課程名稱︰偏微分方程導論
課程性質︰數學系大三必修課程
課程教師︰陳俊全
開課學院：理學院
開課系所︰數學系
考試日期︰2018年06月29日(五)
考試時限：08:10-10:00，共計110分鐘
試題 :
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)