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The axes are drawn as indicated%in Fig. 1-4. Q-Q The temperature 0 when the steady state has *J -r— +*x been reached satisfies the equation FIG. 1-4. 2a)]. It follows that, when the boundary conditions (1), (2), and (3) are fitted, 00 Ane-n7ixlbsin 0= Σ (nnylb). n=l From condition (4), we have 00 0O= Σ An sin (ηπγlb). n=l The method of Fourier series then gives (see Appendix § 3) A2r+1 = 40o/{(2r + 1)π} A2r = 0 r = 0,1, 2, 3 , . . r= 1,2,3,... 25 8. LAPLACE'S EQUATION IN TWO DIMENSIONS Hence Θ= T i T ^ r V e~(2r T + 1)πφ sin i(2r + l )*ylh) r= 0 If the plate is of finite length / the solution may also be obtained by the above method and is A _ 400 y n ^ sin {(2r + l)nyjb) sinh {(2r + 1)π(/ - *)/6} (2r + 1) sinh {(2r + \)nl\b) for Θ = 0 at x = I.

The corresponding value of φ is then . nx . ny . πζ = ΑΛΛΛ sm — sin -— sin — , a o c where the value of Aln is dependent on the power output of the reactor. 11. A fourth order partial differential equation So far only partial differential equations of the second order have been discussed. Higher order equations of special kinds arise in a number of physical problems and these can often be solved by the method of separation of the variables. Consider the following problem. Example 15. Transverse vibrations of a straight beam of uniform cross section The displacement y in the free vibration of a beam satisfies the equation 11.

6), gives results very much like those for the wave equation. For this reason, it is evidently unnecessary to reproduce the analysis here. However, the method of solving the diffusion or heat flow equations will be demonstrated here and in later chapters by working out particular problems. Example 10. Flow of heat in a rod of finite length The temperature 0 at any point of a rod of length / satisfies the equation a20 _ i a# ä* 2 ~ ~^~dt at any time /. The end x = 0 is maintained at zero temperature and the end x = I is insulated.

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