Chebyshev rational spectral and pseudospectral methods on a semi‐infinite interval

Guo, Bing; Shen, Jie; Wang, Zhongqing

doi:10.1002/nme.392

Cited by 71 publications

(28 citation statements)

References 16 publications

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“…Furthermore, we approximate a function using rational Chebyshev–Gauss–Radau points. The well‐known Chebyshev polynomials are orthogonal in the interval [ − 1,1] in terms of the weight function

ρ (y) = \frac{1}{\sqrt{1 - x^{2}}}

and can be calculated via the following recurrence formula:

T_{0} (x) = 1, T_{1} (x) = x

T_{n + 1} (x) = 2 x T_{n} (x) - T_{n - 1} (x), n = 1, 2, 3, \dots,

The new basis functions denoted by R n ( x ) are represented by in interval Ω = [0, ∞ )

R_{n} (x) = T_{n} ((), \frac{x - 1}{x + 1}) .

Note that R n ( x ) is the eigenfunction of the singular Sturm–Liouville problem:

(x + 1) \sqrt{x} \frac{d}{d x} ((), (x + 1) \sqrt{x} \frac{d}{d x} R_{n} (x)) + n^{2} R_{n} (x) = 0, x \in normalΩ, n ⩾ 1 .

Thus, rational functions satisfy the following conditions:

R_{0} (x) = 1, R_{1} (x) = \frac{x - 1}{x + 1},

…”

Section: Rational Chebyshev Functionsmentioning

confidence: 99%

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Analysis of unsteady stagnation‐point flow over a shrinking sheet and solving the equation with rational Chebyshev functions

Foroutan

Ebadian

Najafzadeh

2016

Math Methods in App Sciences

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This paper investigates the nonlinear boundary value problem, resulting from the exact reduction of the Navier–Stokes equations for unsteady laminar boundary layer flow caused by a stretching surface in a quiescent viscous incompressible fluid. We prove existence of solutions for all values of the relevant parameters and provide unique results in the case of a monotonic solution. The results are obtained using a topological shooting argument, which varies a parameter related to the axial shear stress. To solve this equation, a numerical method is proposed based on a rational Chebyshev functions spectral method. Using the operational matrices of derivative, we reduced the problem to a set of algebraic equations. We also compare this work with some other numerical results and present a solution that proves to be highly accurate. Copyright © 2016 John Wiley & Sons, Ltd.

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ρ (y) = \frac{1}{\sqrt{1 - x^{2}}}

and can be calculated via the following recurrence formula:

T_{0} (x) = 1, T_{1} (x) = x

T_{n + 1} (x) = 2 x T_{n} (x) - T_{n - 1} (x), n = 1, 2, 3, \dots,

The new basis functions denoted by R n ( x ) are represented by in interval Ω = [0, ∞ )

R_{n} (x) = T_{n} ((), \frac{x - 1}{x + 1}) .

Note that R n ( x ) is the eigenfunction of the singular Sturm–Liouville problem:

(x + 1) \sqrt{x} \frac{d}{d x} ((), (x + 1) \sqrt{x} \frac{d}{d x} R_{n} (x)) + n^{2} R_{n} (x) = 0, x \in normalΩ, n ⩾ 1 .

Thus, rational functions satisfy the following conditions:

R_{0} (x) = 1, R_{1} (x) = \frac{x - 1}{x + 1},

…”

Section: Rational Chebyshev Functionsmentioning

confidence: 99%

“…Suppose N is any positive integer, thus we have:

{frakturR}_{N} = S p a n {R_{0}, R_{1}, \dots, R_{N}} .

Guo et al . introduced rational Chebyshev–Guass–Radau points. Let

y_{j} = c o s (\frac{2 j}{2 N + 1}) π, j = 0, 1, \dots, N

be the N + 1 roots of the polynomial T N ( x ) + T N + 1 ( x ).…”

Section: Rational Chebyshev Functionsmentioning

confidence: 99%

Analysis of unsteady stagnation‐point flow over a shrinking sheet and solving the equation with rational Chebyshev functions

Foroutan

Ebadian

Najafzadeh

2016

Math Methods in App Sciences

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“…Accordingly, we should approximate them in the corresponding different weighted spaces. However, the commonly used orthogonal families of rational functions are induced by the Legendre or Chebyshev polynomials, see [2,8,[11][12][13][14] and the references therein. Thus, their weight functions are fixed, which might not be most appropriate in some cases.…”

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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“…The main advantage of this approach is that standard spectral approximation results can be used for the analysis, but its main disadvantage is that the mapped equation is usually very complicated and its analysis is often cumbersome. In the second approach, we do not transform the equation, but we approximate its solution using a new family of orthogonal functions {p k (h(x; λ))}, which are obtained by applying the mapping y = h(x; λ) to classical orthogonal polynomials {p k (y)} (see, for instance, [6,18,16]) and which are suitable for capturing the localized rapid variations in the solution of the given problem. The analysis of this approach will require approximation results by using the new family of orthogonal functions.…”

mentioning

confidence: 99%

Error Analysis for Mapped Legendre Spectral and Pseudospectral Methods

Shen¹,

Wang²

2004

SIAM J. Numer. Anal.

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Abstract. A general framework is introduced to analyze the approximation properties of mapped Legendre polynomials and of interpolations based on mapped Legendre-Gauss-Lobatto points. Optimal error estimates featuring explicit expressions on the mapping parameters for several popular mappings are derived. These results not only play an important role in numerical analysis of mapped Legendre spectral and pseudospectral methods for differential equations but also provide quantitative criteria for the choice of parameters in these mappings.

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Chebyshev rational spectral and pseudospectral methods on a semi‐infinite interval

Cited by 71 publications

References 16 publications

Analysis of unsteady stagnation‐point flow over a shrinking sheet and solving the equation with rational Chebyshev functions

Analysis of unsteady stagnation‐point flow over a shrinking sheet and solving the equation with rational Chebyshev functions

Generalized Jacobi rational spectral method on the half line

Error Analysis for Mapped Legendre Spectral and Pseudospectral Methods

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