Tempered fractional Jacobi-Müntz basis for image reconstruction application and high-order pseudospectral tempered fractional differential matrices

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 fractional-order 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. The suggested matrices' accuracy and efficiency are investigated from the point of view of the L2, L∞-norms errors, the maximum absolute error matrix ‖EN,My‖∞ of the two-dimensional problems, and the fast rate of spectral convergence. Finally, numerical experiments are carried out to show the exponential convergence, applicability, effectiveness, speed, and potential of the suggested matrices.