Complex Variables and Partial Differential Equations (3130005)

Determine the Mobius transformation which maps ${\mathrm{z}}_1 = 0, {\mathrm{z}}_2 = 1, {\mathrm{z}}_3 = \infty$ into $, {\mathrm{w}}_1 = -1, {\mathrm{w}}_2 = -\mathrm{i}, {\mathrm{w}}_3 = 1$.

Expand $\mathrm{f}\left(\mathrm{z}\right)=\frac{1}{ \left(\mathrm{z}-1\right)\left(\mathrm{z}+2\right) }$ valid for the region $\left(\mathrm{I}\right) \left|\mathrm{z}\right|<1$, $\left(\mathrm{II}\right) 1<\left|\mathrm{z}\right|<2$ and $\left(\mathrm{III}\right)\left|\mathrm{z}\right|>2$.

Evaluate ${\int }_{\mathrm{C}}\left(\mathrm{x}-\mathrm{y}+\mathrm{i} {\mathrm{x}}^2\right) d\mathrm{z}$ ; Where $\mathrm{C}$ is a straight line from $\mathrm{z} = 0 \mathrm{to} \mathrm{z} = 1 + \mathrm{i}$.

Check whether the following functions are analytic or not at any point,

Expand Laurent series of $\mathrm{f}\left(\mathrm{z}\right) = \frac{ 1 - {\mathrm{e}}^{\mathrm{z}} }{\mathrm{z}}$ at $\mathrm{z}= 0$and identify the singularity.

If $\mathrm{f}\left(\mathrm{z}\right) = \mathrm{u} + \mathrm{iv}$ , is an analytic function,prove that $\left(\frac{{\partial }^2}{\partial {\mathrm{x}}^2} + \frac{{\partial }^2}{\partial {\mathrm{y}}^2}\right) {\left|\mathrm{Re} \mathrm{f}\left(\mathrm{z}\right)\right|}^2 = 2 {\left|{\mathrm{f}}^{\text{'}}\left(\mathrm{z}\right)\right|}^2$ .

Evaluate the following:

$\left(\mathrm{i}\right) {\int }_{\mathrm{C}} \frac{ \mathrm{z} + 3 }{\mathrm{z} - 1}d\mathrm{z}$ ; Where C is the circle $\left(\mathrm{a}\right) \left|\mathrm{z}\right| = 2 \left(\mathrm{b}\right) \left|\mathrm{z}\right| = \frac{ 1 }{2}$.

Solve : ${\mathrm{x}}^2 \mathrm{p} + {\mathrm{y}}^2 \mathrm{q} = \left(\mathrm{x} + \mathrm{y}\right)\mathrm{z}$

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

Solve $\frac{{\partial }^2\mathrm{z}}{ \partial {\mathrm{x}}^2 } + 2\frac{{\partial }^2\mathrm{z}}{ \partial \mathrm{x}\partial \mathrm{y} } + \frac{{\partial }^2\mathrm{z}}{ \partial {\mathrm{y}}^2 } = {\mathrm{e}}^{2\mathrm{x} + 3\mathrm{y}}$.

Find the general solution of partial differential equation ${\mathrm{u}}_{\mathrm{xx}} = 9{\mathrm{u}}_{\mathrm{y}}$ using method of separation of variables.

Using method of separation of variables, solve $\frac{\partial \mathrm{u}}{ \partial \mathrm{x} } = 2\frac{\partial \mathrm{u}}{ \partial \mathrm{t} } + \mathrm{u}$.

Solve $\mathrm{z}\left(\mathrm{xp} - \mathrm{yq}\right) = {\mathrm{y}}^2 - {\mathrm{x}}^2$.

A string of length $\mathrm{L} = \mathrm{\pi }$ has its ends fixed at $\mathrm{x} = 0$ and $\mathrm{x} = \mathrm{\pi }$. At time $\mathrm{t} = 0$, the string is given a slope defined by $\mathrm{f}\left(\mathrm{x}\right) = 50\mathrm{x} (\mathrm{\pi } - \mathrm{x})$, then it is released. Find the deflection of the string at any time $\mathrm{t}$.

Solve ${\mathrm{p}}^3 + {\mathrm{q}}^3 = \mathrm{x} + \mathrm{y}$.

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

Derive the one dimensional wave equation that governs small vibration of an elastic string. Also state physical assumptions that you make for the system.