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Regularity theory and high order numerical methods for the (1D)-fractional Laplacian. (English) Zbl 1409.65111

The authors study the regularity theory and higher-order numerical methods for the (1D) fractional Laplacian. They first provide the problem as an integral equation, analyze the boundary singularity and produce a diagonal form for the single-interval problem. Using Gegenbauer eigenfunctions and the associated expansions, the Sobolev and analytic regularity results for the solution \(u\) are obtainded, including a weighted-space version of the Sobolev lemma. Also, by utilizing Gegenbauer expansions in conjunction with Nyström discretizations and taking into account the analytic structure of the edge singularity, the authors present a highly accurate and efficient numerical solver for fractional-Laplacian equations posed on a union of finitely many one-dimensional intervals. The sharp error estimates presented indicate that the proposed algorithm is spectrally accurate, with convergence rates that only depend on the smoothness of the right-hand side. In particular, convergence is exponentially fast for analytic right-hand sides. A variety of numerical results presented demonstrate the character of the proposed solver; the new algorithm is significantly more accurate and efficient than those resulting from previous approaches.

MSC:

65R20 Numerical methods for integral equations
35B65 Smoothness and regularity of solutions to PDEs
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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