Polynomial Approximation of Differential Equations

Funaro, Daniele

doi:10.1007/978-3-540-46783-0

Cited by 322 publications

(313 citation statements)

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“…Moreover, the dependence of c on α and β are described explicitly in [22]. Noted that Funaro [11] obtained a similar result for integer µ ≥ 0 and s = 0, while Babuška and Guo [3] derived the result for real α = β and 0 ≤ s ≤ µ.…”

Section: Projection Errors In Non-uniformly Weighted Spacesmentioning

confidence: 84%

Irrational approximations and their applications to partial differential equations in exterior domains,

Guo

Shen

2006

Adv Comput Math

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A family of orthogonal systems of irrational functions on the semi-infinite interval is introduced. The proposed orthogonal systems are based on Jacobi polynomials through an irrational coordinate transform. This family of orthogonal systems offers great flexibility to match a wide range of asymptotic behaviors at infinity. Approximation errors by the basic orthogonal projection and various other orthogonal projections related to partial differential equations in unbounded domains are established. As an example of applications, a Galerkin approximation using the proposed irrational functions to an exterior problem is analyzed and implemented. Numerical results in agreement with our theoretical estimates are presented.Keywords rational and irrational functions · spectral method · semi-infinite interval · exterior problems Mathematics Subject Classifications (2000) 65N35 · 65N22 · 65F05 · 41A20

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Section: Projection Errors In Non-uniformly Weighted Spacesmentioning

confidence: 84%

Irrational approximations and their applications to partial differential equations in exterior domains,

Guo

Shen

2006

Adv Comput Math

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“…Several spectral methods were proposed for their numerical simulations, see [1,[3][4][5][6] and the references therein. The first method is to use Hermite and Laguerre approximations.…”

Section: Introductionmentioning

confidence: 99%

Generalized Jacobi rational spectral method on the half line

Guo

2011

Adv Comput Math

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We introduce an orthogonal system on the half line, induced by generalized Jacobi functions. Some basic results on the generalized Jacobi rational approximation are established, which play important roles in the related spectral method. As an example of applications, the rational spectral method is proposed for partial differential equations of degenerate type. Its convergence is proved. Numerical results demonstrate its efficiency.Keywords Generalized Jacobi rational approximation · Spectral method on the half line · Applications Mathematics Subject Classification (2010) 65M70 · 41A30 · 35K65 Communicated by Zhongying Chen.

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“…We see from (1.1) that one of parameters of the second Jacobi polynomial is twice greater than the degree of the first one, and tends to infinity as the mode l goes to infinity. As a consequence, the usual results on one-dimensional Jacobi approximation established in [2,4,10,11] cannot be applied directly, since they are of the form…”

Section: Introductionmentioning

confidence: 99%