A fractional spectral method with applications to some singular problems
The authors introduce a novel type of projection operators that map into a class of generalized polynomials with non-integer exponents. Working in certain weighted Sobolev spaces, the mapping properties and -- in particular -- the approximation rates of these projections are investigated. These operators are then used as building blocks for solvers for fractional order differential, integral and integro-differential equations. It turns out that they can, for suitably chosen parameters, give rise to exponential convergence rates even in cases where the exact solutions exhibit the types of nonsmooth behaviour at the initial point that one usually finds in differential equations of fractional order. Moreover, it is shown that it is possible to employ such projection operators to more standard problems like a classical elliptic differential equation. In this case it may also happen that the given data lead to nonsmooth behaviour of the solution that can be compensated by the authors' approach in a way that drastically improves the convergence rate.
scientific article; zbMATH DE number 6816259
scientific article