Sec3-4_hints

# Sec3-4_hints - • We solved the homogeneous heat equation...

This preview shows page 1. Sign up to view the full content.

Math 351 Spring, 2008 Hints for problems in section 3.4. Problem 3.4.9 If q ( x, t ) = 0, then the PDE is the homogeneous heat equation with homogeneous Dirich- let boundary conditions, and the solution would have the form u ( x, t ) = X n =1 B n ( t ) sin ± nπx L ² So, use this as a guess for the solution for this PDE which is not homogeneous. That is, take partial derivatives u t and u xx and substitute into the PDE. What to do about the forcing term? Assuming q ( x, t ) is piecewise smooth, it has a Fourier sine series: q ( x, t ) = n X n =1 c n ( t ) sin ± nπx L ² where c n ( t ) = ? Substitute the Fourier sine series for q ( x, t ), and all terms in the PDE (including the forcing term) are similar. Use this to ﬁnd the ODE that involves B n ( t ). Problem 3.4.12 This problem is the non-homogeneous heat equation with homogeneous Neumann bound- ary conditions.
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: • We solved the homogeneous heat equation with homogeneous Neumann boundary con-ditions in the practice problems for section 2.4. From this, guess a solution of the form u ( x, t ) = ∞ X n =0 A n ( t ) * eigenfunctions Proceed as in the previous problem to get an ODE of the form: dA n ( t ) dt + k ± nπ L ² 2 A n ( t ) = c n ( t ) • Show that c n ( t ) = e-t if n = 0 e-2 t if n = 3 if n 6 = 0 , 3 • Verify that with these three choices of c n ( t ), the solution of the ODE is: n 6 = 0 , 3 : A n ( t ) = A n (0) e-k ( nπ/L ) 2 t n = 0 : A ( t ) = A (0) + 1-e-t n = 3 : A 3 ( t ) = A 3 (0) e-k ( nπ/L ) 2 t + e-2 t-e-k ( nπ/L ) 2 t k (3 π/L ) 2-2 1...
View Full Document

## This note was uploaded on 09/02/2008 for the course MAE 105 taught by Professor Neiman-nassat during the Summer '07 term at UCSD.

Ask a homework question - tutors are online