Main Page Chapter 1 Chapter 2 Chapter 3 Chapter 4 List of Topics

EXERCISE

1. Verify that the functions

(a)

(b)

are solutions of equation

SOLUTION :

(a)

\

Hence

So, is a solution of equation

(b) Do yourself.

2. Solve the partial differential equations

(a)

(b)

SOLUTION

(a) Do yourself.

(b) ……………………(1)

Let,

\ (1) becomes

or,

Integrating w.r. to y,

\

or,

Integrating w.r. to x,

\ ,

where

WAVE EQUATION

T = tension

r = density

3. Find the possible values of a and b in such that it satisfies the wave equation. Also show that and are solution of wave equation.

SOLUTION : Do yourself.

4. Find the solution of the wave equation corresponding to the triangular initial deflection (i.e. initial displacement)

(a)

(b)

(c)

(d)

and initial velocity zero i.e., g(x) = 0

HEAT EQUATION

K = thermal diffusion

k = thermal conductivity (constant for one material)

s = the specific heat

r = density

5. Show that is a solution of heat equation.

SOLUTION : Do yourself

6.

(a) Find the temperature u(x,t) in a laterally insulated copper bar 80 cm long if the initial temperature is and the ends are kept at 0°C. How long will it take for the maximum temperature in the bar to drop to 50°C? Physical data for copper : density (r) = 8.92 gm/cm³, specific heat (s) = 0.092 cal/gm°C and thermal conductivity (k) = 0.95 cal/cm sec °C.

(b) Solve the problem in (a), when the initial temperature is and other data are as before.

7. Find the temperature in a laterally bar of length L whose ends are kept at temperature 0°C , assuring that the initial temperature is

(a)

(b)

8. Find a solution of each in example (6) , (7) assuming that both ends of the bar are insulated (instead of kept at temperature 0).

LAPLACE’S EQUATION

9. Show that the functions

(a)

(b)

are solution of Laplace's equation.

10. Two semi-infinite conducting plates are arranged at an angle f_{0 ,} as shown in figure. One plate is charged to 10 volts and the other to V₀ volts. A gap at the tip insulates one plate from the other. Find the potential V in the region 0<V<f₀.

SOLUTION :

4. (d) The wave equation is

………………….(1)

Given boundary conditions are

and

We know solution of (1) is

…………………….(2)

Now,

and

p = n , n = 1, 2, 3, ………

Again,

Hence from (2)

Hence …………………(3)

Now,

\

\

Hence from (3) the required solution is,

SOLUTION

6. (a) The heat equation is

…………………….(1)

where

The boundary conditions are

Since both ends of the bar kept at 0°C, hence temperature at any point on the bar is

……………………(2)

\

Hence,

\ From (2)

\ The maximum temperature in the bar is

,

Now,

\

Now, if,

then

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