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Unformatted text preview: Determine the value of γ for which there is an equilibrium solution (10 points) and find it (10 points). 2. Suppose u ( x, t ) is a solution of the problem u t ( x, t ) = ku xx ( x, t ) + q ( x ) , < x < L, t > , u x (0 , t ) = α, u x ( L, t ) = β, t > u ( x, 0) = f ( x ) , < x < L. Show that ∫ L u ( x, t ) dx = ( k ( β- α ) + ∫ L q ( x ) dx ) t + C, where C is a constant. In addition: (a) Use this to explain why ∫ L q ( x ) dx =- k ( β- α ) is a necessary condition for equilibrium. (b) Determine the constant C . V. Eigenfunction Expansions. Expect at least one problem similar to the problems of homeworks 9 and 10. Many of you could have the joy of doing those exercises for the first time! Do not deprive yourselves of that joy. VI. Fourier transforms. Homeworks 11 and 12 are the relevant ones. Having had only one disastrous quiz and no exam on this topic, expect some questions that will carry weight in the exam. I’ll make a copy of tables A2, A3, A4 of the textbook available, so there is no need to have any of that on your “cheat sheet.” Here are some additional, maybe more conceptual exercises. 1. By differentiating both sides of the equation ˆ f ( ω ) = 1 √ 2 π ∫ ∞ −∞ e ixω f ( x ) dx, establish that F [ xf ( x )]( ω ) =- i d ˆ f dω ( ω ), (assuming convergence of the improper integrals). Use this and formula 14 in A2 to find the Fourier transform of f ( x ) = xe −| x | . 2. Show that if y is a real constant, then F [ e − ixy f ( x )]( ω ) = ˆ f ( ω + y ) . 3. Show that if y is a real constant, then F [ f ( x + y )]( ω ) = e − iωy ˆ f ( ω ) . 4. Compute the Fourier transform of f ( x ) = x ( x- 2) 2 + 4 . (Use Table A2!) 5. Let f ( x ) be defined for-∞ < x < ∞ and be even. Show that ˆ f ( ω ) = F C [ f ]( ω ) for ω > . 4 6. Let f ( x ) be defined for-∞ < x < ∞ and be odd. Show that ˆ f ( ω ) = i F S [ f ]( ω ) for ω > . 7. A function f is defined by f ( x ) = ∫ ∞ −∞ e − y 2 ( x- y ) 2 + 9 dy. Compute ˆ f ( ω ). Use Table A2, of course. VII. And let’s not forget D’Alembert’s formula!! Expect Jean le Rond to make a cameo appearance....
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- Spring '13
- Boundary value problem, Partial differential equation, regular Sturm-Liouville problem