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Small time approximations of Green's functions for one‐dimensional heat conduction problems with convective boundaries
Author(s) -
Yen D.H.Y.,
Beck J.V.
Publication year - 2003
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200310026
Subject(s) - mathematics , convolution (computer science) , thermal conduction , dimensionless quantity , mathematical analysis , separation of variables , integrable system , series (stratigraphy) , dirichlet distribution , gravitational singularity , approximations of π , physics , boundary value problem , mechanics , thermodynamics , paleontology , machine learning , artificial neural network , computer science , biology
Small time approximations are obtained for Green's functions in one‐dimensional heat conduction problems with convective boundaries. The method of images similar to those for cases with Dirichlet or Neumann boundaries is used. Multiple reflections in this method generate infinite series representations for the Green's functions whose general terms involve convolution integrals with integrable singularities. We truncate the series after single and double reflections ( n = 1, 2) and transform the convolution integrals to known closed‐form expressions. Numerical results are presented for n = 1, 2 and compared with exact results obtained using the classical separation of variables. It is seen that results with n = 1, 2 provide accurate approximations for dimensionless time α t / L 2 up to 0.3 and complementary to those obtained by separation of variables.