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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