Finite difference methods for two-dimensional fractional dispersion equation

By Mark M. Meerschaert, Hans-Peter Scheffler, Charles Tadjeran. 2006

433 citations recorded by the source corpus.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

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Citations in the corpus

PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs (2019, methodology): “To validate the above hypothesis, we replace the PINN with the finite difference method [22] (FDM, grid size: x × y = 20 × 20, time step δt = 0.”
A Simple Solver for the Fractional Laplacian in Multiple Dimensions (2018, methodology): “…and fast preconditioners based on multigrid, but their scheme uses the so-called “coordinate fractional Laplacian” that takes a tensor product of one-dimensional operators and is not equivalent to (1) (see also related ﬁnite-diﬀerence approaches with diﬀerent operators [36,37,51,58]).”
Numerical approximation of Lévy-Feller diffusion equation and its probability interpretation (2007, methodology): “[25] derived practical numerical methods to solve two-dimensional fractional dispersion equations with variable coefficients on a finite domain and obtain the first order accuracy in space and time.”
Some applications of fractional order calculus (2010, methodology): “The shifted Grünwald-Letnikov definition of fractional order differ-integral is given as follows: [17] SGL aD α t x(t) = lim h→0 1 hα t/h+1”
A Fast Finite Difference Method for Three-Dimensional Time-Dependent Space-Fractional Diffusion Equations with Fractional Derivative Boundary Conditions (2017, methodology): “We consider the three-dimensional time-dependent space-fractional PDE of order 1 < α, β, γ < 2 that is enclosed with the homogeneous Dirichlet and fractional derivative boundary conditions [1,19,20,31]”
Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann–Liouville derivative (2014, methodology): “Nevertheless, various methods such as Laplace transform [5], Fourier transform [8], variational iteration [12], finite difference [13], Adomian decomposition [14], homotopy perturbation [15,16], and generalized differential transform [17]methods, have been used to solve these equations and have achi”
Algorithms for nonlinear fractional partial differential equations: A selection of numerical methods (2008, background): “When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion (also called superdiffusion) [11], [28].”
A second-order accurate numerical method for the two-dimensional fractional diffusion equation (2007, methodology): “[24] to conclude that the mixed fractional derivative term 1⁄2da;xdb;y ui;j is O(Dx) + O(Dy) uniformly for ðx; yÞ 2 R(2).”
Invariant analysis of time fractional generalized Burgers and Korteweg–de Vries equations (2012, methodology): “As a consequence, several ad hoc methods such as the Finite difference method [16], Adomian decomposition method [17], Variational iteration method [18], Homotopy perturbation method [19], Laplace transformmethod [20], Fourier transformmethod [14], Generalized differential transformmethod [21]”
Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients (2015, methodology): “In the last decade or so, extensive research has been carried out on the development of numerical methods for fractional partial differential equations, including finite difference method [11–13], finite element methods [14, 15], and spectral methods [16].”
Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2 (2006, result): “In [17], the results in [16] are extended to a more general FADE in 1-D with space and time-varying coefficients, and in [ 15 ] extended further to a special case (M is concentrated on the coordinate axes) of the 2-D FADE, but again the coefficients can vary with space and time.”