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Boundary integral solutions of the heat equation
Author:
E. A. McIntyre
Journal:
Math. Comp. 46 (1986), 71-79, S1
MSC:
Primary 65N99; Secondary 45L10, 65M99, 65R20
MathSciNet review:
815832
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The Boundary Integral Method (BIM) has recently become quite popular because of its ability to provide cheap numerical solutions to the Laplace equation. This paper describes an attempt to apply a similar approach to the (time-dependent) heat equation in two space variables.
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Christopher
T. H. Baker, The numerical treatment of integral equations,
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(57 #7079)
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L. Blue, Boundary integral solutions of Laplace’s
equation, Bell System Tech. J. 57 (1978), no. 8,
2797–2822. MR 508234
(80a:65240)
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Y. Chang, C. Kang & D. Chen, "The use of fundamental Green's functions for the solution of problems of heat conduction in anisotropic media," Internat. J. Heat Mass Transfer, v. 16, 1973, pp. 1905-1918.
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T. Cruse & F. Rizzo, eds., Boundary-Integral Equation Method: Computational Applications in Applied Mechanics, American Society of Mechanical Engineers, 1975.
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de Boor, A practical guide to splines, Applied Mathematical
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(80a:65027)
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M. Delves (ed.), Numerical solution of integral equations,
Clarendon Press, Oxford, 1974. A collection of papers based on the material
presented at a joint Summer School in July 1973, organized by the
Department of Mathematics, University of Manchester, and the Department of
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(57 #4551)
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P. Fox, ed., The PORT Mathematical Subroutine Library, Bell Telephone Laboratories, 1976.
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G. Mikhlin, Integral equations and their applications to certain
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Press Book, The Macmillan Co., New York, 1964. MR 0164209
(29 #1508)
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Ben
Noble, Instability when solving Volterra integral equations of the
second kind by multistep methods, Conf. on Numerical Solution of
Differential Equations (Dundee, 1969) Springer, Berlin, 1969,
pp. 23–39. MR 0273859
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G. Polozhii, Equations of Mathematical Physics, Hayden, 1967.
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D. Shippy, "Application of the boundary-integral equation method to transient phenomena in solids," in [4].
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N. Schryer, A Tutorial on Galerkin's Method, using B-Splines, for Solving Differential Equations, Bell System Technical Memorandum 77-1274-1.
- [1]
- C. T. H. Baker, The Numerical Treatment of Integral Equations, Oxford Univ. Press, London, 1977. MR 0467215 (57:7079)
- [2]
- J. L. Blue, "Boundary integral solutions of Laplace's equation," Bell System Tech. J., v. 57, No. 8, 1978, pp. 2797-2822. MR 508234 (80a:65240)
- [3]
- Y. Chang, C. Kang & D. Chen, "The use of fundamental Green's functions for the solution of problems of heat conduction in anisotropic media," Internat. J. Heat Mass Transfer, v. 16, 1973, pp. 1905-1918.
- [4]
- T. Cruse & F. Rizzo, eds., Boundary-Integral Equation Method: Computational Applications in Applied Mechanics, American Society of Mechanical Engineers, 1975.
- [5]
- C. de Boor, A Practical Guide to Splines, Springer-Verlag, Berlin and New York, 1978. MR 507062 (80a:65027)
- [6]
- L. Delves & J. Walsh, Numerical Solution of Integral Equations, Oxford Univ. Press, London, 1974. MR 0464624 (57:4551)
- [7]
- P. Fox, ed., The PORT Mathematical Subroutine Library, Bell Telephone Laboratories, 1976.
- [8]
- A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N. J., 1964. MR 0181836 (31:6062)
- [9]
- N. Ghosh, On the Convergence of the Boundary Element Method, Ph. D. Thesis, Cornell University, 1982.
- [10]
- S. McKee & H. Brunner, "The repetition factor and numerical stability of Volterra integral equations," Comput. Math. Appl., v. 6, 1980, pp. 339-347. MR 604097 (82g:65064)
- [11]
- J. McKenna, private discussions.
- [12]
- S. Mikhlin, Integral Equations and Their Application to Certain Problems in Mechanics, Mathematical Physics and Technology, 2nd English ed., Macmillan, New York, 1964. MR 0164209 (29:1508)
- [13]
- B. Noble, "Instability when solving Volterra integral equations of the second kind by multistep methods," in Conference on the Numerical Solutions of Differential Equations, Lecture Notes in Math., no. 109, Springer-Verlag, Berlin, 1969, pp. 23-39. MR 0273859 (42:8735)
- [14]
- G. Polozhii, Equations of Mathematical Physics, Hayden, 1967.
- [15]
- D. Shippy, "Application of the boundary-integral equation method to transient phenomena in solids," in [4].
- [16]
- N. Schryer, A Tutorial on Galerkin's Method, using B-Splines, for Solving Differential Equations, Bell System Technical Memorandum 77-1274-1.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025-5718-1986-0815832-6
PII:
S 0025-5718(1986)0815832-6
Keywords:
Boundary integrals,
heat equation,
fundamental solutions,
thermal potentials,
Volterra integral equations,
Galerkin's method,
B-splines,
quadrature methods
Article copyright:
© Copyright 1986 American Mathematical Society
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