4.1. Separation of variables (the first blood)


# Chapter 4. Separation of variables and Fourier Series

In this Chapter we consider simplest separation of variables problems, arising simplest eigenvalue problems and corresponding Fourier series.

## 4.1. Separation of variables (the first blood)

Consider IBVP for 1D-wave equation on $(0,l)$: \begin{align} & u_{tt}-c^2 u_{xx}=0,&& 0< x< l \label{eq-4.1.1}\\[3pt] & u|_{x=0}=u|_{x=l}=0, \label{eq-4.1.2}\\[5pt] & u|_{t=0}=g(x) &&u_t|_{t=0}=h(x).\label{eq-4.1.3} \end{align}

### Separation of variables

Let us skip temporarily initial conditions (\ref{eq-4.1.3}) and consider only (\ref{eq-4.1.1})--(\ref{eq-4.1.2}) and look for a solution in a special form $$u(x,t)= X(x) T(t) \label{eq-4.1.4}$$ with unknown functions $X(x)$ on $(0,l)$ and $T(t)$ on $(-\infty,\infty)$.

Remark 1. We are looking for non-trivial solution $u(x,t)$ which means that $u(x,t)$ is not identically $0$.

Therefore neither $X(x)$ nor $T(t)$ could be identically $0$ either.

Plugging (\ref{eq-4.1.4}) into (\ref{eq-4.1.1})--(\ref{eq-4.1.2}) we get \begin{align*} & X(x) T''(t)=c^2 X''(x)T(t), \\[3pt] & X(0)T(t)=X(l)T(t)=0, \end{align*} which after division by $X(x)T(t)$ and $T(t)$ become \begin{align} & \frac{ T''(t)}{T(t)}=c^2 \frac{X''(x)}{X(x)}, \label{eq-4.1.5}\\[3pt] & X(0)=X(l)=0. \label{eq-4.1.6} \end{align} Recall, neither $X(x)$ nor $T(t)$ are identically $0$.

In (\ref{eq-4.1.5}) the l.h.e. does not depend on $x$ and the r.h.e. does not depend on $t$ and since we have an identity we conclude that

Remark 2. Both expressions do not depend on $x,t$ and therefore they are constant.

This is a crucial conclusion of the separation of variables method. We rewrite it as two equalities \begin{equation*} \frac{T''(t)}{T(t)}=-c^2\lambda,\qquad \frac{ X''(x)}{X(x)}=-\lambda, \end{equation*} which in turn we rewrite as (\ref{eq-4.1.7}) and (\ref{eq-4.1.8}): \begin{align} &X'' +\lambda X=0, \label{eq-4.1.7}\\[3pt] & X(0)=X(l)=0, \tag{\ref{eq-4.1.6}}\\[5pt] & T''+ c^2\lambda T=0.\label{eq-4.1.8} \end{align}

### Egenvalue problem

Consider BVP (for ODE) (\ref{eq-4.1.7})--(\ref{eq-4.1.6}). We need to find its solution $X(x)$ which is not identically $0$.

Definition 1. Such solutions are called eigenfunctions and corresponding numbers $\lambda$ eigenvalues (compare with eigenvectors and eigenvalues.)

Proposition 1. (\ref{eq-4.1.7})--(\ref{eq-4.1.6}) has eigenvalues and eigenfunctions \begin{align} & \lambda_n = \frac{\pi^2n^2}{l^2}&& n=1,2,\ldots, \label{eq-4.1.9}\\[3pt] &X_n(x)=\sin (\frac{\pi n x}{l}). \label{eq-4.1.10} \end{align}

Proof. (\ref{eq-4.1.9}) is a 2-nd order linear ODE with constant coefficients and to solve it one needs to consider characteristic equation $$k^2+\lambda =0 \label{eq-4.1.11}$$ and therefore $k_{1,2}=\pm \sqrt{-\lambda}$ and $X= Ae^{\sqrt{-\lambda}x} + Be^{-\sqrt{-\lambda}x}$ (provided $\lambda\ne 0$). Plugging into $X(0)=0$ and $X(l)=0$ we get \begin{align*} &A \qquad +B\qquad = 0,\\ &Ae^{\sqrt{-\lambda}l} + B e^{-\sqrt{-\lambda}l}=0 \end{align*} and this system has a non-trivial solution $(A,B)\ne 0$ if and only if its determinant is $0$: \begin{equation*} \left| \begin{matrix} 1 & 1\\ e^{\sqrt{-\lambda}l} & e^{-\sqrt{-\lambda}l}\end{matrix}\right|= e^{-\sqrt{-\lambda}l}-e^{\sqrt{-\lambda}l}=0 \iff e^{2\sqrt{-\lambda}l} \iff 2\sqrt{-\lambda}l=2\pi in \end{equation*} with $n=1,2,\ldots$. Here we excluded $n=0$ since $\lambda \ne 0$ and both $n$ and $-n$ lead to the same $\lambda$ and $X$. The last equation is equivalent to (\ref{eq-4.1.9}). Then $k_{1,2}=\frac{\pi n}{l} i$.

Meanwhlie $B=-A$ anyway and we get $X=2Ai \sin (\frac{\pi n x}{l})$ i.e. (\ref{eq-4.1.10}) as factor does not matter.

So far we have not covered $\lambda=0$. But then $k_{1,2}=0$ and $X=A+Bx$ and plugging into (\ref{eq-4.1.6}) we get $A=A+Bl=0 \implies A=B=0$ and $\lambda=0$ is not an eigenvalue.

### Simple solutions

After eigenvalue problem has been solved we plug $\lambda=\lambda_n$ into (\ref{eq-4.1.8}): $$T''+ (\frac{c\pi n}{l})^2 T=0 \label{eq-4.1.12}$$ which is also a 2-nd order linear ODE with constant coefficients.

Characteristic equation $k^2+ (\frac{c\pi n}{l})^2=0$ has solutions $k_{1,2}=\pm \frac{c\pi n}{l}i$ and therefore $$T_n(t)=A_n \cos(\frac{c\pi n}{l}t ) + BA_n \sin (\frac{c\pi n}{l}t ) \label{eq-4.1.13}$$ and finally \begin{multline} u_n(x,t)= \underbrace{\bigl( A_n \cos(\frac{c\pi n}{l}t )+ B_n \sin (\frac{c\pi n}{l}t )\bigr)}_{=T_n(t)} \cdot \underbrace{\sin (\frac{\pi n x}{l})}_{=X_n(x)}\\ n=1,2,\ldots \label{eq-4.1.14}\qquad\end{multline}

This simple solution (\ref{eq-4.1.14}) represents a standing wave

which one can decompose into a sum of running waves

and the general discussion of standing waves could be found in wikipedia

### General solutions

The sum of solutions of \ref{eq-4.1.1})-(\ref{eq-4.1.2}) is also a solution: $$u(x,t)= \sum_{n=1}^\infty \bigl( A_n \cos(\frac{c\pi nt}{l} )+ B_n \sin (\frac{c\pi nt}{l} )\bigr) \cdot \sin (\frac{\pi n x}{l}). \label{eq-4.1.15}$$ We have an important question to answer:

Have we covered all solutions of (\ref{eq-4.1.1})-(\ref{eq-4.1.2})? -- Yes, we did but we need to justify it.

Plugging (\ref{eq-4.1.15}) into (\ref{eq-4.1.3})1,2 we get respectively \begin{align} &\sum_{n=1}^\infty A_n \sin (\frac{\pi n x}{l})=g(x),\label{eq-4.1.16}\\ &\sum_{n=1}^\infty \frac{c\pi n}{l} B_n \sin (\frac{\pi n x}{l} )=h(x). \label{eq-4.1.17} \end{align}

How to find $A_n$ and $B_n$? Do they exist? Are they unique?

What we got are Fourier series (actually $\sin$-Fourier series). And we consider their theory in the several next sections.