PYQs

1) Given the one-dimensional wave equation ${\displaystyle \frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{t}^2}}=c^2{\displaystyle \frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{x}^2}}$; $t>0$, where $c^2={\displaystyle \frac{T}{m}}$, $T$ the constant tension in the string and $m$ is the mass per unit length of the string.
i) Find the appropriate solution of the wave equation.
ii) Find also the solution under the conditions $y(0,t)=0$, $y(l,t)=0$ for all $t$ and ${\left[{\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }t}}\right]}_{t=0}=0$, $y(x,0)=a\sin {\displaystyle \frac{\pi x}{t}}$, $0<x<l$, $a>0$.

[2017, 20M]

2) Find the deflection of a vibrating string (length $=\pi$, ends fixed, ${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}}$=${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}})$ corresponding to zero initial velocity and initial deflection. $f(x)=k(\sin x-\sin 2x)$.

[2014, 15M]

3) Solve ${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}}={\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$, $0<x<1$, $t>0$, given that: i) $u(x,0)=0$, $0\le x\le 1$
ii) ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}(x,0)=x^2$, $0\le x\le 1$
iii) $u(0,t)=u(1,t)=0$, for all $t$

[2014, 15M]

4) A tightly stretched string with fixed end points $x=0$ and $x=l$ is initially at rest in equilibrium position. If it is set vibrating by giving each point a velocity $\lambda x(l-x)$, find the displacement of the string at any distance $x$ from one end at any time $t$.

[2013, 20M]

5) A string of length $l$ is fixed at its ends. The string from the mid-point is pulled up to a height $k$ and then released from rest. Find the deflection $y(x,t)$ of the vibrating string.

[2012, 20M]

6) A tightly stretched string has its ends fixed at $x=0$ and $x=1$. At time $t=0$, the string is given a shape defined by $f(x)=\mu x(l-x)$, where $\mu$ is a constant, and then released. Find the displacement of any point $x$ of the string at time $t>0$.

[2009, 30M]

7) The deflection of a vibrating string of length $l$, is governed by the partial differential equation $u_{tt}=C^2{u}_{xx}$. The ends of the string are fixed at and $x=0$ and $l$. The initial velocity is zero. The initial displacement is given by $u(x,0)=\{\begin{array}{ll}{\displaystyle \frac{x}{l}},& 0<x<{\displaystyle \frac{l}{2}}\\ {\displaystyle \frac{1}{l}}(l-x),& {\displaystyle \frac{l}{2}}<x<l\end{array}$ Find the deflection of the string at any instant of time.

[2006, 30M]

8) A uniform string of length $l$, held tightly between $x=0$ and $x=l$ with no initial displacement, is struck at $x=a$, $0<a<l$, with velocity $v_0$. Find the displacement of the string at any time $t>0$.

[2004, 30M]

9) Find the deflection $u(x,t)$ of a vibrating string, stretched between fixed points $(0,0)$ and $(3l,0)$ corresponding to zero initial velocity and following initial deflection:
$f(x)=\{\begin{array}{ll}{\displaystyle \frac{hx}{l}}& \text{ when }0\le x\le 1\\ {\displaystyle \frac{h(3l-2x)}{l}}& \text{ when }l\le x\le 2l\\ {\displaystyle \frac{h(x-3l)}{l}}& \text{ when }2l\le x\le 3l\end{array}$
where $h$ is a constant.

[2003, 30M]

1) A thin annulus occupies the region $0<a\le r\le b$, $0\le \theta \le 2\pi$. The faces are insulated.Along the inner edge the temperature is maintained at $0^{\circ }$, while along the outer egde the temperature is held at $T=Kcos{\displaystyle \frac{\theta }{2}}$, where $K$ is a constant. Determine the temperature distribution in the annulus.

[2018, 20M]

2) Find the temperature $u(x,t)$ in a bar of silver of lengtant cross section of area 1$c{m}^2$. Let density $p=10.6g/c{m}^3$, thermal conductivity $K=1.04/(cm{\sec }^{\circ }C)$ and specific heat $\sigma =0.056/g^{\circ }C$ the bar is perfectly isolated laterally with ends kept at $0^{\circ }C$ and initial temperature $f(x)=\sin (0.1\pi x{)}^{\circ }C$ note that $u(x,t)$ follows the head equation $u_t=c^2{u}_{xx}$ where $c^2=k/(\rho \sigma )$.

[2016, 20M]

3) The edge $r=\alpha$ of a circular plate is kept at temperature $f(\theta )$. The plate is insulated so that there is no loss of heat from either surface. Find the temperature distribution in steady state.

[2012, 20M]

4) Obtain temperature distribution $y(x,t)$ in a uniform bar of unit length whose one end is kept at ${10}^0$ and the other end is insulated. Also it is given that $y(x,0)=1-x$, $0<x<1$.

[2011, 20M]

5) Solve the following heat equation
$\begin{array}{ll}{u}_t-u_{xx}=0,& 0<x<2,t>0\\ u(0,t)=u(2,t)=0& t>0\\ u(x,0)=x(2-x),& 0\le x\le 2\end{array}$.

[2010, 20M]

6) Find the steady state temperature distribution in a thin rectangular plate bounded by the lines $x=0$, $x=a$, $y=0$ and $y=b$. The edges and $x=0$, $x=a$ and $y=0$ are kept at temperature zero while the edge $y=b$ is kept at ${100}^{\circ }\mathrm{C}$.

[2008, 30M]

7) Solve the equation ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}=c^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$ by separation of variables method subject to the conditions $u(0,t)=0=u(l,t)$, for all $t$ and $u(x,0)=f(x)$ for all $x$ in $[0,l]$.

[2007, 30M]

8) The ends $A$ and $B$ of a rod 20 $\mathrm{c}\mathrm{m}$ long have the temperature at ${30}^{\circ }\mathrm{C}$ and ${80}^{\circ }\mathrm{C}$ until steady state prevails. The temperatures of ends are changed to ${40}^{\circ }\mathrm{C}$ and ${60}^0\mathrm{C}$ respectively. Find the temperature distribution in the rod at time $t$.

[2005, 30M]

9) Solve: $\begin{array}{r}{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}={\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}},0<x<l,t>0\\ u(0,t)& =u(l,t)=0\\ u(x,0)& =x(l-x),0\le x\le t\end{array}$.

[2002, 30M]

1) Solve ${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}}=0,0\le x\le a,0\le y\le b$ satisfying the boundary conditions $u(0,y)=0$, $u(x,0)=0$, $u(x,b)=0$, ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}(a,y)=T{\sin }^3{\displaystyle \frac{\pi y}{a}}$.

[2011, 20M]

2) Solve $u_{xx}+u_{yy}=0$ in $D$ where $D=\{(x,y):0<x<a,0<y<b\}$ is a rectangle in a plane with the boundary conditions:
$u(x,0)=0$, $u(x,b)=0$, $0\le x\le a$
$u(0,y)=g(y)$, $u_x(a,y)=h(y)$, $0\le y\le b$

[2007, 30M]