Chebyshev Polynomials as Basis Functions

2018 ◽  
pp. 43-62
Author(s):  
George Rawitscher ◽  
Victo dos Santos Filho ◽  
Thiago Carvalho Peixoto
Keyword(s):  
Chebyshev Polynomials ◽  
Basis Functions

An Algorithm for the Approximate Solution of the Fractional Riccati Differential Equation

2019 ◽  
Vol 20 (6) ◽  
pp. 661-674
Author(s):  
S. S. Ezz-Eldien ◽  
J. A. T. Machado ◽  
Y. Wang ◽  
A. A. Aldraiweesh
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Chebyshev Polynomials ◽  
Unknown Function ◽  
Numerical Approach ◽  
Continuous Functions ◽  
Basis Functions ◽  
Real Constant ◽  
Riccati Differential Equation ◽  
Suggested Approach

AbstractThis manuscript develops a numerical approach for approximating the solution of the fractional Riccati differential equation (FRDE): $$\begin{align*}D^{\mu}&u(x)+a(x) u^2(x)+b(x) u(x)= g(x),\quad 0\leq \mu \leq 1,\quad 0\leq x \leq t,\\&u(0)=d,\end{align*}$$where u(x) is the unknown function, a(x), b(x) and g(x) are known continuous functions defined in [0,t] and d is a real constant. The proposed method is applied for solving the FRDE with shifted Chebyshev polynomials as basis functions. In addition, the convergence analysis of the suggested approach is investigated. The efficiency of the algorithm is demonstrated by means of several examples and the results compared with those given using other numerical schemes.

A Sylvester-Based IMEX Method via Differentiation Matrices for Solving Nonlinear Parabolic Equations

2013 ◽  
Vol 14 (4) ◽  
pp. 1001-1026 ◽  
Author(s):  
Francisco de la Hoz ◽  
Fernando Vadillo
Keyword(s):  
Chebyshev Polynomials ◽  
Three Dimensional ◽  
Nonlinear Parabolic Equations ◽  
Basis Functions ◽  
Hermite Functions ◽  
Nonlinear Parabolic ◽  
Sinc Functions ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral ◽  
Rational Chebyshev

AbstractIn this paper we describe a new pseudo-spectral method to solve numerically two and three-dimensional nonlinear diffusion equations over unbounded domains, taking Hermite functions, sinc functions, and rational Chebyshev polynomials as basis functions. The idea is to discretize the equations by means of differentiation matrices and to relate them to Sylvester-type equations by means of a fourth-order implicit-explicit scheme, being of particular interest the treatment of three-dimensional Sylvester equations that we make. The resulting method is easy to understand and express, and can be implemented in a transparent way by means of a few lines of code. We test numerically the three choices of basis functions, showing the convenience of this new approach, especially when rational Chebyshev polynomials are considered.

scholarly journals Collocation with Chebyshev polynomials for Symm's integral equation on an interval

1992 ◽  
Vol 34 (2) ◽  
pp. 199-211 ◽  
Author(s):  
I. H. Sloan ◽  
E. P. Stephan
Keyword(s):  
Integral Equation ◽  
Chebyshev Polynomials ◽  
Basis Functions ◽  
Singular Function ◽  
Polynomial Space ◽  
The Novel ◽  
Negative Power ◽  
Collocation Points ◽  
Chebyshev Points ◽  
Symm’S Integral Equation

AbstractA collocation method for Symm's integral equation on an interval (a first-kind integral equation with logarithmic kernel), in which the basis functions are Chebyshev polynomials multiplied by an appropriate singular function and the collocation points are Chebyshev points, is analysed. The novel feature lies in the analysis, which introduces Sobolev norms that respect the singularity structure of the exact solution at the ends of the interval. The rate of convergence is shown to be faster than any negative power of n, the degree of the polynomial space, if the driving term is smooth.

Spectral analysis with a minimal basis functions approach for quantification of ligand-receptor dynamic PET study

2005 ◽  
Vol 25 (1_suppl) ◽  
pp. S634-S634 ◽  
Author(s):  
Yun Zhou ◽  
Weiguo Ye ◽  
James R Brasic ◽  
Mohab Alexander ◽  
John Hilton ◽  
...  
Keyword(s):  
Spectral Analysis ◽  
Basis Functions ◽  
Dynamic Pet ◽  
Minimal Basis

Material Identification in Presence of Metal for Baggage Screening

2020 ◽  
Vol 2020 (14) ◽  
pp. 294-1-294-8
Author(s):  
Sandamali Devadithya ◽  
David Castañón
Keyword(s):  
Atomic Number ◽  
Effective Atomic Number ◽  
Simulated Data ◽  
Dual Energy ◽  
Basis Functions ◽  
Total Variation Regularization ◽  
Ct Scanner ◽  
Edge Preserving ◽  
Baggage Screening ◽  
Dual Energy Imaging

Dual-energy imaging has emerged as a superior way to recognize materials in X-ray computed tomography. To estimate material properties such as effective atomic number and density, one often generates images in terms of basis functions. This requires decomposition of the dual-energy sinograms into basis sinograms, and subsequently reconstructing the basis images. However, the presence of metal can distort the reconstructed images. In this paper we investigate how photoelectric and Compton basis functions, and synthesized monochromatic basis (SMB) functions behave in the presence of metal and its effect on estimation of effective atomic number and density. Our results indicate that SMB functions, along with edge-preserving total variation regularization, show promise for improved material estimation in the presence of metal. The results are demonstrated using both simulated data as well as data collected from a dualenergy medical CT scanner.

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