WebThe Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of … WebAug 11, 2024 · I am studying Chebyshev Polynomials to solve some differential equations. I found in the literature that if you have a function being expanded in Chebyshev polynomials such as then you can also expand its derivatives as with the following relation being $c_k=2$ for k=0 and 1 if k>0.
(PDF) The Chebyshev Difference Equation - ResearchGate
WebMar 23, 2012 · Our scheme is as follows. Given a grid function v defined on the Chebyshev points, we obtain a discrete derivative w in two steps: • Let p be the unique polynomial of degree ≤ N with p ( x j ) = υ j , 0 ≤ j ≤ N . • Set w j = p ′ ( x j ) . WebMay 10, 2024 · The CSEM used herein consists in applying the Chebyshev spectral method in every element. It is a combination of the Chebyshev spectral method and FEM. The Chebyshev spectral method uses Chebyshev polynomials as the basis for spatial discretization. We expand the solution in Chebyshev series. sheriff hospital
Spectral collocation in space and time for linear PDEs
WebMar 1, 2024 · In this study, the orthonormal piecewise Bernoulli functions are generated as a new kind of basis functions. An explicit matrix related to fractional integration of these functions is obtained. An efficient direct method is developed to solve a novel set of optimal control problems defined using a fractional integro-differential equation. The presented … WebGiven a grid function ude ned on the Chebyshev points, we obtain a discrete derivative win two steps: Let pbe the unique polynomial of degree Nwith p(xj) = uj;0 j N. Set wj = … WebApr 22, 2013 · The Chebyshev derivative matrix at the quadrature points, , is an matrix given by. , , , for , and for and , where for and . The discrete Laplacian is given by , … sheriff hoss mack