D,2 • MATHEMATICAL FORMULATION 

 series in terms of the eigenfunctions 



F{x, y) = Yj ^"^''*^^' y^ 



n=.l 



where, in view of the orthogonaHty property, 



// pcFcpndxdy 



G 



An = ^. (2-7) 



jj pdpldxdy 



G 



Actually, the function F{x, y), representing the initial temperature dis- 

 tribution in the domain, seldom satisfies the boundary conditions, the 

 usual form being F{x, y) = const. However, this is not a serious diffi- 

 culty, for there always exists a function F{x, y) which satisfies all con- 

 ditions of the theorem and is, nevertheless, arbitrarily close to F{x, y). 

 This can be seen immediately if ^ = F{x, y) is regarded as the equation 

 of a surface. The only consequence is that, in general, there will not be 

 uniform convergence but rather convergence "in the mean," that is: 



n 



Y\m jjyY AmiPm{x, y) - F{x, y)J dxdy = 



G m = l 



which is sufficient for all practical purposes (cf. [8, Vol. 1, p. 43]). 



The practical use of the Fourier series solution (Eq. 2-6) for transient 

 temperature calculations is contingent on rapid convergence of the series, 

 which is assured for sufficiently large times t by virtue of the decreasing 

 exponential factors e"^-'. At small times, however, the convergence may 

 become slow and calculations with Eq. 2-6 may become cumbersome. In 

 this case, special approximate procedures, based on Laplace transform, 

 source and image methods, etc. [6], prove to be more convenient. We shall 

 have occasion to employ such procedures in the limit of small times when 

 use of the Fourier series is not practical (cf. Art. 5). 



It is seen then that the general problem reduces to the solution of the 

 eigenvalue problem, or what is usually known as the third boundary- 

 value problem in mathematical physics. It may be worthwhile to mention 

 that the boundary-value problem can always be formulated as an equiva- 

 lent isoperimetric problem in the calculus of variations, designated by 

 Riemann as Dirichlet's principle (cf. [8, Vol. 2, Chap. 7]). In problems 

 of vibration of beams, membranes, and plates, in aeroelastic problems, etc., 

 this principle, also known as the Rayleigh-Ritz method, is a very powerful 

 tool of analysis. However, in the case of heat conduction this method does 

 not prove to be so useful. 



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