PYQs

1) Given the one-dimensional wave equation ∂2y∂t2=c2∂2y∂x2; t>0, where c2=Tm, 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 [∂y∂t]t=0=0, y(x,0)=asinπxt, 0<x<l, a>0.

[2017, 20M]

2) Find the deflection of a vibrating string (length =π, ends fixed, ∂2u∂t2=∂2u∂x2) corresponding to zero initial velocity and initial deflection. f(x)=k(sinx−sin2x).

[2014, 15M]

3) Solve ∂2u∂t2=∂2u∂x2, 0<x<1, t>0, given that: i) u(x,0)=0, 0≤x≤1
ii) ∂u∂t(x,0)=x2, 0≤x≤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 λx(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)=μx(l−x), where μ 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 utt=C2uxx. 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)={xl,0<x<l21l(l−x),l2<x<l 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 v0. 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)={hxl when 0≤x≤1h(3l−2x)l when l≤x≤2lh(x−3l)l when 2l≤x≤3l
where h is a constant.

[2003, 30M]

1) A thin annulus occupies the region 0<a≤r≤b, 0≤θ≤2π. The faces are insulated.Along the inner edge the temperature is maintained at 0∘, while along the outer egde the temperature is held at T=Kcosθ2, 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 1cm2. Let density p=10.6g/cm3, thermal conductivity K=1.04/(cmsec∘C) and specific heat σ=0.056/g∘C the bar is perfectly isolated laterally with ends kept at 0∘C and initial temperature f(x)=sin(0.1πx)∘C note that u(x,t) follows the head equation ut=c2uxx where c2=k/(ρσ).

[2016, 20M]

3) The edge r=α of a circular plate is kept at temperature f(θ). 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 100 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
ut−uxx=0,0<x<2,t>0u(0,t)=u(2,t)=0t>0u(x,0)=x(2−x),0≤x≤2.

[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∘C.

[2008, 30M]

7) Solve the equation ∂u∂t=c2∂2u∂x2 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 cm long have the temperature at 30∘C and 80∘C until steady state prevails. The temperatures of ends are changed to 40∘C and 600C respectively. Find the temperature distribution in the rod at time t.

[2005, 30M]

9) Solve: ∂u∂t=∂2u∂x2,0<x<l,t>0u(0,t)=u(l,t)=0u(x,0)=x(l−x),0≤x≤t.

[2002, 30M]

1) Solve ∂2u∂x2+∂2u∂y2=0,0≤x≤a,0≤y≤b satisfying the boundary conditions u(0,y)=0, u(x,0)=0, u(x,b)=0, ∂u∂x(a,y)=Tsin3πya.

[2011, 20M]

2) Solve uxx+uyy=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≤x≤a
u(0,y)=g(y), ux(a,y)=h(y), 0≤y≤b

[2007, 30M]