课程名称︰偏微分方程式一 课程性质︰必选 课程教师︰陈逸昆 开课学院：理学院 开课系所︰数学系 考试日期︰2019年01月08日(二) 考试时限：10:20-12:10，共计110分钟 试题 : 　　　　　　　　　　　　　　PDE, Fall 2018 Final Exam DEP.＿＿＿＿＿＿＿＿＿　NAME＿＿＿＿＿＿＿＿＿＿＿＿ ID NUMBER＿＿＿＿＿＿＿＿ 1. Write down an explicit formula for a solution of n 　　　　　　　　　　　u - △u - u = f, in |R × (0,∞), t n u = g, on |R × {t=0}. (20%) 2. Let u solve the initial-value problem for the wave equation in one dimension u_t - u_xx = 0, in |R × (0,∞), u = g, u_t = h, on |R × {t=0}. Suppose supports of both g and h are [0,8]. The kinetic energy is ∞ 2 ∞ 2 k(t):=∫u_t(x,t)dx, and the potential energy is p(t):=∫u_x(x,t)dx. -∞ -∞ (a). Find the support of u. (10%) (b). Prove k(t)+p(t) is constant in time. (10%) (c). Show that there exist M>0 such that k(t) = p(t) whenever t>M. Find the best M. (10%) 3. Solve the following partial differential equation 1 2 2 ---(u_x + u_y) = u 2 with Cauch data u(cosθ,sinθ)=1,　0≦θ≦2π by characteristic method. (25%) 4. Find an entropy solution to the Cauchy problem: u_t + uu_x = 0, in |R × (0,∞), u(x,0) = g(x), on |R × {t=0}, where 1 + x, on [-1,0], g(x) = 1 - x, on (0,1], 0, otherwise. Indicate the location of shock waves, if any, and check that the entropy condition holds. (25%) --

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