Least squares finite-element solution of a fractional order two-point boundary value problem

Abstract

In this paper, a theoretical framework for the least squares finite-element approximation of a fractional order differential equation is presented. Mapping properties for fractional dimensional operators on suitable fractional dimensional spaces are established. Using these properties existence and uniqueness of the least squares approximation is proven. Optimal error estimates are proven for piecewise linear trial elements. Numerical results are included which confirm the theoretical results.

Variational formulation for the stationary fractional advection dispersion equation

“…Finite difference have also been applied to construct numerical approximation [9]. Aside from [5] we are not aware of any other papers in the literature which investigate the Galerkin approximation and associated error analysis for the FADE.…”

Infinitely many solutions for boundary value problems arising from the fractional advection dispersion equation

“…The interest in the physical models containing fractional differential operators is mainly due to the fact that they can describe the diffusion phenomenon more accurately in complex dynamic systems involving anomalous diffusion [5], [11], [10]. In [11], Fix and Roop set up a model for contaminant transport of ground-water flow, and prove the existence and uniqueness of the least squares approximation for a steady-state fractional advection-dispersion equation (FADE for short) with no advection terms by the least squares finite element analysis.…”

Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations

“…We note that Definitions (4) and (5) are not equivalent [17]. For the deterministic space-fractional partial differential equations where the space-fractional derivative is defined by (5), or the Riemann-Liouville space-fractional derivative, or the Caputo space-fractional derivative, many numerical methods are available, for example, finite difference methods [18][19][20][21][22][23][24][25][26][27][28][29][30], finite element methods [14,[31][32][33][34][35][36][37][38][39][40] and spectral methods [41,42]. For the deterministic space-fractional partial differential equations where the space-fractional derivative is defined by (4), some numerical methods are also available, for example the matrix transfer method (MTT) [21,22,43] and the Fourier spectral method [44].…”