課程名稱︰偏微分方程導論 課程性質︰數學系必修 課程教師︰林紹雄 開課學院：理學院 開課系所︰數學系 考試日期︰2013年03月15日，08:10-09:10 考試時限：60分鐘 是否需發放獎勵金：是 試題 : 　　　　　　Math 2206 (Introduction to PDE) Quiz No.2 (3/15/2013) Solve the following problems. Please write down your computational or proof st- eps clearly on the answer sheets. In the following, (R_+)^2 = {(x,t)∈R^2|t>0}. A. (30 points) Consider the initial-value problem for the Burger's equation 　　u_t + ( (u^2)/2 )_x = 0 for (x,t)∈(R_+)^2 with the initial data 　　u(x,0) = 0 for x 不屬於[0,1], and (x,0) = -x for x∈[0,1]. 　　Find a piecewise smooth weak solution defined for all t>0 whose jump 　　discontinuities (if any) satisfy the shock wave conditions. B. (15 points) Consider the initial value problem a(u) u_x + u_t =0 with u(x,0) 　　=h(x), where a(u) and h(x) are C1 functions. Prove that there exists a cla- 　　ssical solution u∈C1((R_+)^2) whcih is continous up to t = 0 iff a(h(X)) 　　is a nondecreasing function of x. C. (25 points) show that u_(xy) + y u_(yy) + sin(x+y) = 0 is a hyperbolic type 　　equation. Find its characteristic cures, and reduce the equation into its 　　normal form, and find the general solutions of the equation. --

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