7,488 research outputs found
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
Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
Over the past few decades, there has been substantial interest in evolution
equations that involving a fractional-order derivative of order
in time, due to their many successful applications in
engineering, physics, biology and finance. Thus, it is of paramount importance
to develop and to analyze efficient and accurate numerical methods for reliably
simulating such models, and the literature on the topic is vast and fast
growing. The present paper gives a concise overview on numerical schemes for
the subdiffusion model with nonsmooth problem data, which are important for the
numerical analysis of many problems arising in optimal control, inverse
problems and stochastic analysis. We focus on the following aspects of the
subdiffusion model: regularity theory, Galerkin finite element discretization
in space, time-stepping schemes (including convolution quadrature and L1 type
schemes), and space-time variational formulations, and compare the results with
that for standard parabolic problems. Further, these aspects are showcased with
illustrative numerical experiments and complemented with perspectives and
pointers to relevant literature.Comment: 24 pages, 3 figure
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