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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︰化工系
考試日期（年月日）︰101/4/20
考試時限（分鐘）：110(後延至130)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
2
(1)(25%)Find the solution T(x,y) of Laplace equation (▽ T=0) in the
semi-infinite strip (y≧0, 0≦x≦a), also satisfying the boundary conditions:
T(0,y)=T(a,y)=0, y＞0
T(x,0)=f(x), 0≦x≦a
and the additional condition that T(x,y)→0 as y→∞
∞ 0
│ │
│ │
│ │
│ │
│ │
y-axis │ │
│ │
│ │
│ │
│ │
└────────┴─── x-axis
0 f(x) a
2
δu δ u
(2)(25%)Find the solution u(x,t) of the PDE:-----=(a^2)-------+x^2 subject to
δt 2
δx
the boundary conditions:
u(0,t)=T1, u(L,t)=T2, t>0
Where T1, T2 are given constant, and the initial condition:u(x,0)=f(x)
2
(3)(25%)Find the solution T(r,θ) of Laplace eqution in a disc with ▽ T=0 →
2 2
δ T 1 δT 1 δ T
(-------) + -----(-----) + -------(-------)=0, 0≦r≦R. The distribution of T
2 r δr r^2 2
δr δθ
at perimeter(r=R) is prescribed as T(R,θ)=1+2cosθ+3cos2θ+4sinθ
(The given image of the problem is a circle whose radius is R, and the values
of T(R,θ) is equal to f(θ) )
(4)(25%)Find the solution T(r,θ,z) of the Laplace equation ▽ T=0 →
2 2 2
δ T 1 δT 1 δ T δ T
(-------) + -----(-----) + -------(-------) + (-------)=0 in a semi-cylinder
2 r δr r^2 2 2
δr δθ δz
as shown,(A image of semi-cylinder with a radius R and length L )
satisfying the boundary conditions:
δT
T(r,θ,0)=-----(r,θ,L)=0, T(r,0,z)=T(r,π,z)=T1, T(R,θ,z)=T2,
δz
.T(0,θ,z)=finite
General solution of Bessel differential equation:
(x^2)y"+x'y+((λx)^2-μ^2)y=0, y(x)=c1J (λx)+c2Y (λx)
μ μ
General solution of modified Bessel differential equation:
(x^2)y"+x'y-((λx)^2+μ^2)y=0, y(x)=c1I (λx)+c2K (λx)
μ μ
The graph of these four kinds of Bessel function is given.
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※ 編輯: tsf73 來自: 218.167.193.186 (04/20 21:09)