「一、这无穷多个本征函教都是极值函数 【证】8 8y cd.c 2/y (Sy)adc 级变分81=2sycd 2/(8y/)2d 因为泛函阴的二级变分恒取正值,所以这些极 值函数均使泛函取极小
Isoperimetric Problem (continued) Variational Formulism of Differential Equation (Approximation Method (Rayleigh-Ritz) Examples !ùáõ ¼êÑ´4¼ê =y> δI[y] = Z 1 0 δy 02 xdx = 2Z 1 0 y 0 (δy) 0xdx ?C© δ 2 I[y] = 2Z 1 0 δ y 0δy 0 xdx = 2Z 1 0 (δy 0 ) 2xdx > 0 ϼI[y]?C©ð§¤±ù 4 ¼êþ¦¼4 C. S. Wu 1lù C©{ÐÚ
「一、这无穷多个本征函教都是极值函数 【证】8 8y cd.c 2/y (Sy)adc 级变分81=2sycd 2/(8y/)2d 因为泛函I的二级变分恒取正值,所以这些极 值函数均使泛函取极小
Isoperimetric Problem (continued) Variational Formulism of Differential Equation (Approximation Method (Rayleigh-Ritz) Examples !ùáõ ¼êÑ´4¼ê =y> δI[y] = Z 1 0 δy 02 xdx = 2Z 1 0 y 0 (δy) 0xdx ?C© δ 2 I[y] = 2Z 1 0 δ y 0δy 0 xdx = 2Z 1 0 (δy 0 ) 2xdx > 0 ϼI[y]?C©ð§¤±ù 4 ¼êþ¦¼4 C. S. Wu 1lù C©{ÐÚ
仁二、这无穷个本征值正好也就是泛函的极值 【证】将方程(0)+Ay=0乘以极值函 数(x),积分
Isoperimetric Problem (continued) Variational Formulism of Differential Equation (Approximation Method (Rayleigh-Ritz) Examples !ùá ÐÒ´¼4 =y>ò§ xy0 0 + λxy = 0¦±4¼ êy(x)§È© λ Z 1 0 y 2xdx = − Z 1 0 y xy0 0 dx = −y · xy0 1 0 + Z 1 0 y 02 xdx = Z 1 0 y 02 xdx âå^ Z 1 0 xy2dx = 1§ÒU λ = Z 1 0 y 02 xdx C. S. Wu 1lù C©{ÐÚ
仁二、这无穷个本征值正好也就是泛函的极值 【证】将方程(xy)+y=0乘以极值函 数y(x),积分 A/ ward y(ry)dr
Isoperimetric Problem (continued) Variational Formulism of Differential Equation (Approximation Method (Rayleigh-Ritz) Examples !ùá ÐÒ´¼4 =y>ò§ xy0 0 + λxy = 0¦±4¼ êy(x)§È© λ Z 1 0 y 2xdx = − Z 1 0 y xy0 0 dx = −y · xy0 1 0 + Z 1 0 y 02 xdx = Z 1 0 y 02 xdx âå^ Z 1 0 xy2dx = 1§ÒU λ = Z 1 0 y 02 xdx C. S. Wu 1lù C©{ÐÚ
仁二、这无穷个本征值正好也就是泛函的极值 【证】将方程(xy)+y=0乘以极值函 数y(x),积分 d r yay) dT
Isoperimetric Problem (continued) Variational Formulism of Differential Equation (Approximation Method (Rayleigh-Ritz) Examples !ùá ÐÒ´¼4 =y>ò§ xy0 0 + λxy = 0¦±4¼ êy(x)§È© λ Z 1 0 y 2xdx = − Z 1 0 y xy0 0 dx = −y · xy0 1 0 + Z 1 0 y 02 xdx = Z 1 0 y 02 xdx âå^ Z 1 0 xy2dx = 1§ÒU λ = Z 1 0 y 02 xdx C. S. Wu 1lù C©{ÐÚ