scholarly journals Approximation Properties of Chebyshev Polynomials in the Legendre Norm

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3271
Author(s):  
Cuixia Niu ◽  
Huiqing Liao ◽  
Heping Ma ◽  
Hua Wu
Keyword(s):  
Numerical Analysis ◽  
Spectral Method ◽  
Chebyshev Polynomials ◽  
One Dimension ◽  
Weighted Norm ◽  
Interpolation Operator ◽  
Approximation Properties ◽  
Chebyshev Points ◽  
Chebyshev Interpolation ◽  
Chebyshev Spectral Method

In this paper, we present some important approximation properties of Chebyshev polynomials in the Legendre norm. We mainly discuss the Chebyshev interpolation operator at the Chebyshev–Gauss–Lobatto points. The cases of single domain and multidomain for both one dimension and multi-dimensions are considered, respectively. The approximation results in Legendre norm rather than in the Chebyshev weighted norm are given, which play a fundamental role in numerical analysis of the Legendre–Chebyshev spectral method. These results are also useful in Clenshaw–Curtis quadrature which is based on sampling the integrand at Chebyshev points.

Multidomain Chebyshev spectral method for 3-D dc resistivity modeling

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1616-1623 ◽  
Author(s):  
Shengkai Zhao ◽  
Matthew J. Yedlin
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Spectral Method ◽  
Direct Current ◽  
Chebyshev Polynomials ◽  
Difference Method ◽  
Dc Resistivity ◽  
Chebyshev Spectral Method ◽  
The Finite Difference Method ◽  
Resistivity Modeling

We use the multidomain Chebyshev spectral method to solve the 3-D forward direct current (dc) resistivity problem. We divided the whole domain into a number of subdomains and approximate the potential function by a separate set of Chebyshev polynomials in each subdomain. At an interface point, we require that both the potential and the flux be continuous. Numerical results show that for the same accuracy the multidomain Chebyshev spectral method is 2 to 260 times faster than the finite‐difference method.

scholarly journals New Solutions for System of Fractional Integro-Differential Equations and Abel’s Integral Equations by Chebyshev Spectral Method

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Hassan A. Zedan ◽  
Seham Sh. Tantawy ◽  
Yara M. Sayed
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Integral Equations ◽  
Spectral Method ◽  
Operational Matrix ◽  
Numerical Results ◽  
Efficient Technique ◽  
Test Problems ◽  
Chebyshev Spectral Method

Chebyshev spectral method based on operational matrix is applied to both systems of fractional integro-differential equations and Abel’s integral equations. Some test problems, for which the exact solution is known, are considered. Numerical results with comparisons are made to confirm the reliability of the method. Chebyshev spectral method may be considered as alternative and efficient technique for finding the approximation of system of fractional integro-differential equations and Abel’s integral equations.

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