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A Problem in the Linear Flow of Heat discussed from the point of view of the Theory of Integral Equations

Published online by Cambridge University Press:  20 January 2009

H. S. Carslaw
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In the first section of Kneser's book on Integral Equations and their Applications to Mathematical Physics, he applies that theory to the solution of some of the problems which arise in the Linear Flow of Heat. The object of this paper is to illustrate Kneser's use of Integral Equations in the Mathematical Theory of the Conduction of Heat by the discussion of one of the classical problems of Linear Flow which he leaves untouched.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , February 1911 , pp. 97 - 106
Copyright
Copyright © Edinburgh Mathematical Society 1911

References

1 Kneser, , Die Integralgleichungen und ihre Anwendungen in der Mathematischen Physik (Braunschweig, 1911).Google Scholar

2 Cf., for example, Carslaw, Fourier's Series and Integrals and the Mathematical Theory of the Conduction of Heat, §105, p. 270. In future this book will be referred to as Fourier's Series.

3 Cf. Fourier's Series, p. 230.

4 The properties of these funcions are worked oat at length in the discussion of the problem in Fourier's Series, §105.

5 We use the notation for F(b) – F(a).

6 The Green's Functions employed in the applications of Integral Equations to the Conduction of Heat must not be confused with those to which the same term was applied in the case of Instantaneous Point Sources. As a matter of fact, the new Green's Functions can be obtained from the old by integration : and the results given in my paper in these Proceedings [Vol. XXI., p. 40, 1903], and in Fourier's Series, Chapter XVIII., can be used in the work on Integral Equations.

7 For a general proof, of. Kneser, loc. cit., p. 6.

8 Cf. Bôcher, , An Introduction to the Study of Integral Equations (Camb. Math. Tracts, No. 10), p. 68, where a more exact statement of this fundamental theorem is given.Google Scholar

Also Kneser, loc. cit., pp. 33–4.

9 Proc. Edin. Math. Soc., Vol. XXI., p. 40, 1903. Also Fourier's Series, p. 386.

10 Cf. Kneser, loc. cit. p. 20.