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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04月13日，09:00-12:00
考試時限：180分鐘
是否需發放獎勵金：是
試題 :
Math 2206 (Intorduction to PDE) Mtdterm (4/13/2013)
There are problems A to F with a total of 150 points. Please write dwon your
computational or proof steps clearly on the answer sheets.
A. Solve the following PDEs explicitly, or in implicit forms, or in series
forms. c in (b),(c),(d) is a positive constant.
2 2 2
(a) (10 points) Solve x ((u_x) + (u_y) ) = 1 with Cauchy data u(x,0) = 0.
2 2
u_tt = c (u_xx) + xt for x > 0, t > 0,
(b) (12 points) Solve u(x,0) = 0, (u_t)(x,0) = x for x ≧0,
u(0,t) = t^2 for t ≧0.
(c) (12 points) Solve u_tt = (c^2)*(u_xx + u_yy) on G ×(0,∞) with the
initial condtions u(x,y,0) = 0 , (u_t)(x,y,0) = (sin x)^2 * sin(2y),
and the boundary conditions (u_x)(0,y,t) = (u_x)(π,y,t) = 0,
u(x,0,t) = u(x,π,t) = 0, where G = {(x,y)| 0 < x < π, 0 < y < π}.
(d) (11 points) Find a function u(x,y,z,t) = v(r,t)
with r = √(x^2 + y^2 + z^2) such that
u_tt = (c^2)△u for r > 0, t > 0,
u(x,y,z,0) = 0, (u_t)(x,y,z,0) = 0 for r > 0,
lim 4π(r^2) * (v_r)(r,t) = g(t),
r->0+
where g(t) is a given C^2 function satisfying g(0) = g'(0) = g''(0) = 0.
B. (20 points) Find a piecewise smooth weak solution u(x,t) to the Buger's
equation (u_t) + (u^2 /2)_x = 0 defined in R ×[0,∞) such that the
discontinuities of u satisfy the shock entropy condition, and u has initial
condition u(x,0) = 0 for x 不屬於[0,1], and u(x,0) = -x^2 for x∈[0,1].
C. Consider the equation
(u_xx) - 4(u_xy) + 4(u_yy) + 3(u_x) - 6(u_y) - u - 1 = 0.
(a) (4 points) Find all the charateristic curves. What type does this
equation belong?
(b) (4 points) Transform the equation into the canonical form by the method
of characteristics.
(c) (6 points) Find the general solutions of the equation.
(d) (6 points) Given the Cauchy data u(x,-2x) = 0, (u_x)(x,-2x) = φ(x),
where φ(x) is C^2 function. Is this problem always solvable? If no,
find conditions of φ(x) such that this problem has a solution.
D. Consdier the n-dimensional wave equation with dissipation
u_tt + α(u_t) = △u for x∈R^n, t > 0,
u(x) = g(x) , (u_t)(x) = h(x) for x∈R^n, at t = 0.
where g and h are C^2 functions, and α≧0 is a constant.
(a) (15 points) Let (x0, t0) ∈ R^n ×(0,∞) (i.e. t0 > 0). If g = h = 0 on
the closed ball bar(B_(t0)) (x0) = {x∈R^n| |x-x0| ≦t0 }, prove that
u(x0,t0) = 0.
(b) (5 points) Use (a) to prove that the systme has at most one solution.
E. (15 points) Consider the string vibration problem (where k is a constant):
u_tt = u_xx for 0 < x < L, t > 0,
u(0,t) = 0 , (u_tt)(L,t) = -k(u_x)(L,t) for t≧0.
Write down the corresonding eigenvalue problem. Find all eigenvalues, and
the corresponding eigenfunctions. Determine their multiplicity.
F. Determine which of the following statements is true. Prove your answer.
Each has 10 points.
(a) The equation y*(u_x) + x*(u_y) = 0 with Cauchy data u(0,y) = e^(-y^2)
has intinitely many solutions defined in the whole plane.
(b) Let u(x,y,z,t) solve the 3-d wave equation in R^3 ×[0,∞) such that
u(x,y,z,0) = 0, and (u_t)(x,y,z,0) = φ(x,y,z) for all (x,y,z)∈R^3,
where φ is nonzero C^1 function which is 0 outside the ball
x^2 + y^2 + z^2 < 1. Then for each fixed (x,y,z)∈R^3, u(x,y,z,t)≠0
for arbitrarily large t > 0. However, for each fixed t > 0, u = 0
outside the some ball.
(c) The initial boundary value problem (where α is a constant):
u_tt = u_xx for x > 0, t > 0,
u(x,0) = (u_t)(x,0) = 0 for x≧ 0,
(u_t)(0,t) + α(u_x)(0,t) = 0 for t≧ 0
has the unique solution u(x,t) = 0.
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