課程名稱︰化學數學一
課程性質︰必修
課程教師︰陸駿逸
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰2008/01/18
考試時限（分鐘）：150分鐘
是否需發放獎勵金：yes
（如未明確表示，則不予發放）
試題 :
1. (10pts) Use the operator factorization method to find the general solution
of the PDE
(d^2)u(x,y) (d^2)u(x,y) (d^2)u(x,y)
────── - 2────── - 3────── = 0 (這裡的d是偏微分)
dx^2 dxdy dy^2
2. (10pts) Show that the operator ▽^2 can be expressed as
1 d d 1 d^2 d^2
─ ─(r─) + ── ─── + ── (這裡的d也是偏微分)
r dr dr r^2 dθ^2 dz^2
in the cylindrical coordinate.
Hint: You may use the Gauss' theorem.
3. (10pts) Given the formula
1 d dφ 1 d dφ
(▽^2)φ = ── ─(r^2*──) + ───── ──(sinθ──) +
r^2 dr dr r^2*sinθ dθ dθ
1 (d^2)φ
────── ──── = 0 (同上，這裡的d還是指偏微分)
(r*sinθ)^2 dψ^2
Assume that a solution has the particular form φ(r,θ,ψ) =
Ｒ(r)Θ(θ)Φ(ψ) where Ｒ(r), Θ(θ), and Φ(ψ) are three unknown
functions. Determine suitable ODEs for Ｒ(r), Θ(θ), and Φ(ψ).
4. (30pts) Use the method of the separation variable to solve the PDE
dφ(x,t) (d^2)φ(x,t)
──── = ──────
dt dx^2
dφ
where 0≦x≦π. The boundary conditions ──(x=0,t) = 0, and the
dx
x^2
initial condition φ(x,t=0) = ── - 1. (照慣例，d還是指偏微分)
π^2
(a) Find the particular solution of the PDE which is of the form
φ(x,t) = f(t)g(x). Determine the suitable solution of f(t)
and g(x) where the boundary conditions at x = 0 and x = π are
satisfied.
(b) Assume that the general solution is the linera combination of
the particular solutions found above. Find the solution which
also obeys the initial condition at t = 0.
Hint: You can use the half range cosine series here.
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