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Algebraic Interpolation
Published in Victor S. Ryaben’kii, Semyon V. Tsynkov, A Theoretical Introduction to Numerical Analysis, 2006
Victor S. Ryaben’kii, Semyon V. Tsynkov
We therefore conclude that in contradistinction to the previous case (2.18), the Lebesgue constants may, in fact, grow slowly rather than rapidly, as they do on the non-equally spaced nodes (2.20). As such, even the high-degree interpolating polynomials in this case will not be overly sensitive to perturbations of the input data. Interpolation nodes (2.20) are known as the Chebyshev nodes. They will be discussed in detail in Chapter 3.
Time-dependent lowest term estimation in a 2D bioheat transfer problem with nonlocal and convective boundary conditions
Published in Inverse Problems in Science and Engineering, 2021
Fermín S. V. Bazán, Mansur I. Ismailov, Luciano Bedin
In this section, we will derive an efficient numerical approach for solving the bioheat model (1)–(5) that we will use in the numerical treatment of the inverse problem of interest. For this purpose, we will use the CPS method in which the discretization of spatial derivatives is done using the Chebyshev differentiation matrix, giving rise to a system of time-dependent ordinary differential equations that can be solved in in different ways. For the sake of clarity, before describing details of the proposed approach, let us introduce some generalities and notation about Chebyshev's differentiation matrix. Let denote a vector with function values as entries, where are the so called Chebyshev nodes (also known as Chebyshev–Gauss–Lobatto nodes) within the interval , and let the associated Chebyshev differentiation matrix be denoted by D. Assuming that the rows and columns of D are indexed from 0 to n, the entries of D are given by [34] where if i = 0 or i = n and otherwise. Provided v is sufficiently smooth, it is well known that matrix D yields highly accurate approximations to , , simply by taking , , and so on [33,34]. Further, to approximate the derivative at Chebyshev nodes distributed within an interval , observe that introducing the change of variable , , and setting , we have . This implies that the Chebyshev differentiation matrix with nodes within is simply times matrix D defined in (16).