RATIONAL SOLUTIONS FOR THE TIME-FRACTIONAL DIFFUSION EQUATION

A uniform rational approximation of the Mittag-Leffler function is derived which serves as a global approximant, accounting for both the Taylor series for small arguments and asymptotic series for large arguments. Using a spectral decomposition of the time-fractional diffusion equation, applying Laplace and Fourier transforms, in conjunction with the rational approximation, yields a simple closed form solution to this equation. Furthermore, the approximation is inverted to yield a global rational approximation of the inverse Mittag-Leffler function. Last, we apply Mellin transforms to solve a time-fractional boundary value problem.