令t(x,τ)=X(x)*T(τ),代入方程,得:
X*T'=aT*X''
令-r=T'/aT=X''/X
則T'+raT=0,X''+rX=0,且X'(0)=0,-λX'(δ)=h[X(δ)-X(∞)]
當r<=0時,X(x)=C1*e^[√(-r)x]+C2*e^[-√(-r)x]
X'=√(-r)C1*e^[√(-r)x]-√(-r)C2*e^[-√(-r)x]
X'(0)=√(-r)C1-√(-r)C2=0,得:C1=C2
即X(x)=C*e^[√(-r)x]+C*e^[-√(-r)x]
X'=√(-r)C*e^[√(-r)x]-√(-r)C*e^[-√(-r)x]
-λ√(-r)C*{e^[√(-r)δ]-e^[-√(-r)δ]}=hC*{e^[√(-r)δ]-e^[-√(-r)δ]-∞}
等式左邊為有界量,右邊{e^[√(-r)δ]-e^[-√(-r)δ]-∞}為無窮量,所以C=0
所以X(x)=0
當r>0時,X(x)=C1*cos(√r*x)+C2*sin(√r*x)
X'=-C1*√r*sin(√r*x)+C2*√r*cos(√r*x)
X'(0)=C2*√r=0,得:C2=0
即X(x)=C*cos(√r*x)
X'=-C*√r*sin(√r*x)
λC*√r*sin(√r*δ)=hC*[cos(√r*δ)-cos(√r*∞)]
等式左邊為定值,右邊[cos(√r*δ)-cos(√r*∞)]為不定值,所以C=0
所以X(x)=0
綜上所述,X(x)=0,即t(x,τ)=X(x)*T(τ)=0