3,888 research outputs found
Differential quadrature method for space-fractional diffusion equations on 2D irregular domains
In mathematical physics, the space-fractional diffusion equations are of
particular interest in the studies of physical phenomena modelled by L\'{e}vy
processes, which are sometimes called super-diffusion equations. In this
article, we develop the differential quadrature (DQ) methods for solving the 2D
space-fractional diffusion equations on irregular domains. The methods in
presence reduce the original equation into a set of ordinary differential
equations (ODEs) by introducing valid DQ formulations to fractional directional
derivatives based on the functional values at scattered nodal points on problem
domain. The required weighted coefficients are calculated by using radial basis
functions (RBFs) as trial functions, and the resultant ODEs are discretized by
the Crank-Nicolson scheme. The main advantages of our methods lie in their
flexibility and applicability to arbitrary domains. A series of illustrated
examples are finally provided to support these points.Comment: 25 pages, 25 figures, 7 table
A Class of Second Order Difference Approximation for Solving Space Fractional Diffusion Equations
A class of second order approximations, called the weighted and shifted
Gr\"{u}nwald difference operators, are proposed for Riemann-Liouville
fractional derivatives, with their effective applications to numerically
solving space fractional diffusion equations in one and two dimensions. The
stability and convergence of our difference schemes for space fractional
diffusion equations with constant coefficients in one and two dimensions are
theoretically established. Several numerical examples are implemented to
testify the efficiency of the numerical schemes and confirm the convergence
order, and the numerical results for variable coefficients problem are also
presented.Comment: 24 Page
An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains
In this paper, we propose a novel unstructured mesh control volume method to
deal with the space fractional derivative on arbitrarily shaped convex domains,
which to the best of our knowledge is a new contribution to the literature.
Firstly, we present the finite volume scheme for the two-dimensional space
fractional diffusion equation with variable coefficients and provide the full
implementation details for the case where the background interpolation mesh is
based on triangular elements. Secondly, we explore the property of the
stiffness matrix generated by the integral of space fractional derivative. We
find that the stiffness matrix is sparse and not regular. Therefore, we choose
a suitable sparse storage format for the stiffness matrix and develop a fast
iterative method to solve the linear system, which is more efficient than using
the Gaussian elimination method. Finally, we present several examples to verify
our method, in which we make a comparison of our method with the finite element
method for solving a Riesz space fractional diffusion equation on a circular
domain. The numerical results demonstrate that our method can reduce CPU time
significantly while retaining the same accuracy and approximation property as
the finite element method. The numerical results also illustrate that our
method is effective and reliable and can be applied to problems on arbitrarily
shaped convex domains.Comment: 18 pages, 5 figures, 9 table
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