The Heat Equation—Tychonov’s solution
The most common point of view towards the heat equation is that of
solving the initial value problem: given $u(x, 0)$ for all
$x\in\R$ try to find the corresponding solution $u(x, t)$ to the
heat equation $u_t = u_{xx}$ for all $x\in\R$ and $t\gt0$.
Tychonov observed that there is a simple formula for the solution
to the heat equation if, instead of prescribing the initial value,
one prescribes the value of the solution at $x=0$.
Let $J\subset\R$ be some interval, and suppose we have a function
$\phi\in C^\infty(J)$ that satisfies
\begin{equation}
|\phi^{(k)}(t)| \leq C A^k k^{\theta k} \tag{1}
\end{equation}
for some $\theta\in(1,2)$ and some $A\gt0$. Functions that
satisfy this condition are said to be
of
Gevrey
class $\theta$. One such function is given by
\begin{equation}
\phi(t) =
\begin{cases}
e^{-1/t^2} & (t\gt0) \\ 0 & (t\leq 0)
\end{cases} \tag{2}
\end{equation}
which satisfies the Gevrey condition (1) with $\theta=\frac32$
(although it takes some work to prove that.)
Given the function $\phi$ we define
\begin{equation}
u(x, t) = \sum_{k=0}^\infty \phi^{(k)}(t) \frac{x^{2k}}{(2k)!}.
\tag{3}
\end{equation}
Assuming that the series converges and that we are allowed to
differentiate it term by term, we find
\[
u_t = \sum_{k=0}^\infty \phi^{(k+1)}(t) \frac{x^{2k}}{(2k)!}
\text{ and }
u_{xx} = \sum_{k=1}^\infty \phi^{(k)}(t) \frac{x^{2(k-1)}}{(2(k-1))!}.
\]
It follows that $u_t = u_{xx}$, i.e. $u$ is a solution of the heat equation.
To prove convergence of the series we use our assumption (1) on the
size of the derivatives $\phi^{(k)}(t)$. If $|x|\leq L$, then we get
\[
\left|\frac{\phi^{(k)}(t) x^{2k}} {(2k)!}\right|
\leq C (AL^2)^k \frac{k^{\theta k}} {(2k)!} \stackrel{\rm def}= M_k.
\]
Applying the ratio test we see that
\[
\lim_{k\to\infty}\frac{M_{k+1}} {M_k}
=\lim_{k\to\infty} ML^2 \frac{(k+1)^\theta}{(2k+2)(2k+1)} \left(1+\frac{1} {k}\right)^{\theta k}
=ML^2 e^\theta \lim_{k\to\infty} \frac{(k+1)^\theta}{(2k+2)(2k+1)} = 0,
\]
where we have used $\theta\lt2$ in the very last step.
Therefore the Weierstrass M–test implies that the series for
$u$ converges uniformly for $t\in J$, $|x|\leq L$, for any finite
$L$. The same arguments apply to the series for $u_t$ and $u_{xx}$,
and indeed all higher derivatives of $u$, so we see that we are
allowed to differentiate the series for $u$ term by term, and $u$ is
indeed a solution to the heat equation which is defined for $t\in J$, $x\in\R$.