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Table Of Integrals Series And Products

Fractional Differential Equations

Zeros of orthogonal polynomials

What is the fractional Laplacian? A comparative review with new results

Orthogonal approximations in Sobolev spaces stability and convergence spectral methods and pseudospectral methods spectral methods for multi-dimensional and high order problems mixed spectral methods combined spectral methods spectral methods on the spherical surface.

Spectral Methods and Their Applications

We propose in this paper an efficient and accurate numerical method for the spectral fractional Laplacian equation using the Caffarelli–Silvestre extension. In particular, we propose several strategies to deal with the singularity and the additional dimension associated with the extension problem: (i) reducing the $$d+1$$ dimensional problem to a sequence of d-dimension Poisson-type problems by using the matrix diagonalizational method; (ii) resolving the singularity by applying the enriched spectral method in the extended dimension. We carry out rigorous analysis for the proposed numerical method, and provide abundant numerical examples to verify the theoretical results and illustrate effectiveness of the proposed method.

An Efficient and Accurate Numerical Method for the Spectral Fractional Laplacian Equation

Usual spectral methods are very effective for problems with smooth solutions, but their accuracy will be severely limited if solution of the underlying problems exhibits singular behavior. We develop in this paper enriched spectral-Galerkin methods (ESG) to deal with a class of problems for which the form of leading singular solutions can be determined. Several strategies are developed to overcome the ill conditioning due to the addition of singular functions as basis functions, and to efficiently solve the approximate solution in the enriched space. We validate ESG by solving a variety of elliptic problems with weakly singular solutions.

Enriched Spectral Methods and Applications to Problems with Weakly Singular Solutions

We present two new classes of orthogonal functions, log orthogonal functions (LOFs) and generalized log orthogonal functions (GLOFs), which are constructed by applying a $\log$ mapping to Laguerre polynomials. We develop basic approximation theory for these new orthogonal functions and apply them to solve several typical fractional differential equations whose solutions exhibit weak singularities. Our error analysis and numerical results show that our methods based on the new orthogonal functions are particularly suitable for functions which have weak singularities at one endpoint, and can lead to exponential convergence rate, as opposed to low algebraic rates if usual orthogonal polynomials are used.

Log orthogonal functions: approximation properties and applications

In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), 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, and show that with an appropriate choice of the parameters in GJFs, the resulted linear systems can be sparse and well-conditioned. 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. The idea and results presented in this paper will be useful to deal with more general FDEs associated with Riemann-Liouville or Caputo fractional derivatives.

Generalized Jacobi Functions and Their Applications to Fractional Differential Equations

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Papers

An Efficient and Accurate Numerical Method for the Spectral Fractional Laplacian Equation

Enriched Spectral Methods and Applications to Problems with Weakly Singular Solutions

Log orthogonal functions: approximation properties and applications

Log orthogonal functions: approximation properties and applications

Generalized Jacobi Functions and Their Applications to Fractional Differential Equations