Summary: Laplace transform is useful in equation solving. By definition, Laplace transform transforms a function $f(x)$ defined on $x\geq 0$ into another function by calculating the convolution
$$ F(s) = \int_0^\infty e^{-s x} f(x) dx, $$ or symbolically,
$$ F(s) = \mathcal L_s (f(x)). $$ A table of important Laplace transforms can be found on mathworld.wolfram.com. Here we steal some of the commonly used.
$$ \begin{align} \mathcal L_s (1) =& \frac{1}{s}\\ \mathcal L_s (x^n) =& \frac{n!

Summary: Green’s Function Method In this section, we demonstrate Green’s function method of solving differential equations.
Cable Equation The cable equation is written as 1
\begin{equation} \frac{\partial}{\partial t} u(t,x) = \frac{\partial^2}{\partial x^2} u(t,x) - u(t,x) + i_{e}(t,x), \label{eqn-cable-equation-potential} \end{equation}
or
\begin{equation} \frac{\partial}{\partial t} i(t,x) = \frac{\partial^2}{\partial x^2} i(t,x) - i(t,x) + \frac{\partial}{\partial x} i_e (t,x), \label{eqn-cable-equation-current} \end{equation}
where $t$, $x$, $i$, $i_e$ are all renormalized unit less quantities. For the meaning and definition of them, ref.

Summary: Fourier Serious and Continuous Fourier Transform A function $f(x)$ defined on $x\in [-L, L]$ can be decomposed into Fourier series
$$ f(x) = \sum_{-\infty}^\infty A_n e^{i n k_0 x }, $$ where
$$ k_0 = 2 \pi/ 2 L = \pi/L. $$ A continuous Fourier transform is not very different from Fourier series for equation solving. A function $f(x)$ can be written as a convolution of another function $g(x)$,
$$ \begin{equation} f(x) = \int_{-\infty}^\infty g(t) e^{2\pi i x t} dt, \label{eqn-fourier-transform-original-1} \end{equation} $$ where $f(x)$ is the Fourier transform of $g(t)$.