课程名称︰偏微分方程导论 课程性质︰数学系大三必修课程 课程教师︰陈俊全 开课学院：理学院 开课系所︰数学系 考试日期︰2018年06月29日(五) 考试时限：08:10-10:00，共计110分钟 试题 : Choose 4 from the following 6 problems 1. Use the Fourier series method to solve the wave equation u_tt - u_xx = 0 for 0 < x < 2 with u(0,t) = 0, u(2,t) = 0, u(x,0) = 1, u_t(x,0) = x. 2. Find the harmonic function in square {0 < x < 1, 0 < y < 1} with the boundary conditions u(x,0) = 0, u_y(x,0) = x, u(0,y) = y, u_x(1,y) = 0. 3. Let x = r cosθ, y = r sinθ. Solve u_xx + u_yy = 0 in the disk {(x,y)| x^2+y^2≦4} with the boundary condition u = 1+cos2θ on r=2. 4. Let D = {(y1,y2,y3)∈|R^3 | y3 > 0}. By the Green representation Theorem, the bounded solution of the problem Δu = f in D, u = h on ∂D is given by ∂G(y,z) u(z) = ∫h(y)---------dS(y) + ∫f(y)G(y,z)dy, ∂D ∂ ν D where G(y,z) is the Green function. (a) Find the Green function G(y,z). (b) Show that z3 h(y) u(z) = -----∫ ---------dS(y) 2π ∂D |y-z|^3 if f = 0. 5. (a) Prove the mean value property in three dimensions: the average value of any harmonic function over a sphere equals its value at the center. (b) Use the mean value property to prove the strong maximum principle for a harmonic function in a bounded connected domain. 6. Let u(x1,x2,x3,t) be the solution of the three-dimensional wave equation u_tt - c^2Δu = 0 in |R^3 x (0,∞) with u(x,0) = ψ(x) and u_t(x,0) = φ(x). Use the Kirchhoff formula for the solution 1 ∂ 1 u(x,t) = --------- ∫ φ(y)dS(y) + ----[-------- ∫ ψ(y)dS(y)] 4πc^2t |y-x|=ct ∂t 4πc^2t |y-x|=ct to derive the formula for the solution of the one-dimensional wave equation. (Hint: Assume u depends on x1 only.) --

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