We will cover following topics

Equation Of A Vibrating String

Let \(u(x,t)\) represents the wave equation of a vibrating string, which satisfies the following equation and the initial conditions:

\[\dfrac{\partial^{2} u}{\partial t^{2}}=c^{2} \dfrac{\partial^{2} u}{\partial x^{2}}\] \[u(x, 0)=f(x) \quad \dfrac{\partial u}{\partial t}(x, 0)=g(x)\] \[u(0, t)=0 \quad u(L, t)=0\]

Then, the solution to this PDE is given by:

\[u(x, t)=\sum_{n=1}^{\infty}\left[A_{n} \cos \left(\dfrac{n \pi c t}{L}\right) \sin \left(\dfrac{n \pi x}{L}\right)+B_{n} \sin \left(\dfrac{n \pi c t}{L}\right) \sin \left(\dfrac{n \pi x}{L}\right)\right]\]

where

\[A_{n}=\dfrac{2}{L} \int_{0}^{L} f(x) \sin \left(\dfrac{n \pi x}{L}\right) d x\]

and

\[B_{n}=\dfrac{2}{n \pi c} \int_{0}^{L} g(x) \sin \left(\dfrac{n \pi x}{L}\right) d x, \quad n=1,2,3, \dots\]

Heat Equation

The heat equation with boundary values is given by:

\[\dfrac{\partial u}{\partial t}=k \dfrac{\partial^{2} u}{\partial x^{2}}\] \[u(x, 0)=f(x) \quad u(0, t)=0 \quad u(L, t)=0\]

The solution to this equation is given by:

\[u(x, t)=\sum_{n=1}^{M} B_{n} \sin \left(\dfrac{n \pi x}{L}\right) \mathrm{e}^{-k\left(\dfrac{n \pi}{L}\right)^{2} t}\]

where

\[B_{n}=\dfrac{2}{L} \int_{0}^{L} f(x) \sin \left(\dfrac{n \pi x}{L}\right) d x, \quad n=1,2,3, \ldots\]

Laplace Equation

Let us consider the 2-D Laplace equation \(\nabla^{2} u=\dfrac{\partial^{2} u}{\partial x^{2}}+\dfrac{\partial^{2} u}{\partial y^{2}}=0\) with boundary conditions given below:

\[u(0, y)=g_{1}(y) \quad u(L, y)=g_{2}(y)\] \[u(x, 0)=f_{1}(x) \quad u(x, H)=f_{2}(x)\]

In these types of problems, we will use separation of variables method to solve for the solution \(u\).

PYQs

Equation Of A Vibrating String

1) Given the one-dimensional wave equation \(\dfrac{\partial^{2} y}{\partial t^{2}}=c^{2} \dfrac{\partial^{2} y}{\partial x^{2}}\); \(t > 0\), where \(c^{2}=\dfrac{T}{m}\), \(T\) 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 \(\left [ \dfrac{\partial y}{\partial t}\right]_{t=0}=0\), \(y(x, 0)=a \sin \dfrac{\pi x}{t}\), \(0< x< l\), \(a>0\).

[2017, 20M]

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

[2014, 15M]

3) Solve \(\dfrac{\partial^{2} u}{\partial t^{2}}=\dfrac{\partial^{2} u}{\partial x^{2}}\), \(0< x< 1\), \(t > 0\), given that: i) \(u(x, 0)=0\), \(0 \leq x \leq 1\)
ii) \(\dfrac{\partial u}{\partial t}(x, 0)=x^{2}\), \(0 \leq x \leq 1\)
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 \(u_{t t}=C^{2} u_{x x}\). The ends of the string are fixed at and \(x=0\) and \(l\). The initial velocity is zero. The initial displacement is given by \(u(x,0)=\left \{ \begin{array}{ll}{\dfrac{x}{l},} & {0 < x < \dfrac{l}{2}} \\ {\dfrac{1}{l}(l-x),} & {\dfrac{l}{2} < x < l}\end{array}\right.\) 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 \((3 l, 0)\) corresponding to zero initial velocity and following initial deflection:
\(f(x)=\left \{ \begin{array}{ll}{\dfrac{h x}{l}} & {\text { when } 0 \leq x \leq 1} \\ {\dfrac{h(3 l-2 x)}{l}} & {\text { when } l \leq x \leq 2 l} \\ {\dfrac{h(x-3 l)}{l}} & {\text { when } 2 l \leq x \leq 3 l}\end{array}\right.\)
where \(h\) is a constant.

[2003, 30M]

Heat Equation

1) A thin annulus occupies the region \(0<a\leq r\leq b\), \(0\leq \theta \leq 2\pi\). 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 \(T=Kcos\dfrac{\theta}{2}\), 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 \(p=10.6 g / c m^{3}\), thermal conductivity \(K=1.04 /\left(c m \sec ^{\circ} C\right)\) 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 \(u_{t}=c^{2} u_{x x}\) 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
\(\begin{array}{ll}{u_{t}-u_{x x}=0,} & {0 < x< 2, t > 0} \\ {u(0, t)=u(2, t)=0} & {t > 0} \\ {u(x, 0)=x(2-x),} & {0 \leq x \leq 2}\end{array}\).

[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 \(\dfrac{\partial u}{\partial t}=c^{2} \dfrac{\partial^{2} u}{\partial x^{2}}\) 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 \(\mathrm{cm}\) 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: \(\begin{aligned} \dfrac{\partial u}{\partial t}=\dfrac{\partial^{2} u}{\partial x^{2}}, 0 < x < l, t > 0 \\ u(0, t) &=u(l, t)=0 \\ u(x, 0) &=x(l-x), 0 \leq x \leq t \end{aligned}\).

[2002, 30M]

Laplace Equation

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

[2011, 20M]

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

[2007, 30M]

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