\chapter{Elementary solution methods}




\section{Fourier method}


The most elementary solution method I know of is \emph{separation of variables}, which basically makes a general yet bold ansatz that is an attempt to split our PDE into ODEs with respect to one variable each, then prays to the math gods that things work out.

This solution method has a chance at being succesful when the RHS (or any 1 of the sides) of the PDE reduces to a constant after we make our ansatz because this allows us to split the PDE into multiple ODEs. Here is the general method.

\begin{itemize}
	\item Assume the ansatz $f(x,y) = X(x)Y(y)$ and substitute into equation
	\item Separate variables to both sides and equate both sides to a constant $\pm \lambda$
	\item Solve each problem using SL theory or otherwise and then substitute the solutions into the original ansatz; the general solution is a linear combination of these eigenfunctions
	\item Equate the general solution to the final boundary condition and find the projection.
\end{itemize}


Why such an ansatz? Many PDEs arise naturally from physics, and in some contexts waves were experimentally found to obey initial conditions and this ansatz (think that there is one "initial profile" such that the function along other dimensions is a scaling of this profile).


\section{Fourier transform method}

Using the ansatz (guess) that a solution can be represented as the Fourier transform of some function $u$ is quite effective.

Fourier studied a PDE now hailed as the heat equation, and he tried to find new solutions to it based on the trivial , which were sinudoids. His idea was basically to consider convolutions of some unknown function with these elementary solutions to find further solutions.

Fourier conjectured that (sufficiently nice) periodic functions could always be represented as something called a Fourier series, and that non-periodic function (again, sufficiently nice ones, whatever that means) could still be represented by a 'Fourier integral'. Though such statements could not be verified until much later, Fourier's steadfast beliefs ultimately paid off, producing legitimate solutions to the heat equation.

Fourier analysis warrant much attention due to its usefulness in PDEs as well as signal processing, and though we do not discuss the theory here, you can read about it in Fourier Analysis. To those already familiar with the theory, I offer you a neat way to solve PDEs with Fourier analysis.


\section{Heat equation}


\[ u_{t}=cu_{xx}\]

\subsection{Heat kernel}


\begin{definition}[Heat kernel]
\end{definition}

This serves as a fundamental solution to the heat equation.


\subsection{Heat equation on circle}


Is also known as the 1D diffusion equation; it describes how heat is transferred in a wire. If we furthermore assume that the spacial function when conducting separation of variables

The solution method is mostly similar, however we now assume that the spacial function is periodic. Though the solutin looks similar, we arrive at a slightly different version of the heat kernel.



\section{Fundamental solutions of PDEs}

The theory of extending fundamental solutions (Green's functions) is also applicable to the theory of PDEs.

This is one way to solve the heat equation; one finds its fundamental solution (the heat kernel) and then considers solutions as colvolutions with this kernel!

\[\mathcal{L}u=f\]
\[\mathcal{L}G=\delta\]
\[u(x,t)= \int G(x,t)f(x)dt\]


\section{Laplace method}




\section{Euler's equation}
\[u(x,y) = f(x+ \lambda_1 y) + g(x+\lambda_2 y)\]

When the PDE is nonparabolic, the trick is to use a "generic" linear transform of variables, and then decide this transform more concretely in order to reduce the complexity as much as possible.


When parabolic

\[u(x,y) = f(x+ \lambda y) + yg(x+\lambda y)\]




\section{Wave equation}
General solution
d'Alembert formula