By William A. Day
J-B. J. FOURIER'S immensely influential treatise Theorie Analytique de los angeles Chaleur [21J, and the next advancements and refinements of FOURIER's rules and techniques by the hands of many authors, offer a hugely profitable thought of warmth conduction. in line with that thought, the expansion or decay of the temperature e in a accomplishing physique is ruled via the warmth equation, that's, via the parabolic partial differential equation Such has been the impact of FOURIER'S thought, which needs to eternally stay the classical thought in that it units the traditional opposed to which all different theories are to be measured, that the mathematical research of warmth conduction has grow to be considered as being virtually identicalt with the research of the warmth equation, and the reader won't must be reminded that extensive analytical learn has t yet no longer completely; witness, for instance, these theories which might substitute the warmth equation by way of an equation which means a finite pace of propagation for the temperature. The reader is talked about the item [9] of COLEMAN, FABRIZIO, and OWEN for the derivation of such an equation from sleek Continuum Thermody namics and for references to previous paintings during this path. viii creation amply established that the warmth equation enjoys many homes of serious curiosity and elegance.
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Additional info for Heat Conduction Within Linear Thermoelasticity
Example text
We begin by deriving expansions for the mean square temperature and the mean square heat flux which are valid when a = 0; this, of course, is the classical case but the result would seem not to have been noticed before its publication in [16]. If we put + i)Fwfi, F(w)¢(1 - x, w, 0) + G(w)¢(x, w, 0) (F(w) sinh Po(1 - x) + G(w) sinh Pox) cosech Po, Po(w) = pew, 0) = (1 0 0 (x, w) = = the sum 8(x, t) = Re L0 0 (x, w) exp(iwt) is the trigonometric solution of the heat equation which satisfies the boundary conditions 8lx=o = J, 81x=1 = g.
G(z, y, (1 + a)s) dz - G(z, z*, (1 1] + a)s) dz dz* Since we know G explicitly we can determine the right-hand side and, indeed, we find, after further calculation, that a R(x, y, s) - (1 + a)G(x, y, (1 + a)s) = - - (1 - F(x, s, a»F(y, s, a) s 2 ' 1+ J (1 a + a)s + a)s tanh tJ(1 where F(y, s, a) = sinh(J(1 + a)s· y) + sinh(J(l+ a)s· (1 sinh J(1 + a)s - y» . The asymptotic behaviour of F is F(y, s, a) ~ exp( - J(1 + a)s' dey»~ as s -+ 00, where dey) is the distance introduced earlier, and so we conclude that R(x, y, s) - (1 + a)G(x, y, (1 + a)s) ~ - ~ exp( -J(1 + a)s' dey»~ as s -+ 00.
2 and that E! satisfies the inequality 2E'1 + tl (a~2:tr Integration with respect to 2E(T) + t dx dt ~ h,2. yields the estimates tT Ll (:~) 2dx dt ~ 2E(0) + f: h2 dt, f: fo! 4). 2) we conclude that f: Edt is bounded above by a constant multiple of 1 + fOT(h 2 + h,2) dt. In the same way, the integral fOT El dt 52 4. Approximation by Heat Equation or Integro-differential Equation is bounded above by a constant multiple of 1+ foT(h l2 + h"2) dt and, since h' and h" are square-integrable on [0, 00), we conclude that E 1 is integrable on [0, 00).
