m264508 - Nurul Hanim Hashim WeBWorK assignment 6 due...

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Unformatted text preview: Nurul Hanim Hashim WeBWorK assignment 6 due 6/2/08 at 11:55 PM. where bn (b) Determine the Fourier cosine series for the function gx e 5x defined for 0 x π: § ¦ $ ©§ ¦ £ MATH264 May 2008 gx £ a0 where a0 and an £ 7.(2 pts) Solve the heat equation § ¥ ¥ ¤ 2 ¦ £ 1 if 4 x 0 x if 0 x 4 Then the Fourier series of f is a0 an bn £ £ £ ¡ 4.(2 pts) Let f be a periodic function of period 10 such that fx x2 for 5 x 5. Then the Fourier series of f is a0 an bn £ £ £ ¡ Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 1 2 5 " $ § ¦ fx nπx ∑ bn sin 3 n1 ∞ £ #§ ¦ £ #§ ¦ £ ¨§ ¦ gx b1 t bn t n1 , and for all n 1. § ¦ §¦ " §¦ £ ¨§ ¤ The solution is u x t ¦ gx ¡ 5.(3 pts) (a) Determine the Fourier sine series for the function fx x2 defined for 0 x 3: ¥ ¥ £ #§ ¦ ∑ bn t ∞ sin nx , where § § ¥ ¥ where a0 § ¥ ¥ ¦3¤ ¥ ¦¤ ¦ ¤ ¥ £ ¨§ ¤ ¦ £¨§ ¤ ¦ £ '§ ¤ ¦ n1 2 ¤ ¥ ¥ ¤ £ ut uxx 30x 0 x π t 0 u0t 00t∞ uπt 00t∞ ux0 sin x 0 x π ¡ ∑ an cos " ∞ 8.(2 pts) Find a formal solution to the initial-boundary value problem nπx 5 bn sin nπx 5 £ ©§ ¦ and an t n1 £ § ¦ §¦ " £ 4§ ¤ ¦ uxt a0 ¡ where a0 ∑ an t ∞ cos nx , where a0 § ¥ ¥ e7x § ¥ ¥ nπx ∑ an cos 4 n1 " ¡ ∞ nπx bn sin 4 § ¥ ¥ ¦3¤ ¦ ¤ ¦ ¤ £ ¨§ ¤ ¦ £ ¨§ ¤ ¦ £ ¨§ ¤ ¦ ! ¥ £ § ¦ fx 4uxx with boundary conditions ux 0 t ux π t ux0 0 0 " uxt 3.(2 pts) Let f be a periodic function of period 8 defined by ! ¥  £ ¨§ ¤ ¦ k0 ut t 00 x π 0t∞ 0t∞ 0xπ £ 1§ ¦ § § ¦ ¦ §¦ £ 0§ ¤ is u x t ¦ § sin 2k ¡ π ¥ ∑ bk t ∞ 1x, § ¥ ¥ 00t∞ u0t uπt 00t∞ ux0 8 sin x sin 6x 0x § § ¥ ¦§ ¤ ¦ ¥ ¥ ¤ ¦ ¥ ¥ ¤ ¦ ¡ £ ¨§ ¤ £ § ¤ ¦ ¦ £ ¨§ ¤¦ § § ¥ ¥ ¥ ¥ ¦)(§ ¦ ¤ ¦ ¤ ¦ ¤ £ ¨§ ¤ ¦ £ ¨§ ¤ ¦ £ '§ ¤ ¦ 9uxx ut subject to the Dirichlet boundary conditions £ if 0 x π 2 x π if π 2 x π Then the solution to the heat equation 9uxx ut with Dirichlet boundary conditions u0t uπt ux0 00t∞ 00t∞ fx 0xπ £ ¥ ! & £ ©§ ¦ 6.(2 pts) Let f x & ! ¥ x " ¡  λy 0 0 x 8 y0 0y8 0 has a non-trivial solution. ,n 1 2 3 λn Your formula should give the eigenvalues in increasing order. The eigenfunctions to the eigenvalue λn are yn Cn where Cn is an arbitrary constant. 2.(2 pts) Solve the heat equation ¥  ¤ £ ©§ ¥ ¤ ¦ ¤ ¤ ¤ £ £ y ¥ ¥ % £ ¨§ ¦ 1.(2 pts) Determine the values of λ (eigenvalues) for which the boundary-value problem ¡ ¢ ¥ n1 ∑ an cos nx ∞ £ ¨§ ¦ ¡ !  £ £ #§ ¦ £ where bk t ...
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