Questions and Answers

PART-A 

QUESTIONS AND ANSWERS

1. What conditions are assumed in deriving the one dimensional wave equation? 

See Page No. 3.13

2. What are the various solutions of

See Page No. 3.11

3. A string is stretched and fastened to two points l apart. Motion is started by displacing the string into the form from which it is released at time t = 0. Formulate this problem as the boundary value problem.

Solution: The displacement function y (x, t) is the solution of the wave equation.

The boundary and initial conditions are

4. A string of length 2 l stretched to a constant tension T, is fastened at both the ends and hence fixed. The mid point of the string is taken to a height b and then released from rest in that position.

It is desired to It solve for transverse vibrations of the string. Write the governing equation and the corresponding conditions.

See Page No. 3.24

5. What is the constant a2 in the wave equation

utt = a2 uxx?

(OR)

In the wave equation what does c2 stand for?

Solution :

6. State the suitable solution of the one dimensional heat equation

Solution: 

7. State the governing equation for one dimensional heat equation and necessary conditions to solve the problem.

Solution: The one dimensional heat equation is

where u (x, t) is the temperature at time t at a point distant x from the left end of the rod.

The boundary conditions are

(i) u (0,t) = k1°C for all t ≥ 0

(ii) u (l,t) = k2°C for all t ≥ 0

 (l being the length of the one dimensional rod)

The initial condition is

(iii) u (x, 0) = f(x), 0 < x < 1

8. Write all variable separable solutions of the one dimensional heat equation u1 = α2 uxx 

Solution :

9. Write down the diffusion problem in one-dimension as a boundary value problem in two different forms.

Solution :

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

Sólution :

(i) Heat flows from higher to lower temperature.

(ii) The rate at which heat flows across any area is proportional to the area and to the temperature gradient normal to the curve. This constant of proportionality is known as the thermal conductivity (k) of the material. It is known as Fourier law of heat conduction.

11. Write any two solutions of the Laplace equation Uxx + Uyy = 0 involving exponential terms in x or y.

Sol.

12. In steady state conditions, derive the solution of one dimensional heat flow equation.

Solution: The p.d.e. of unsteady one dimensional heat flow is

In steady state condition, the temperature u depends only on x and not on time t

The general solution is u = ax + b where a, b are arbitrary.

13. Write the boundary conditions and initial conditions for solving the vibration of string equation, if the string is subjected to initial displacement f(x) and initial velocity g(x). 

Solution: The wave equation is

14. Write down the governing equation of two dimensional steady state heat conduction.

Solution: The required equation is

15. The ends A and B of a rod of length 10 cm have their temperature kept at 20°C and 70°C. Find the steady state temperature distribution on the rod.

Solution:

16. Write down the three possible solutions of Laplace equation in two-dimensions.

Solution :

Laplace equation in two-dimensions is V2 u =

17. Write down the partial differential equation governing the transverse vibrations of an elastic string.

Solution :

18. Explain the various variables dominating the wave equation.

Solution: The wave equation is

y → displacement

x → distance of the point on the string from fixed point along x direction

t → time

19. Classify the partial differential equation

Solution :

20. Classify the partial differential equation

Solution :

⸫ the given equation is hyperbolic equation.

21. Classify the partial differential equation uxx + 2uxy + uyy = 0

Solution :

22. Classify the partial differential equation

fxx + 2fxy + 4fyy = 0

Solution :

Given fxx + 2fxy + 4fyy = 0

23. Classify the partial differential equation

uxx + uxy + uyy = 0

Solution : 

⸫ The given equation is an elliptic equation.