The Laguerre-Hermite spectral methods for the time-fractional sub-diffusion equations on unbounded domains
The paper studies the numerical solution of the following time-fractional sub-diffusion equations (FSDEs) on unbounded domains: \[ _0^C\partial_t^{\alpha}u(x,t)-\partial_x^{2}u(x,t)+u(x,t)=f(x,t),\qquad x \in \mathbb{R},\, t\in \Lambda, \] subject to the following conditions: \[ \lim\limits_{|x|\to \infty} u(x,t) = 0,\quad u(x,0)=\phi(x), \] where \(0 < \alpha < 1\) and \(\Lambda = (0,+\infty)\). The authors use the Laguerre-Hermite spectral Galerkin and collocation methods to solve the above problem. The generalized associated Laguerre functions of the first kind (GALFs-1) and Hermite functions (HFs) are introduced as basis functions in time and spatial space, respectively. Some basic properties and definitions of the confluent hypergeometric functions, generalized associated Laguerre functions, Hermite functions, fractional calculus, and Laplace transform are given. In the time domain, the fractional derivative of the basis functions has derived in the Galerkin method by using the Laplace transformation. The collocation method expands the solution in terms of suitable global basis functions. The formulation of the Laguerre-Galerkin method and the spectral collocation methods based on GALFs-1 are given and the convergence behavior of numerical solutions is tested by some simple test problems. The error analysis is carried out separately on time and space domains, and also the time-space error is estimated. Lastly, some numerical results are given to validate the methods.
generalized associated Laguerre functions
scientific article; zbMATH DE number 7137009
scientific article