Week 12

Tutorial 12: Partial Differential Equations

Question PDF

TUTORIAL 12-PDE-Questions.pdf

542.8KB

Written Solution

w12.pdf

1172.8KB

PDEs solvable as ODEs. Solve the following PDEs like an ODE.

👉

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.