Complex Variables and Partial Differential Equations (3130005)

Question-5c

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

Q5) (c)

→ We know that, the solution of heat equation

\frac{\partial^{2} u}{\partial t^{2}} = c^{2} \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}}}