課程名稱︰偏微分方程式導論 課程性質︰系必修 課程教師︰黃信元 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰ 2012/6/19 考試時限（分鐘）：110 mins 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : A (20 pts) Solve the diffusion equation with variable disspation 2 u －u + bt u = 0 for －∞ < x < ∞ t xx with u(x,0) = φ(x), where b > 0 is a constant. 3 bt /3 (Hint. You may derive an equation for v(x,t) = e u(x,t).) B (20 pts) Separate variables for the equation tu = u + 2u, 0 < x < l, t > 0 t xx with the boundary conditions u(0,t) = u(l,t) = 0. Show that there are infinite number of solutions that satisfy the initial condition u(x,0) = 0. C (20 pts) Let f(x) = exp[-a|x|], a > 0. Compute the Fourier transform of f(x). D (20 pts) Define { 0 x < 0 u(x) = { { sin x x ≧ 0 Show that u + u = δ in the sense of distributions, where δ denotes the xx 0 delta function. n ＿ E (20 pts) Let Ω be an open, bounded connected set in R . Let f be in C(Ω), 2 and suppose u in C (Ω╳[0,∞)) is a solution of 1 { u = △u in Ω╳[0,∞) { t { u(x,t) = 0 on ∂Ω╳[0,∞) { u(x,0) = f(x) in Ω Prove lim u(x,t) = 0. t->∞ 1 ＿ (Hint: You may use the energy method and the fact that, for g in C (Ω) with g| = 0, there exists a constant C such that ∂Ω ||g|| 2 ≦ C ||▽g|| 2 .) L (Ω) L (Ω) --

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