208 research outputs found
Gauss-Jacobi-type quadrature rules for fractional directional integrals
Fractional directional integrals are the extensions of the RiemannāLiouville fractional integrals from one- to multi-dimensional spaces and play an important role in extending the fractional differentiation to diverse applications. In numerical evaluation of these integrals, the weakly singular kernels often fail the conventional quadrature rules such as NewtonāCotes and GaussāLegendre rules. It is noted that these kernels after simple transforms can be taken as the Jacobi weight functions which are related to the weight factors of GaussāJacobi and GaussāJacobiāLobatto rules. These rules can evaluate the fractional integrals at high accuracy. Comparisons with the three typical adaptive quadrature rules are presented to illustrate the efficacy of the GaussāJacobi-type rules in handling weakly singular kernels of different strengths. Potential applications of the proposed rules in formulating and benchmarking new numerical schemes for generalized fractional diffusion problems are briefly discussed in the final remarking section.postprin
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
Structure-Preserving Discretization of Fractional Vector Calculus using Discrete Exterior Calculus
Fractional vector calculus is the building block of the fractional partial
differential equations that model non-local or long-range phenomena, e.g.,
anomalous diffusion, fractional electromagnetism, and fractional
advection-dispersion. In this work, we reformulate a type of fractional vector
calculus that uses Caputo fractional partial derivatives and discretize this
reformulation using discrete exterior calculus on a cubical complex in the
structure-preserving way, meaning that the continuous-level properties
and
hold exactly on the
discrete level. We discuss important properties of our fractional discrete
exterior derivatives and verify their second-order convergence in the root mean
square error numerically. Our proposed discretization has the potential to
provide accurate and stable numerical solutions to fractional partial
differential equations and exactly preserve fundamental physics laws on the
discrete level regardless of the mesh size.Comment: 25 pages, 4 figure
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