Sec3-4_hints

Sec3-4_hints - We solved the homogeneous heat equation with...

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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.
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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 n L 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 ( n/L ) 2 t n = 0 : A ( t ) = A (0) + 1-e-t n = 3 : A 3 ( t ) = A 3 (0) e-k ( n/L ) 2 t + e-2 t-e-k ( n/L ) 2 t k (3 /L ) 2-2 1...
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