Hermite Spectral Methods for Fractional PDEs in Unbounded Domains

DOI10.1137/16M1097109zbMath1373.65075OpenAlexW2755519002MaRDI QIDQ5357973

Zhiping Mao, Jie Shen

Publication date: 18 September 2017

Published in: SIAM Journal on Scientific Computing (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1137/16m1097109

zbMATH Keywords

unbounded domainerror estimatespectral methodfractional PDEsfractional nonlinear Schrödinger equationsfractional advection-diffusion equationsHermite polynomials/functionsHermite-collocationHermite-Galerkin methods

Mathematics Subject Classification ID

Spectral, collocation and related methods for boundary value problems involving PDEs (65N35) Error bounds for boundary value problems involving PDEs (65N15) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) NLS equations (nonlinear Schrödinger equations) (35Q55) Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs (65M60) Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs (65M70) Error bounds for initial value and initial-boundary value problems involving PDEs (65M15) Fractional partial differential equations (35R11)

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Uses Software

• OPQ

Cites Work

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• Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations
• Fractional Sturm-Liouville eigen-problems: theory and numerical approximation
• A new spectral method for numerical solution of the unbounded rough surface scattering problem
• Hermite spectral and pseudospectral methods for nonlinear partial differential equation in multiple dimensions
• The accurate solution of Poisson's equation by expansion in Chebyshev polynomials
• Hermite pseudospectral approximations. An error estimate
• Direct solution of partial difference equations by tensor product methods
• Connection coefficients between orthogonal polynomials and the canonical sequence: An approach based on symbolic computation
• Generalized Jacobi functions and their applications to fractional differential equations
• Spectral Methods
• Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials
• The Hermite Spectral Method for Gaussian-Type Functions
• Numerical study of fractional nonlinear Schrödinger equations
• Fractional Spectral Collocation Method