Complex Variables and Partial Differential Equations (3130005)

Question-5b-OR

BE | Semester-3 Winter-2019 | 26-11-2019

Q5) (b)

Find the temperature in the thin metal rod of length $L$ with both the ends insulated and initial temperature is $\sin \frac{πx}{L}$ .

→we know that the solution of heat equation

\frac{\partial u}{\partial t} = k \frac{\partial^{2} u}{\partial x^{2}}

is given by

u (x, t) = \sum_{n = 1}^{\infty} B_{n} \sin (\frac{nπx}{L}) e^{- \frac{n^{2} π^{2} kt}{L^{2}}}

Using $u (x, 0) = \sin \frac{nπ}{L}$ , we get

u (x, 0) = \sum_{n = 1}^{\infty} B_{n} \sin (\frac{nπx}{L})

$\Rightarrow \sin \frac{nπ}{L} = B_{1} \sin \frac{nπ}{L} + B_{2} \sin \frac{2 nπ}{L} + B_{3} \sin \frac{3 nπ}{L} + . . .$

\Rightarrow B_{1} = 1, B_{2} = 0, B_{3} = 0, . . .

So, the solution is

u (x, t) = \sin \frac{πx}{L} e^{- \frac{π^{2} kt}{L^{2}}}

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