A spring-damping regularization of the Fourier sine series solution to the inverse Cauchy problem for a 3D sideways heat equation

Free torsional vibration of cracked carbon nanotubes with elastic torsional boundary conditions is studied. Eringen’s nonlocal elasticity theory is used in the analysis. Two similar rotation functions are represented by two Fourier sine series. A coefficient matrix including torsional springs and crack parameter is derived by using Stokes’ transformation and nonlocal boundary conditions. This useful coefficient matrix can be used to obtain the torsional vibration frequencies of cracked nanotubes with restrained boundary conditions. Free torsional vibration frequencies are calculated by using Fourier sine series and compared with the finite element method and analytical solutions available in the literature. The effects of various parameters such as crack parameter, geometry of nanotubes, and deformable boundary conditions are discussed in detail.

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On a Tauberian theorem for the $L\sp{1}$-convergence of Fourier sine series

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1983-0691274-0 ◽

1983 ◽

Vol 88 (1) ◽

pp. 34-34

Author(s):

William O. Bray

Keyword(s):

Tauberian Theorem ◽

Sine Series ◽

Fourier Sine ◽

Fourier Sine Series

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Reducing the near boundary errors of nonhomogeneous heat equations by boundary consistent methods

IMA Journal of Applied Mathematics ◽

10.1093/imamat/hxaa029 ◽

2020 ◽

Author(s):

Chein-Shan Liu ◽

Chih-Wen Chang

Keyword(s):

Heat Equations ◽

Series Method ◽

Sine Series ◽

Fourier Cosine ◽

Cosine Series ◽

Fourier Sine ◽

Fourier Cosine Series ◽

Fourier Sine Series ◽

Gibbs Phenomena ◽

Nonhomogeneous Heat

Abstract In the paper, we point out a drawback of the Fourier sine series method to represent a given odd function, where the boundary Gibbs phenomena would occur when the boundary values of the function are non-zero. We modify the Fourier sine series method by considering the consistent conditions on the boundaries, which can improve the accuracy near the boundaries. The modifications are extended to the Fourier cosine series and the Fourier series. Then, novel boundary consistent methods are developed to solve the 1D and 2D heat equations. Numerical examples confirm the accuracy of the boundary consistent methods, accounting for the non-zeros of the source terms and considering the consistency of heat equations on the boundaries, which can not only overcome the near boundary errors but also improve the accuracy of solution about four orders in the entire domain, upon comparing to the conventional Fourier sine series method and Duhamel’s principle.

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