I have a question on how to apply Green's function in the solution of Heat equation. Assuming I want to solve $$\frac{\partial^{2} T}{\partial x^{2}}+\frac{1}{k} g(x, t)=\frac{1}{\alpha} \frac{\partial T}{\partial t} \quad \text { in } \quad 0<x<\infty, \quad t>0$$
$$\begin{array}{ll} \text { BC1: } & T(x=0, t)=f(t) \\ \text { IC: } & T(x, t=0)=F(x) \end{array}$$ If we assume having a Dirichlet boundary condition and evaluate it into Green function solution equation, given by
$$\begin{aligned} T(x, t)=&\left.\int_{L^{\prime}} G\left(x, t \mid x^{\prime}, \tau\right)\right|_{\tau=0} F\left(x^{\prime}\right) x^{\prime P} d x^{\prime} \\ &+\alpha \int_{\tau=0}^{t} \int_{L^{\prime}} G\left(x, t \mid x^{\prime}, \tau\right) \frac{1}{k}g\left(x^{\prime}, \tau\right) x^{\prime P} d x^{\prime} d \tau \\ &+\alpha \left\{\left.\int_{\tau=0}^{t}\left[ G\left(x, t \mid x^{\prime}, \tau\right)\right]\right|_{x^{\prime}=0} \frac{1}{k} f(\tau) d \tau\right\} \end{aligned}$$ Where Green's function here is $$G\left(x, t \mid x^{\prime}, \tau\right)=\frac{1}{[4 \pi \alpha(t-\tau)]^{1 / 2}}\left\{\exp \left[-\frac{\left(x-x^{\prime}\right)^{2}}{4 \alpha(t-\tau)}\right]-\exp \left[-\frac{\left(x+x^{\prime}\right)^{2}}{4 \alpha(t-\tau)}\right]\right\}$$
Then the third term will be $0$ which implies no effect of Dirichlet boundary condition. Why this problem arises and do I have to transform non-homogeneous boundary condition to homo-boundary condition and apply Green's function to get the result?
Thanks in advance for the help.