Complex Variables and Partial Differential Equations (3130005)

Winter-2019

Question-4c

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

Q4) (c)

Solve : $\frac{\partial u}{\partial t} = k \frac{\partial^{2} u}{\partial x^{2}}$ for the condition of heat along rod without radiation subject to the conditions $(i) \frac{\partial u}{\partial t} = 0$ for $x = 0$ and $x = L$ & $(ii) u = Lx - x^{2}$ at $t = 0$ for all $x$ .

Solution of heat equation:

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

$B_{n} = \frac{2}{L} \int_{0}^{L} f (x) \sin (\frac{nπx}{L}) d x$

B_{n} = \frac{2}{L} \int_{0}^{L} (Lx - x^{2}) \sin (\frac{nπx}{L}) d x

$B_{n} = \frac{2}{L} {[(Lx - x^{2}) \{\frac{- \cos (\frac{nπx}{L})}{\frac{nπ}{L}}\} - (L - 2 x) \{\frac{- \sin (\frac{nπx}{L})}{\frac{n^{2} π^{2}}{L^{2}}}\} + (- 2) \{\frac{\cos (\frac{nπx}{L})}{\frac{n^{3} π^{3}}{L^{3}}}\}]}_{0}^{L}$

B_{n} = \frac{2}{L} [- \frac{2 L^{3}}{n^{3} π^{3}} \cos nπ + \frac{2 L^{3}}{n^{3} π^{3}}] = = \frac{2}{L} \times \frac{2 L^{3}}{n^{3} π^{3}} [1 - \cos nπ]

B_{n} = \frac{4 L^{2}}{n^{3} π^{3}} [1 - {(- 1)}^{n}]

Solution:

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

u (x, t) = \sum_{n = 1}^{\infty} \frac{4 L^{2}}{n^{3} π^{3}} [1 - {(- 1)}^{n}] \sin (\frac{nπx}{L}) e^{- \frac{n^{2} π^{2} kt}{L^{2}}}

Complex Variables and Partial Differential Equations (3130005)

Q4) (c)

Solve : $\frac{\partial u}{\partial t} = k \frac{\partial^{2} u}{\partial x^{2}}$ for the condition of heat along rod without radiation subject to the conditions $(i) \frac{\partial u}{\partial t} = 0$ for $x = 0$ and $x = L$ & $(ii) u = Lx - x^{2}$ at $t = 0$ for all $x$ .

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Complex Variables and Partial Differential Equations (3130005)

Q4) (c)

Solve : ∂u ∂t = k ∂2u ∂x2 for the condition of heat along rod without radiation subject to the conditions i ∂u ∂t = 0 for x = 0 and x = L & ii u = Lx - x2 at t = 0 for all x.

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Solve : $\frac{\partial u}{\partial t} = k \frac{\partial^{2} u}{\partial x^{2}}$ for the condition of heat along rod without radiation subject to the conditions $(i) \frac{\partial u}{\partial t} = 0$ for $x = 0$ and $x = L$ & $(ii) u = Lx - x^{2}$ at $t = 0$ for all $x$ .