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Isogeometric analysis for time-fractional partial differential equations. (English) Zbl 1450.65123
Summary: We consider isogeometric analysis to solve the time-fractional partial differential equations: fractional diffusion and diffusion-wave equations. Traditional spatial discretization for time-fractional models include finite differences, finte elements, spectral methods, etc. A novel method-isogeometric analysis is used for spatial discretization in this paper. The traditional \(L1\) scheme and \(L2\) scheme are used for time discretization of our models. Isogeometric analysis has potential advantages in exact geometry representations, efficient mesh generation, \(h\)- and \(k\)- refinements, and smooth basis functions. We show stability and a priori error estimates for spatial discretization and the space-time fully discrete scheme. A variety of numerical examples in 2d and 3d are provided to verify theory and show accuracy, efficiency, and convergence of isogeometric analysis based on B-splines and non-uniform rational B-splines.
MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65D07 Numerical computation using splines
26A33 Fractional derivatives and integrals
74S05 Finite element methods applied to problems in solid mechanics
35R11 Fractional partial differential equations
Software:
FODE; GeoPDEs
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