Complex Variables and Partial Differential Equations  (3130005)

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

Solve the p.d.e. $2\mathrm{r} + 5\mathrm{s} + 2\mathrm{t} = 0$

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 $\frac{\partial \mathrm{u}}{ \partial \mathrm{x} } = 2\frac{\partial \mathrm{u}}{ \partial \mathrm{t} } + \mathrm{u}$.

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 the PDE $\left({\mathrm{D}}^2 - \mathrm{D}{\text{'}}^2 + \mathrm{D} - \mathrm{D}\text{'}\right)\mathrm{z} = 0$.

Solve $\left(2{\mathrm{D}}^2 - 5\mathrm{DD}\text{'} + \mathrm{D}{\text{'}}^2\right)\mathrm{z} = 24\left(\mathrm{y} - \mathrm{x}\right)$

 

Solve the p.d.e. ${\mathrm{u}}_{\mathrm{x}} + {\mathrm{u}}_{\mathrm{y}} = 2\left(\mathrm{x} + \mathrm{y}\right)\mathrm{u}$ using method of separation of variables.

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.

Find the solution of wave equation ${\mathrm{u}}_{\mathrm{tt}} = {\mathrm{c}}^2 {\mathrm{u}}_{\mathrm{xx}} , 0 \le \mathrm{x} \le \mathrm{\pi }$ with initial and boundary condition $\mathrm{u}\left(0, \mathrm{t}\right) = \mathrm{u}\left(\mathrm{\pi }, \mathrm{t}\right) = 0 ; \mathrm{t} > 0 , \mathrm{u}\left(\mathrm{x}, 0\right) = \mathrm{k}\left(\sin \mathrm{x} - \sin 2\mathrm{x}\right), {\mathrm{u}}_{\mathrm{t}}\left(\mathrm{x}, 0\right) = 0 ; 0 \le \mathrm{x} \le \mathrm{\pi }, \left({\mathrm{c}}^2 = 1\right)$

Solve $\mathrm{r} + \mathrm{s} + \mathrm{q} - \mathrm{z} = 0$.

Using method of separation of variables, solve $2{\mathrm{u}}_{\mathrm{x}} = {\mathrm{u}}_{\mathrm{t}} + \mathrm{u}, \mathrm{u}\left(\mathrm{x}, 0\right) = 4{\mathrm{e}}^{-3\mathrm{x}}$.

Find the solution of ${\mathrm{u}}_{\mathrm{t}} = {\mathrm{c}}^2 {\mathrm{u}}_{\mathrm{tt}}$ together with the initial and boundary conditions $\mathrm{u}\left(0, \mathrm{t}\right) = \mathrm{u}\left(2, \mathrm{t}\right) = 0 ; \mathrm{t}\ge 0, \mathrm{u}\left(\mathrm{x}, 0\right) = 10 ; 0 \le \mathrm{x} \le 2.$