数学物理方程试题 Laplace变换表 原函数f(t) P1(x)= (n!)2(-n)!(2 (2+1)xP2(x)=(l+1)P1+1(x)+lP1-1(x) erfc P+1(x)=P(x)+(l+1)P1(x) P√P P+1(x)=(2l+1)P4(x)+P-1(x) P(x)=Pl+1(x)-2rP(x)+P-1(x) J(x)= -2aVt 2/Perf kA(k++1(2 注意:除第一题应当直接写出答案外, 其余各题均须写出必要的关键步骤. (20分)写出下列各本征值问题的解: y"(x)+My(x)=0 y(0)=0,y(l)=0; y(x)+Ay(x)=0, y(0)=0,y(l)=0 y"(x)+My(x)=0, y(x)=y(x+2丌)=0
✄ ☎ ✆ ✝ ✞ ✟ ➔ → ✠ ✡ ☛ Pl(x) = X l n=0 1 (n!)2 (l + n)! (l − n)! x − 1 2 n (2l + 1)xPl(x) = (l + 1)Pl+1(x) + lPl−1(x) P 0 l+1(x) = xP 0 l (x) + (l + 1)Pl(x) P 0 l+1(x) = (2l + 1)Pl(x) + P0 l−1 (x) Pl(x) = P0 l+1(x) − 2xP 0 l (x) + P0 l−1 (x) Jν(x) = X∞ k=0 (−) k k!Γ (k + ν + 1) x 2 2k+ν Laplace ➑➒☛ ✾ ✹✸ f(t) ➓ ✹✸ F(p) 1 1 p erfc α 2 √ t 1 p e −α √p 1 √ πα sin 2√ αt 1 p √ p e −α/p 1 √ πt cos 2√ αt 1 √ p e −α/p 1 √ πt e −α 2/4t 1 √ p e −α √p 1 √ πt e −2α √ t 1 √p e −α 2/perfc α √p ☞✌❀ ✏✑✒✓✍✎✖✗✦✧✙✏✚✛ ✜✢✣✓✤✑✦✧★✔✩✪✫✬✭❉ ✱✲ (20 ✶) ✒ ➠ ➋➌✓✔✕✼✖✵P▼ ❀ (1) y 00(x) + λy(x) = 0, y(0) = 0, y(l) = 0; (2) y 00(x) + λy(x) = 0, y 0 (0) = 0, y0 (l) = 0; (3) y 00(x) + λy(x) = 0, y(x) = y(x + 2π) = 0; 10
r2 dy (r) (x)=0 v(±1)有界 25分)求解下列定解问题 -0 00, art ay- array 0,t>0, =0 r\x=1 a cos. 00,Jo(x)为零阶 Bessel函数 五、(20分)用 Laplace变换方法求半无界杆热传导问题的 Green函数: G(a,t:r, t) daddy (a,t; a, t')l 00 本试题用毕收回
(4) d dx (1 − x 2 ) dy(x) dx + λy(x) = 0, y(±1)×✗ . ⑦✲ (25 ✶) ❾▼ ➋➌➇▼ ✖✵❀ ∂ ua ∂xt∂y− 2 ∂ u0 ∂xx∂y= , 0 0, ∂u ∂x x=0 = 0, ∂u ∂x x=l = 0, t > 0, u t=0 = A cos2 πx l , ∂u ∂t t=0 = 0, 0 0 ✛ J0(x) ❖✛ú Bessel ✹✸❉ ➍✲ (20 ✶) ✜ Laplace ➑➒➞➟❾Ð✢ ✗✣✤✥✦✖✵P Green ✹✸❀ ∂G(x, t; x 0 , t0 ) ∂t − κ ∂ G(x,t;x 0 ,t0 ) δ ∂xx∂y= (x − x 0 ) δ(t − t 0 ), 0 0, G(x, t; x 0 , t0 ) x=0 = 0, t > 0, G(x, t; x 0 , t0 ) t=0 = 0, 0 0 ❉ ✔ ❽ ✵ ✜ ✧ ✃ ★ 11
