Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2

By John Paul Roop. 2006

285 citations recorded by the source corpus.

Subfields: Mechanics of Materials, Modeling and Simulation, Numerical Analysis

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Citations in the corpus

fPINNs: Fractional Physics-Informed Neural Networks (2018, neutral): “As an example, we consider the 1D steady-state fractional diffusion problem [64]: ( ∂ ∂x Dα−1 0+ − ∂ ∂x Dα−1 1− ) u(x) = f(x), x ∈ (0, 1), α ∈ (1, 2),”
Adaptive finite element method for fractional differential equations using hierarchical matrices (2016, methodology): “5, 32, 28]. However, as the size of the problem increases, the time required to solve the nal system of equations increases considerably due to the nonlocality of the fractional dierential operators [43, 23, 10, 11, 30, 9, 4, 36, 37, 40, 26]. Generally, there are two main diculties in solving space ”
Time two-mesh algorithm combined with finite element method for time fractional water wave model (2018, background): “[23] J.P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2, J. Comput.”
Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations (2016, background): “In particular, pioneer work with finite element analysis is carried out in [16] and [8], and fast algorithms for some FDEs have been developed in [22,23,14,10], among others.”
A second-order accurate numerical method for the two-dimensional fractional diffusion equation (2007, background): “[12,13,15,21] treat explicit or implicit finite difference methods for space fractional diffusion equations, and [9,29] examine finite element methods for such problems.”
Fourier spectral methods for fractional-in-space reaction-diffusion equations (2014, background): “Even the construction off such matrices presents difficulties, especially in efficiency, see, for example, [14].”
A characteristic difference method for the variable-order fractional advection-diffusion equation (2013, unknown): “[27], Roop [23], Zhou et al.”
The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation (2013, methodology): “Similar to the numerical methods for the classical differential equations, there also exist finite difference methods [3, 10, 12, 13, 20, 24, 26, 27, 31, 34, 51, 52, 54, 55, 62], finite element methods (FEMs) [16, 48, 59, 61], and spectral methods [6, 25, 28] for numerically solving the FDEs.”
TEMPERED FRACTIONAL CALCULUS (2015, methodology): “A wide variety of effective numerical methods have been developed to solve fractional diffusion equations, see for example [18,21,32–34,56,64,68].”
A Fourier method for the fractional diffusion equation describing sub-diffusion (2007, methodology): “Some numerical methods for solving the space or time, or time-space fractional partial differential equations have been proposed [15],[16],[17], [18], [19], [20], [21], [22], [23], [24].”
Numerical approximation of Lévy-Feller diffusion equation and its probability interpretation (2007, methodology): “Roop [31] investigated the computational aspects of finite-element approximation of fractional advection dispersion equation using continuous piecewise polynomial basis functions on a regular triangulation of the bounded domain in R.”
A unified Petrov-Galerkin spectral method for fractional PDEs (2015, background): “, see [47,29]), one can easily show that it satisfies all the properties of a norm since the Riemann–Liouville fractional derivative of a constant is non-zero.”