Laplace transformation solutions for one heat conduction equation
A typical heat conduction equation needs to be solved when taking convection is that a panel with length $L$ is heated. One side at $x=0$ has the convective boundary condition as $\dot{q}^{\prime \prime} = h(T_g-T_s)$, where $h$ is the convective heat transfer coefficient, $T_g$ is the ambient temperature outside and $T_s$ is the temperature at $x=0$. Another boudanry condition is adabatic as $\dot{q}^{\prime \prime} = 0$. The initial condition is that the panel is at $T_{\infty}$.
The governing equation is
\[\frac{\partial^{2} T}{\partial x^{2}}-\frac{1}{\kappa} \frac{\partial T}{\partial t}=0, \quad a<x<b\]where $\kappa$ is the inverse of the thermal diffusivity $\alpha$ for mathmatical further simplications.
Mutiple by $e^{-p t}$ and integrate with respect to $t$ from 0 to $\infty$, which gives the Laplace transformation as,
\[\int_{0}^{\infty} e^{-p t} \frac{\partial^{2} T}{\partial x^{2}} d t-\frac{1}{\kappa} \int_{0}^{\infty} e^{-p t} \frac{\partial T}{\partial t} d t=0\]For Laplace transformation,
\[L\left\{\frac{\partial v}{\partial t}\right\}=p L\{v\}-v_{0}\] \[L\left\{\frac{\partial^{n} v}{\partial x^{n}}\right\}=\frac{\partial^{n} \tilde{v}}{\partial x^{n}}\]That turns the governing equation as,
\[\frac{d^{2} [T]}{d x^{2}}-\frac{p}{\kappa} [T]=-\frac{1}{\kappa} T_{0}(x)\]where $[T]$ is the Laplace transformed variable.
In this way, the partial differential equation is reduced to the ordinary differential equation.
Write for shortness that $q^2 = \frac{p}{\kappa}$, the equation will become,
\[\frac{d^{2} [T]}{d x^{2}}-q^2 [T]=-\frac{1}{\kappa} T_{0}(x)\]Two solutions exist for the ordinary equation as $B^{q x}$ and $A e^{-q x}$. Bacasue the temperature is bounded as $x \rightarrow \infty$, $A e^{-q x}$ is taken.
The boundary condition at $x = 0$ for the subsidiary equation, the equation after Laplace transformation, is,
\[\frac{d [T]}{d x}-h [T]=\frac{h T_g}{p}\]The solution satisfies the boundary condition is,
\[[T]=\frac{h T_{g} e^{-q x}}{p(q+h)}\]The inverse Laplace transformation for it is,
\[v=V \operatorname{erfc} \frac{x}{2 \sqrt{(\kappa t)}}-V e^{h x+h^{2} \kappa l} \operatorname{erfc}\left(\frac{x}{2 \sqrt{(} \kappa t)}+h \sqrt{\kappa t}\right)\]
