Tempered Fractional Sturm--Liouville EigenProblems

Abstract

Continuum-time random walk is a general model for particle kinetics, which allows for incorporating waiting times and/or non-Gaussian jump distributions with divergent second moments to account for Lévy flights. Exponentially tempering the probability distribution of the waiting times and the anomalously large displacements results in tempered-stable Lévy processes with finite moments, where the fluid (continuous) limit leads to the tempered fractional diffusion equation. The development of fast and accurate numerical schemes for such nonlocal problems requires a new spectral theory and suitable choice of basis functions. In this study, we introduce two classes of regular and singular tempered fractional Sturm--Liouville problems of two kinds (TFSLP-I and TFSLP-II) of order $\nu \in (0,2)$. In the regular case, the corresponding tempered differential operators are associated with tempering functions $p_I(x) = \exp(2\tau) $ and $p_{II}(x) = \exp(-2\tau)$, $\tau \geq 0$, respectively, in the regular TFSLP-I and TFSLP-II, which do not vanish in $[-1,1]$. In contrast, the corresponding differential operators in the singular setting are associated with different forms of $p_I(x) = \exp( 2\tau) (1-x)^{1+\alpha} (1+x)^{1+\beta}$ and $p_{II}(x) = \exp( -2\tau) (1-x)^{1+\alpha} (1+x)^{1+\beta}$, vanishing at $x=\pm 1$ in the singular TFSLP-I and TFSLP-II, respectively. The aforementioned tempered fractional differential operators are both of tempered Riemann--Liouville and tempered Caputo type of fractional order $\mu = \nu/2 \in (0,1)$. We prove the well-posedness of the boundary-value problems and that the eigenvalues of the regular tempered problems are real-valued and the corresponding eigenfunctions are orthogonal. Next, we obtain the explicit eigensolutions to TFSLP-I and -II as nonpolynomial functions, which we define as tempered Jacobi poly-fractonomials. These eigenfunctions are orthogonal with respect to the weight function associated with TFSLP-I and -II. Finally, we introduce these eigenfunctions as new basis (and test) functions for spectrally accurate approximation of functions and tempered-fractional differential operators. To this end, we further develop a Petrov--Galerkin spectral method for solving tempered fractional ODEs, followed by the corresponding stability and convergence analysis, which validates the achieved spectral convergence in our simulations.