ens having the form (26), we 10 so that the last boundary condition gives b. sin

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ens having the form (26), we 10 so that the last boundary condition gives b. sinsinb sin+20 sin 210 in This can be satisfied if we choose 10 10 10 3rx 10 3%of 10 2 20. 10 m, b,--10, 4 or m3 = 40 - Using these in (27) gives the required solution U(x, t) = 50e-9e'ta sin 3nx + 20e-8e4 sin 2xx ._ 10e-32-a sin 4xx (28) Remark 2, For most problems we can anticipate the fact that we must make the choice c =-, since otherwise we do not get the sine terms present in the last Example 1. Thus we need not bother with any choice of the constant other than-2t. This fact can also be deduced from a physical point of view if U represents some physical variable, such as temperature, and t denotes time. This is because if c were positive then as t we would have U co, or as we often say U would be unbounded, and this would contradict conservation principles in nature. Also, if c were zero, U would be independent of time, which is . boundary condition o f Illustrative not the usual case. A EXERCISES of variables" to obtain solutions to each of the following boundarn Use the method of "separation value probiems au au au cx u au Partial Differential Equations In General 555

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Fourier Analysis Wth Applications To Boundary

By Spiegel

Publisher: MacGraw-Hill Publications

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