課程名稱︰工程數學二
課程性質︰必修
課程教師︰李克強
開課學院：工學院
開課系所︰化學工程學系
考試日期（年月日）︰102/04/19
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(一) (25%)
Find the Fourier Seried expansion of f(x)=exp(x) in the interval
[-π,π].
(二) (25%)
Find the solution of the PDE ▽^2 ψ = α^2 ψ, where α^2 is a given
real constant in the domain of a disc with radius R as shown below. The
distribution of ψ along the edge is given as f(θ).
ð^2 ψ ðψ ð^2 ψ
Note that ▽^2 ψ = ──── + ─── + ─────.
ðr^2 r ðr r^2 ðθ^2
(找不到partial的符號就用KK音標的ð代替了)
(附圖就是一個圓的半徑R,r：半徑,θ：角度,就不畫了)
(三) (25%)
Find the solution of the PDE ▽^2 T = 0 in a finite cylinder subject to
given boundary conditions as shown.
ð^2 T ðT ð^2 T
Note that ▽^2 T = ─── + ─── + ─── = 0.
ðr^2 r ðr ðz^2
T1
——————————————
╱╲ ╲
∕ ﹨ ﹨
T2 ∣ ∣ ∣0
﹨ ∕ ∕
╲╱ ╱
——————————————
(看得出是圓柱吧...另外半徑R,圓柱長L)
(四) (25%)
Find the solution of the PDE ▽^2 T = 0 in a rectangular domain subject
to given boundary conditions as shown.
0
———————————————— ↑
∣ ∣∣
∣ ∣∣ ðT
T1 ∣ ▽^2 T = 0 ∣M ── = 0
↑y ∣∣ ðx
∣ x ∣∣
—→—————————————— ↓
←——————L————————→
T2
Note：
Tables of integrals：
exp(ax)
∫exp(ax) sin(bx) dx = ───── [ a sin bx - b cos bx ]
a^2 + b^2
exp(ax)
∫exp(ax) cos(bx) dx = ───── [ a cos bx - b sin bx ]
a^2 + b^2
x sin 2ax
∫sin^2(ax) dx = ─ - ────
2 4a
x sin 2ax
∫cos^2(ax) dx = ─ + ────
2 4a
General solution of Bessel's differential equation：
x^2 y" + xy' + (λ^2 x^2 - μ^2 ) y = 0 => y = c1Jμ(λx) + c2Yμ(λx)
General solution of Modified Bessel's differential equation：
x^2 y" + xy' - (λ^2 x^2 + μ^2 ) y = 0 => y = c1Iμ(λx) + c2Kμ(λx)
(最後附上J0.J1.J2、I0.I1.I2.I3、Y0.Y1.Y2、K0.K1.K2.K3的圖)
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◆ From: 140.112.250.1
※ 編輯: hellersjoke 來自: 140.112.250.1 (06/22 18:46)