[試題] 100下 李克強 工程數學二 期末考

[tsf73 (我是8號)]

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）︰2012/6/22
考試時限（分鐘）：130
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :

1.(30%)

                                            2
Find the solution of the Laplace equation ▽ T = 0 outside the sphere:

r = R with the distribution of T on the surface of the sphere


T = f(Φ) = 1+cos2Φ+cos3Φ


where Φ is indicated in the spherical coordinates as follows:

http://ppt.cc/xLYA


2.(30%)


Solve the following ODE numerically with the finite difference (FD) scheme

accurate to second order:

y" + y' + y = x (y＝y(x), x is [0,1])

Subject to the boundary conditions:

y(0) = 1, y(1) + y'(1) = 1

Use one grid point and compare with the exact solution at x = 0.5,

y(0.5) = 1.146825. What is the percentage of deviation there?


3.(40%)

(A)(20%)

Find the eigenvalues and eigenvectors of the matrix

     ┌                   ┐
     │  -1     0     5   │
     │                   │
     │                   │
A =  │   0     1     0   │
     │                   │
     │                   │
     │   0     0     2   │
     └                   ┘

(B)(20%)

Find A^50 = ?

Appendix

The Laplace Equation in spherical coordinates

The Legendre polynomail of degree n is denoted by P (x), where n = 1,2,3,4,5
                                                   n

One-sided derivative that is corrected to O(h^2)

at both left and right boundary point

Mean value theorem(centered difference formula)


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