(Spectral) Chebyshev collocation methods for solving differential equations

Amodio, Pierluigi; Brugnano, Luigi; Iavernaro, Felice

doi:10.1007/s11075-022-01482-w

Cited by 6 publications

(3 citation statements)

References 27 publications

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“…i.e., (5) with t = h. This clearly explains the use of the Jacobi basis for the expansion (11). Further, we observe that, when α = 1, one retrieves the approach described in [11] for ODE-IVPs.…”

Section: Remarksupporting

confidence: 55%

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A Spectrally Accurate Step-by-Step Method for the Numerical Solution of Fractional Differential Equations

Brugnano,

Burrage,

Burrage

et al. 2024

J Sci Comput

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In this paper we consider the numerical solution of fractional differential equations. In particular, we study a step-by-step procedure, defined over a graded mesh, which is based on a truncated expansion of the vector field along the orthonormal Jacobi polynomial basis. Under mild hypotheses, the proposed procedure is capable of getting spectral accuracy. A few numerical examples are reported to confirm the theoretical findings.

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Section: Remarksupporting

confidence: 55%

“…In this paper, we consider a major improvement of the recent solution approach described in [2,9], based on previous work on Hamiltonian Boundary value Methods (HBVMs) [7,8,11,14] (also used as spectral methods in time [3,5,12,13]), for solving fractional initial value problems in the form:…”

Section: Introductionmentioning

confidence: 99%

A Spectrally Accurate Step-by-Step Method for the Numerical Solution of Fractional Differential Equations

Brugnano,

Burrage,

Burrage

et al. 2024

J Sci Comput

Self Cite

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“…We refer to [68,71] (and, in particular, to the monograph [67] 1 ) for the derivation and analysis of such methods, which have been devised within the framework of the so called line integral methods (see, e.g., the review paper [72]). We mention that generalizations and extensions of such approach have been also considered in [73][74][75][76][77][78][79][80][81][82][83][84][85]. HBVMs have been also considered for the efficient numerical solution of a number of Hamiltonian PDEs (see, e.g., [86][87][88][89][90]), among which the NLSE [91] and Manakov systems [92].…”

Section: Introductionmentioning

confidence: 99%

Recent Advances in the Numerical Solution of the Nonlinear Schrödinger Equation

Barletti,

Brugnano,

Gurioli

et al. 2023

Preprint

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In this review we collect some recent achievements in the accurate and efficient solution of the Nonlinear Schrödinger Equation (NLSE), with the preservation of its Hamiltonian structure. This is achieved by using the energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs) after a proper space semi-discretization. The main facts about HBVMs, along with their application for solving the given problem, are here recalled and explained in detail. In particular, their use as spectral methods in time, which allows efficiently solving the problems with spectral space-time accuracy.

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Numerical solution of FDE-IVPs by using fractional HBVMs: the fhbvm code

Brugnano,

Gurioli,

Iavernaro

2024

Numer Algor

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In this paper we describe the efficient numerical implementation of Fractional HBVMs, a class of methods recently introduced for solving systems of fractional differential equations. The reported arguments are implemented in the Matlab$$^{\copyright } $$ © code , which is made available on the web. An extensive experimentation of the code is reported, to give evidence of its effectiveness.

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(Spectral) Chebyshev collocation methods for solving differential equations

Cited by 6 publications

References 27 publications

A Spectrally Accurate Step-by-Step Method for the Numerical Solution of Fractional Differential Equations

A Spectrally Accurate Step-by-Step Method for the Numerical Solution of Fractional Differential Equations

Recent Advances in the Numerical Solution of the Nonlinear Schrödinger Equation

Numerical solution of FDE-IVPs by using fractional HBVMs: the fhbvm code

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