Tutorial 12: Partial Differential Equations
Question PDF
Written Solution
- PDEs solvable as ODEs. Solve the following PDEs like an ODE.
a)
b)
c)
d)
e)
Solution
Question (a)
Question (b)
Question (c)
Question (d)
Question (e)
- Solve the following PDEs using direct integration: a) b) given that at and . c) given that and . d) given that and .
Solution
Question (a)
Question (b)
Using
Hence,
Using
Question (c)
Using
Hence,
Using
Thus, , where is a constant
Question (d)
Using
Hence,
Using
- Solve the following equation using the method of separation of variable: a) with (General and particular solutions). b) c)
Solution
Question (a)
Let , thus and . Thus,
For ODE,
Thus, the particular solution:
Question (b)
Substituting into the partial differential equation yields .
Separating variables and using the separation constant we obtain
When
so that
A particular product solution of the partial differential equation is
When the differential equations become and , so in this case , and .
Question (c)
Substituting into the partial differential equation yields .
Separating variables and using the separation constant we obtain,
Then,
We consider three cases:
Case 1
If then and . Also, and , so
Case 2
If , then , and . Also, and , so
Case 3
If , then , and . Also, and , so
- Solve the eigenvalue problem a. b. c.
Solution
Question (a)
If satisfy
To make , with , we must choose,
Therefore, the eigenvalue is:
And the associated eigenfunction is:
Question (b)
If satisfy , then
The and is constant,
Thus,
To make with , we must choose,
Thus, the eigenvalue is:
And the eigenfunction is:
Question (c)
If satisfy with
The and is constant, the boundary condition of implies that
Since
Thus,
Hence,
Thus, the eigenvalue is:
And the eigenfunction is:
- Show that the eigenfunctions
are orthogonal on .
Solution
Show that if and are distinct function.
Then,
For Case 1,
If and , where and are distinct positive integers, then,
Using the identity: , with and , then
, , thus, are true.
So, it is orthogonal.
For Case 2,
If and ; and are positive integers. Then,
Using the identity: with and , then
So, it is orthogonal.