ANNA UNIVERSITY – PART A QUESTIONS

Applications of Partial Differential Equation

Model

1. Classify the partial differential equation

( 1 x 2 ) z x x 2 x y z x y + ( 1 y 2 ) z y y + x z x + 3 x 2 y z y 2 z = 0

2. The steady state temperature distribution is considered in a square plate with sides x = 0, y = 0, x = a, y = a. The edge y = 0 is kept at a constant temperature T and the other three edges are insulated. The same state is continued subsequently. Express the problem mathematically.

November 2002

1. Classify the PDE : 5(i) u x x = u y y (ii) u x y = u x u y + x y

2. Write any 2 solutions of laplace eqn involving exponential terms in x or y

April 2003

1. Classify the following equations:

i. 4 u x x + 4 u x y + u y y 6 u x 8 u y 16 u = 0

ii. u x x + u y y = ( u x ) 2 + ( u y ) 2

November 2003

1. Classify the following partial differential equations :

i. y 2 u x x 2 x y u x y + x 2 u y y + 2 u x 3 u = 0

ii. y 2 u x x + u y y + ( u x ) 2 + ( u y ) 2 + 7 = 0

2. An insulated rod of length 60 cm has its ends at A and B is maintained at 20 C and 80 C respectively. Find the steady state solution of the rod.

April 2004

1. Classify : u x x + u y y = 0

2. A rod 30 cm long has its ends A and B kept at 20 C and 80 C respectively until steady state conditions prevail. Find the steady state temperature in the rod.

November 2004

1. What is the constant a 2 in the wave equation u t t = a 2 u x x .

2. State any two laws which are assumed to derive one dimensional heat equation.

April 2005

1. State fourier law of heat conduction.

November 2005

1. What is the basic difference between the solutions of one dimentional wave equation and one dimentional heat equation.

2. In steady state derive the solution of one dimentional heat flow equation.

November 2005

1. Write the steady state heat flow equation in two dimension in Cartesian and polar form.

April 2006

1. A string of length of length 2l is fastened at both ends. The mid point of the string is displaced to a distance ‘b’ and released from rest in this posision.Write the initial conditions.

2. In the one dimentional heat equation h t = α 2 u x x what does α 2 stand for ?

April 2006

1. What are the possible solutions of one dimentional wave equation?

2. In steady state conditions derive the solution of 1 dimentional heat equation.ANNA UNIVERSITY – PART B QUESTIONS

Applications of Partial Differential Equation

Model

1. A taught string of length L is fastened at both ends. The mid point of the string is taken to a height of b and then released from rest in this position. Find the displacement of the string at any time t.

2. A rod 30 cm long has its ends A and B at 20 C and 80 C respectively until steady state condition prevail. The temperature at the end B is then suddenly reduced to 60 C and at the end A is raised to 40 C and maintained so. Find the resulting temperature u( x , t ).

November 2002

1. A tightly stretched string of length l has its ends fixed at x = 0 and x = l. The mid point of the string is then taken to a height h and then released from rest in that position. Obtain an expression for the displacement of the string at any subsequent time.

2. The boundary value problem governing the steady state temperature distribution in a flat, thin, square plate is given by

u x x + u y y = 0 , 0 < x < a ; 0 < y < a u ( x , 0 ) = 0 ,

u ( x , a ) = 4 sin 3 ( x / a ), 0 < x < a, u ( 0 , y ) = 0,

u ( a , y ) , 0 < y < a Find the steady state temp distribution in the plate.

April 2003

1. A tightly stretched flexible string has its ends fixed at x = 0 and x = L. At time t = 0, the string is given a shape defined by f(x) = k x 2 ( L x ) and then released from rest. Find the displacement of any point x of the string at any time t > 0.

2. A rod of length l cm long has its ends A and B at 40 C and 90 C respectively until steady state condition prevail. The temperature at the end B is then suddenly reduced to 40 C and at the end A is raised to 90 C and maintained so. Find the resulting temperature u( x , t ). Also show that the temperature at the mid point of the rod remains unaltered for all time, regardless of the material of the rod.November 2003

1. A tightly stretched string with fixed end points x = 0 and x = l is initially at rest in its equilibrium position. If it is set vibrating by giving each point a velocity k x ( l – x ). Find the displacement of the string at any time..

2. A rectangular plate with insulated surface is 10 cm wide so long compared to its width that it may be considered infinite length. If the temperature along short edge y = 0 is given u ( x , 0 ) = 8 sin ( x / 10 ) when 0 < x < 10, while the two long edges x = 0 and x = 10 as well as the other short edge are kept at 0 C, find the steady state temperature function u( x , y ).

April 2004

1. A string is stretched between two fixed points at a distance 2l apart and the points of the string are given initial velocities v where

v = ( cx / l ) in 0 < x < l Find the displacement of

= ( c / l ) ( 2 l – x ) in l < x < 2 l the string at any time.

2. Find the solution of the one dimensional diffusion equation subject to :

i. u is bounded as t ii. u x = 0 when x = 0 for all t

ii. u x = 0 when x = a t iv. u( x , 0 ) = x ( a – x ) , 0 < x < a

November 2004

1. A tightly stretched flexible string has its ends fixed at x = 0 and x = l is initially in a position given by y( x , 0 ) = y 0 sin 3 ( x / l ) . It is released from rest. Find the displacement of the string at any time t.

2. A rectangular plate with insulated surface is 10 cm wide and so long compared to its width that it may be considered infinite in length. The temperature at short edge x = 0 is given by

u = 20 y for 0 < y < 5

= 20 ( 10 y ) for 5 < y < 10

and all other edges are kept at 0 C. Find the steady state temperature.

April 2005

1. A tightly stretched string of length 2 l has its ends at x = 0 and x = 2l. The mid point of the string is then taken to a height ‘b’ and then released from rest in that position. Obtain an expression for the displacement.

November 2005

1. A rod 10 cm long has its ends A and B at 20 C and 40 C respectively until steady state condition prevail. The temperature at the end B is then suddenly reduced to 10 C and at the end A is raised to 50 C and maintained so. Find the resulting temperature u( x , t ).

November 2005

1. Find the steady state temperature distribution in a rectangular plate of sides ‘a’ and ‘b’ insulated at the lateral surfaces and satisfying the boundary conditions u(0,y) =u(a,y) = 0 for 0 < y < b, u(x,b) = 0 and u(x,0) = (ax x2) for 0 < x < a.

April 2006

1. A tightly stretched string of length l has its ends fixed at x = 0 and x = l. The point x = l / 3 of the string is then taken to a height h and then released from rest in that position. Obtain an expression for the displacement of the string at any subsequent time.

April 2006

1. A tightly stretched string with fixed end points x = 0 and x = 2l is initially at rest in its equilibrium position. If it is set vibrating by giving each point a velocity k x ( 2l – x ). Find the displacement of the string at any time..

2. An infinitely long plate in the form of an area is enclosed between the linesy = 0, y = for the positive values of x. The temperature is zero along the edges y = 0, y = and the edges at infinity. If the edge x = 0 is kept at temperature Ky find the steady state temperature distribution in the plate.