# H8_130 - Math 415 Homework 8 10.5.7 Solve 100 u xx = u t<...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 415 Homework 8 10.5.7 Solve 100 u xx = u t , < x < 1 , t > 0; u (0 ,t ) = 0 = u (1 ,t ) , t > 0; u ( x, 0) = sin2 πx- sin5 πx, ≤ x ≤ 1 . The general solution to this heat equation problem is u ( x,t ) = ∞ X n =1 c n e- 100 n 2 π 2 t sin nπx. Looking at the initial condition, we must have u ( x, 0) = ∞ X n =1 c n sin nπx = sin2 πx- sin5 πx. If two sums of sine functions with similar inputs are equal, then the coefficients must be equal. I.e. c 1 = 0 since there is no sin πx term on the right hand side, c 2 = 1, c 5 =- 1, and c i = 0 for i 6 = 2 or 5. So our final solution is u ( x,t ) = e- 400 π 2 t sin2 πx- e- 2500 π 2 t sin5 πx. 10.5.10 Consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0 ◦ C for all t > 0. Find an expression for the temperature u ( x,t ) if the initial temperature is the function u ( x, 0) = x, for 0 ≤ x < 20 , 40- x for 20 ≤ x ≤ 40 ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

H8_130 - Math 415 Homework 8 10.5.7 Solve 100 u xx = u t<...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online