Unit 6
Applications of Partial Differential Equations (PDE)
Q(1) A tightly stretched string with fixed end points and is initially in a position given by If it is released from rest from this position. Find the displacement y at any distance x from one end at any time t.
Ans-
The differential equation satisfied by is
The initial and boundary conditions are given by
(i)
(ii)
(iii)
(iv)
The most general solution is given by
From condition (i) we get,
From condition (iii) we get,
The most general solution will become
(1)
From Condition (ii)
Equation (1) becomes
Combining all these solutions we get,
(2)
Applying condition (iv)
By using
Comparing we get,
Substitute in equation (2) we get,
Q(2) . Solve: subject to condition
(i)
(ii)
(iii) is odd
(iv) for
Ans-
We have
Step 1: The most general Heat equation is
(1)
Step 2: Use first B.C
Step 3: Equation (1) becomes
(2)
Step 4: Use 2nd B.C
,
Step 5: Equation (2) becomes
Step 6: Take addition of all solutions and
(3)
Step 7: Use I.C
Step 8: To find we have,
Put in eqn (3) we get,
Q(3) Solve : subject to :
Ans-
We have
Step 1: The most general heart equation is
….. (1)
Step 2: Use B.C
Step 3: Equation (1) becomes
(2)
Step 4: Use 2nd B.C
,
Step 5: Equation (2) becomes
Step 6: Take addition of all solutions and
(3)
Step 7: Use I.C.
Step 8: To find
Put in Equation (3) we get,
Q(4) Solve the equation with conditions
(i) when for all .
(ii) when for all values of .
(iii) when for all values of .
(iv) when for
Ans-
Solution:
Here in condition (i), when for all values of , for all z is given
Step 1:
The most general solution is
(1)
Step 2:
Now condition (i) that must remain finite as , this is possible only if
Step 3:
Apply condition (ii)
Step 4: The equation (1) becomes
Step 5:
By condition (iii)
Step 6:
Combining all these solution we get
Applying condition (iv) we have
Which is half range since series
The complete solution is
Q(5) A rectangular plate with insulated surfaces is 10 cm wide and so long compared to its width that may be considered infinite in length without introducing an appreciable error. If the temperature along short edge is given
while the two long and as well as the other short edge are kept at 0C. Find steady state temperature
Ans-
We have to solve the P.D.E
Subject to the conditions
(i)
(ii)
(iii)
(iv)
Step 1: The most general solution is
(1)
Step 2:
By (i) condition,
Step 3:
By (ii) condition
The equation (1) becomes
Step 4: By (iii) condition
Step 5:
Applying condition (iv), we have
By comparing we get,
,
The answer is
Q(6) Solve by method separation of variables
where
Ans-
Given
Let the solution of this equation be S
Solving
Integrate both sides
By solving
Integrating both sides,
When ,
(given)
By comparing
Also
Q(8) An infinity long uniform metal plate is enclosed between lines and for . The temperature is zero along the edges and at infinity. If the edge is kept at a constant temperature . Find the temperature distribution .
Ans-
We have to solve P.D.E.
Subject to the boundary condition.
In condition (iii), when is given
(i)
(ii)
(iii)
(iv)
In condition (iii), when is given
The most general solution is given by
Now condition (iii) u(x , y) must remain as
By condition (i)
By condition (ii)
Apply condition (iv) we have
Which is represented by half range Fourier fine series
For in
The complete solution is
Q(9) Solve the equation where satisfies the following conditions
(i)
(ii)
(iii)
(iv) is finite
Ans:
We have
Step 1: The most general solution is
Step 2: Use 1st B.C
step 3: Equation (1) becomes
Step 4: Use 2nd B.C
,
Step 5: Equation (2) becomes
Step 6: Take addition of all solutions and
Step 7: Use I.C
Step 8: To find we have
Q(10) A string is stretched and fastened to two points apart. Motion is started by displacing by the string in form from which it is released at time t=0 find the displacement from one end.
Ans-
We have,
Subject to the condition:
(i)
(ii)
(iii)
(iv)
The general solution is
By condition (i)
By condition (iii)
Eq (1) becomes
Applying condition (ii) we get
Substituting in equation (2) we get
Combining these solutions we get
Applying condition (iv) we get
By comparing
Final Answer is