1

.

PDE

2

Then

3

4

5

T equation is

And

6

7

𝑢𝑢𝑡𝑡𝑡𝑡 − 𝑐𝑐2𝑢𝑢𝑥𝑥𝑥𝑥 = 𝐻𝐻(𝑥𝑥), 0 < 𝑥𝑥 < 𝐿𝐿, 𝑡𝑡 > 0.

𝑢𝑢(0, 𝑡𝑡) = 0, 𝑢𝑢(𝐿𝐿, 𝑡𝑡) = 0, (3)

𝑢𝑢𝑡𝑡(𝑥𝑥, 0) = 0, 𝑢𝑢(𝑥𝑥, 0) = 0.

8

The new equation is 𝑣𝑣𝑡𝑡𝑡𝑡 = 𝑐𝑐2𝑣𝑣𝑥𝑥𝑥𝑥 is homogeneous .

Applying the conditions 𝑢𝑢(0, 𝑡𝑡) = 𝑢𝑢(𝐿𝐿, 𝑡𝑡) we have

Now the problem is 𝑣𝑣𝑡𝑡𝑡𝑡 = 𝑐𝑐2𝑣𝑣𝑥𝑥𝑥𝑥 , 0 < 𝑥𝑥 < 𝐿𝐿, 𝑡𝑡 > 0.

9

10

11

12

13

Then 𝜓𝜓(𝑥𝑥) = 𝑇𝑇1−𝑇𝑇0𝐿𝐿

𝑥𝑥 + 𝑇𝑇0 .

14

15

𝑋𝑋′′ − 𝜆𝜆𝑋𝑋 = 0, 𝑌𝑌′′ + 𝜆𝜆𝑌𝑌 = 0.

and continue to find X and Y.

16

17

Excercises:

Ex 1.

Ex2.

Ex 3.

Ex 4.

18

Ex 5.

Exercise 7.9 page 265 Q(1,2,3,4,11,12,13)