设介质四周绝热,初始温度为φ(x, y).Solution现在的定解问题是方程为热传导方程,设0< x < a, 0< y < b?u?t?κ??2u?x2+?2u?y2?= 0边界条件?u?x????x=0= 0?u?x????x=a= 0?u?y????y=0= 0?u?y????y=b= 0初始条件u|t=0=φ(x, y)9分离变量,设u(x, y, t) =X(x)Y(y)T(t)代入方程X(x)Y(y)T0(t)?κ[X00(x)Y(y)T(t) +X(x)Y00(y)T(t)]= 0除以κXY T1κT0(t)T(t)=?X00(x)X(x)+Y00(y)Y(y)?=?λX00(x)X(x)=?Y00(y)Y(y)?λ=?μ得到三个常微分方程.令λ=μ+νT0(t) +λκT(t) = 0X00(x) +μX(x) = 0Y00(y) +νY(y) = 0代入齐次的边界条件T(t)X0(0)Y(y) = 0T(t)X0(a)Y(y) = 0T(t)X(x)Y0(0) = 0T(t)X(x)Y0(b) = 0所以X0(0) = 0X0(a) = 0Y0(0) = 0Y0(b) = 0关于X(x)的齐次方程和齐次边界条件及关于Y(y)的齐次方程和齐次边界条件,分别构成了关于X及Y的两个本征值问题.我们需要求解两个本征值问题.先看X(X00+μX= 0X0(0) = 0X0(a) = 0μ= 0时X=Ax+BX0(0) =A= 0X0(a) =A= 0B任意,所以μ0= 0为本征值,取X0= 1为本征函数.μ6= 0时X=Asin√μ x+Bcos√μ x代入边界条件X0(0) =A√μ= 0?A= 0X0(a) =?B√μsin√μ a= 0sin√μ a= 0?√μ a=nπ所以μn=?nπa?2,n= 1,2,3, ...取B= 1Xn= cosnπax10
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