numerical tests to support the theoretical investigations and demonstrate the superior accuracy of the method even when the viscosity parameter → 0, in the absence of any adaptive strategies typically required for adaptive refinements.

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This paper focuses on presenting an accurate, efficient, and fast pseudospectral method to solve tempered fractional differential equations (TFDEs) in both spatial and temporal dimensions. We use a tempered integration matrix that can be employed to evaluate 𝑛-fold tempered integrals of a real function 𝑔 for 𝑛 ∈ ℝ + . Also, it may be used to compute any non-integer order 𝜆 < 0 tempered derivatives of 𝑔. 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 actually 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 examples, the present technique shows many advantages, for instance, high accuracy, a much higher rate of running, fewer computational hurdles and programming, calculating the tempered-derivative/integral of fractional order and its exceptional accuracy in comparison with other competitive numerical schemes. The study includes the elapsed times taken to construct the collocation matrices and obtain the numerical solutions. Also, numerical examination of the produced condition number 𝜅(𝐴) of the resulted linear systems. The accuracy and efficiency of the proposed method are studied from the standpoint of the 𝐿 2 and 𝐿 ∞ -norms error and fast rate of spectral convergence.