High Accuracy Spectral Method For Some Singular Problems

Based on the advantages of Galerkin method and orthogonal polynomials,the usual spectral methods will provide high-order accuracy for problems with smooth solutions.However,they may not work well for problems with singular solutions due to various facts such as corner singularities,non-matching boundary conditions,non-smooth coeffi-cients,which restricts the applications of the spectral method.In order to recovery the high accuracy,we proposed several special spectral methods to deal with some singular problems.We split it into four parts below:1.Generalized Jacobi functions(GJFs)spectral method and their applications to a class of prototype fractional differential equations(FDEs).In this section,we define a new class of generalized Jacobi functions,which is intrinsically related to fractional calculus,and can serve as natural basis functions for properly designed spectral methods for FDEs.We establish spectral approximation results for these GJFs in weighted Sobolev spaces involving fractional derivatives.We construct efficient GJF-Petrov-Galerkin methods for a class of prototypical fractional initial value problems(FIVPs)and fractional boundary value problems(FBVPs)of general order.Moreover,we derive error estimates with convergence rate only depending on the smoothness of data,so truly spectral accuracy can be attained if the data are smooth enough.2.Generalized Laguerre functions(GLFs)spectral method and their applications to Tempered fractional diffusion equations.We review Tempered fractional derivatives originated from the tempered fractional diffusion equations(TFDEs)modeled on the whole space R(see[78]).For numerically solving TFDEs,generalized Laguerre functions were defined and some important properties were proposed to establish the approximate theory.The related prototype tempered fractional differential problems was proposed and solved as the guidance.TFDEs are numerically solved by two domains Laguerre spectral method and the numerical experiments show some properties of the TFDEs and verify the efficiency of the spectral scheme.3.Extended spectral method(ESM)and their applications to several singular prob-lems.The usual spectral methods will provide high-order accuracy for problems with smooth solutions.However,they may not work well for problems with singular solutions,which restricts the applications of the spectral method.For many singular problems,it is possible to determine the forms of a few leading singular terms.It is expected that we can improve the convergence rate by adding these singular terms into the spectral basis.However,the new system with added singular terms is usually ill conditioned and hard to solve.We present a new extended spectral-Galerkin method,to improve the numer-ical efficiency,which allows us to split it into two separated problems:one is to find an approximation for the smooth part by a usual spectral method,the other is to determine an approximation to the singular part with several singular terms.4.High accuracy spectral method for fractional Poisson equation and its Caffarelli-Silvestre extension.In view of the complexity of the fractional Laplace operator,Caffarelli and Silvestre extended the complex d dimensional Poisson equation to the simple d + 1 dimensional integer order problem.Due to the Caffarelli-Silvestre extension problem caused the singularity in the extended direction,we use ESM to efficiently solve the extension problem.