[試題] 105-1 江衍偉 物理數學 期末考
課程名稱︰物理數學
課程性質︰選修
課程教師︰江衍偉
開課學院：電機資訊學院
開課系所︰電信工程學研究所
考試日期（年月日）︰106.1.11
考試時限（分鐘）：170
是否發放獎勵金：是
試題 :
1. Consider a differential equation u"(t) - k^2*u(t) = f(t), 0<t<∞,
with the initial conditions u(0)=u'(0)=0, where k is a real constant and
f(t) is a given function.
(a) Find the Green's function g(t,τ) of this problem, satisfying
g"(t,τ) - k^2g(t,τ) = δ(t,τ). [6%]
(b) Express the solution u(t) in terms of g(t,τ) and f(t), and show that
the causality is satisfied. [4%]
2. (a) Consider the eigenproblem u_n"(x) + k*u_n(x) = λ_n*u_n(x), 0<x<1,
u_n(0)=u_n(1)=0, where k is a nonzero real constant. Find the
eigenfunctions u_n(x) and eigenvalues λ_n. [8%]
(b) By using the eigenfunction expansion method and the result of (a),
solve the following boundary value problem:
u"(x) + k*u(x) = sin(πx), 0<x<1, u(0)=3, u(1)=5, where k is a nonzero
real constant. [8%]
(c) Does your solution in (b) exactly satisfy the boundary conditions at
x=0 and x=1? If no, discuss the reason briefly. [2%]
(d) Is your solution valid for all nonzero real k? If no, discuss the
reason briefly. [2%]
3. Find the function y(x) which makes the following functional stationary:
I = integral[0→1](2y'^2-y'y-5y'+y)dx + [y(0)]^2 - 2y(0) - 2[y(1)]^2 + 4y(1)
(a) when y(0)=2 and y(1)=3. [10%]
(b) when the values y(0) and y(1) are not specified. [10%]
4. (a) For a linear differential operator L, we can consider the Green's
function G(x,ξ;λ) satisfying LG-λG=δ(x-ξ). Discuss briefly the
importance of G in investigating the spectral property of operator L.
[5%]
(b) Why do we seldom consider the residual spectrum for a practical
boundary value problem? Explain briefly. [5%]
(c) Please give a physical or engineering example for which a continuous
spectrum exists, and discuss its physical meaning briefly. [5%]
5. (a) Consider the integral F(z) = integral[0→∞] e^(-t^3)/(t+z^3) dt
for complex valued z. By using Watson's lemma, find the asymptotic
expansion of F(z) in terms of {z^-j},j=0→∞ for |z|→∞. [10%]
(b) If there is no saddle point along the integration path which has been
deformed as much as possible, can we use the method of steepest descent
to asymptotically evaluate the integral? Discuss this briefly. [8%]
6. Consider a three-dimensional Poisson equation ▽^2 u(r) = f(r) within a
finite region V bounded by a closed surface S, on which a Dirichlet boundary
condition is specified. Show that the solution of this problem is unique.
[7%]
7. Solve the following partial differential equation for u(x,t) by the
characteristic method. [10%]
∂u/∂t + ∂/∂x [6u^2-u] = 0, -∞<x<∞, t>0
u(x,0) = 2x + 5, -∞<x<∞
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