This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: problem, we found that a solution of this heat conduction problem is given by u ( x,t ) = ∞ X n =1 c n en 2 π 2 α 2 t L 2 sin nπx L , where the coeﬃcients c n are chosen so that ∞ X n =1 c n sin nπx L = f ( x ) = u ( x, 0) for 0 ≤ x ≤ L. We now know how to ﬁnd the sine series for f ( x ) on 0 ≤ x ≤ L so we are in business! EXAMPLE. Find the solution of the heat conduction problem . 01 u xx = u t , < x < 100 , t > 0; u (0 ,t ) = u (100 ,t ) = 0 , t > 0; u ( x, 0) = 3 sin πx5 sin 4 πx + 2 sin 17 πx 2 , ≤ x ≤ 100 . EXAMPLE. Consider the conduction of heat in a rod 80 cm long whose sides are insulated and whose ends are maintained at 0 ◦ C for all t > 0. Find an expression for the temperature u ( x,t ) if the initial temperature distribution in the rod is given below. Suppose that α 2 = 1. u ( x, 0) = ≤ x < 20 30 20 ≤ x ≤ 60 60 < x ≤ 80 HOMEWORK: SECTION 10.5...
View
Full Document
 Spring '10
 SCHURLE
 Heat, Heat Transfer, Boundary value problem, Joseph Fourier, heat conduction problem

Click to edit the document details