Applie othe f egrie Transom Weat Conductio within Insi 例14.1求解无界杆的热传导问题 0 a2 at ax2 f(x,t)-∞<x<∞t>0 t=0=0 ∞<x<0 说明 在这种无界区间的定解问题中,往往并不明 确列出边界条件 实际上,无界区间,只是一个物理上的抽象 ρ因此,如果要完整地列出定解问题的话,则 还应当有边界条件u+→0 C. S. Wu
Application of Laplace Transform Application of Fourier Transform Other Integral Transforms Heat Conduction within Infinite Rod Wave Propagation within an Infinite String ~14.1 ¦)Ã.\9D¯K ∂u ∂t − κ ∂ 2u ∂x2 = f(x, t) − ∞ < x < ∞ t > 0 u t=0 = 0 − ∞ < x < ∞ `² 3ù«Ã.«m½)¯K¥§ ¿Ø² (Ñ>.^ ¢Sþ§Ã.«m§´ÔnþÄ Ïd§XJ/ѽ)¯K{§K Ak>.^u x→±∞ → 0 C. S. Wu 1où È©C
例14.1求解无界杆的热传导问题 0u02 at ax2 =f(a, t) ∞<x< t>0 u=0=0 ∞<x<0 解作 Laplace变换 令(x,)=U(x,p)=/(,e-dt 把(x、P)看成只是的函数,P是参数,所以 2U(,P 利用初始条件,有 C. S. Wu
Application of Laplace Transform Application of Fourier Transform Other Integral Transforms Heat Conduction within Infinite Rod Wave Propagation within an Infinite String ~14.1 ¦)Ã.\9D¯K ∂u ∂t − κ ∂ 2u ∂x2 = f(x, t) − ∞ < x < ∞ t > 0 u t=0 = 0 − ∞ < x < ∞ ) LaplaceC -u(x, t) ; U(x, p) = Z ∞ 0 u(x, t)e−ptdt rU(x, p)w¤´x¼ê§p´ëꧤ± ∂ 2u ∂x2 ; d 2U(x, p) dx 2 |^Щ^§k ∂u ∂t ; pU(x, p) C. S. Wu 1où È©C
例14.1求解无界杆的热传导问题 0u02 at ax2 =f(a, t) ∞<x< t>0 u=0=0 ∞<x<0 解作 Laplace变换 令(x,)=U(x,p)=/(,e-dt 把U(x,p)看成只是x的函数,卩是参数,所以 a2u dU( 0x2 dr2 利用初始条件,有 U(x,)
Application of Laplace Transform Application of Fourier Transform Other Integral Transforms Heat Conduction within Infinite Rod Wave Propagation within an Infinite String ~14.1 ¦)Ã.\9D¯K ∂u ∂t − κ ∂ 2u ∂x2 = f(x, t) − ∞ < x < ∞ t > 0 u t=0 = 0 − ∞ < x < ∞ ) LaplaceC -u(x, t) ; U(x, p) = Z ∞ 0 u(x, t)e−ptdt rU(x, p)w¤´x¼ê§p´ëꧤ± ∂ 2u ∂x2 ; d 2U(x, p) dx 2 |^Щ^§k ∂u ∂t ; pU(x, p) C. S. Wu 1où È©C
例14.1求解无界杆的热传导问题 0u02 at ax2 =f(a, t) ∞<x< t>0 u=0=0 ∞<x<0 解作 Laplace变换 A-u(c, t)=U(a,p)=/u(a, t)e"dt 把U(x,p)看成只是x的函数,p是参数,所以 a2u dU( 0x2 dr2 利用初始条件,有三pU(x,p)
Application of Laplace Transform Application of Fourier Transform Other Integral Transforms Heat Conduction within Infinite Rod Wave Propagation within an Infinite String ~14.1 ¦)Ã.\9D¯K ∂u ∂t − κ ∂ 2u ∂x2 = f(x, t) − ∞ < x < ∞ t > 0 u t=0 = 0 − ∞ < x < ∞ ) LaplaceC -u(x, t) ; U(x, p) = Z ∞ 0 u(x, t)e−ptdt rU(x, p)w¤´x¼ê§p´ëꧤ± ∂ 2u ∂x2 ; d 2U(x, p) dx 2 |^Щ^§k ∂u ∂t ; pU(x, p) C. S. Wu 1où È©C
例14.1求解无界杆的热传导问题 0u02u at ax2 f(x,t)-∞<x<∞t>0 0 ∞<x< 解 再进一步令f(x,t)=F(x,p) 定解问题就变成P(xP) d-U(. p) F(. p) 0的条件下即可解得(见书,例10
Application of Laplace Transform Application of Fourier Transform Other Integral Transforms Heat Conduction within Infinite Rod Wave Propagation within an Infinite String ~14.1 ¦)Ã.\9D¯K ∂u ∂t − κ ∂ 2u ∂x2 = f(x, t) − ∞ < x < ∞ t > 0 u t=0 = 0 − ∞ < x < ∞ ) 2?Ú-f(x, t) ; F(x, p) ½)¯KÒC¤ pU(x, p) − κ d 2U(x, p) dx 2 = F(x, p) 3U x→±∞ → 0^e=) (Ö§~10.8) U(x, p) = 1 2 1 √κp Z ∞ −∞ F(x 0 , p) exp − r p κ |x − x 0 | dx 0 C. S. Wu 1où È©C