Finite difference schemes for the tempered fractional Laplacian

Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian

“…The convergence rates are lower than desired ones because of the regularity of the exact solution u. These results are similar to the ones in one dimension [25].…”

“…For the tempered fractional Laplacian (1.1), the existing numerical methods at present are mainly analyzed in one dimension. Among them, [26] presents a Riesz basis Galerkin method for the tempered fractional Laplacian and gives the well-posedness proof of the Galerkin weak formulation and convergence analysis; [25] proposes a finite difference scheme and proves that the accuracy depends on the regularity of the exact solution on Ω rather than the regularity on the whole line. So far, its seems that there are no numerical analysis and implementation discussion on (1.1) in two dimension.…”

On tempered fractional calculus with respect to functions and the associated fractional differential equations

“…[16][17][18] Tempered fractional calculus is so important that it has been rediscovered at least twice under different names, as generalised proportional fractional calculus 19 and as substantial fractional calculus. [20][21][22] Other variants of fractional calculus using the same tempered concept have also been studied, such as tempered versions of the Riesz derivative and fractional Laplacian, [23][24][25] which are of great interest to mathematicians.…”