Let the Dirichlet problem  for the colloidal substance be given by linear fragment oscillation time on a unit  interval of time.
\begin{eqnarray}\label{eqn1-1}
\left\{
\begin{array}{lll}
\displaystyle\frac{\partial^2 u(x, t)}{\partial t^2}=\frac{\partial^2u(x, t)}{\partial x^2}, \quad x\in (0; \pi), \quad t \in (0; 1),\\
\displaystyle u(0, t)=0, \quad t\in [0; 1], \quad u(1, t)=0, \quad t\in [0; 1];\\
\displaystyle u(x, 0) = f(x), \quad x\in [0; \pi], \quad u(x, 1)=h(x), \quad x\in [0; \pi],\\
\end{array}\right.
\end{eqnarray}
where $u(x, t)$ is the deviation of the linear fragment rod from the equilibrium position.

We are looking for a solution to the problem (\ref{eqn1-1}).
\begin{definition}
A solution to the boundary value problem (\ref{eqn1-1}) \cite{Sveshnikov}, p. 254 is function $u(x, t)$ such that:\\
1. $u(x,t)$ is continuous with first derivatives on set $t\in [0; 1], x\in [0; \pi],$\\
2. $u(x,t)$ has continuous second order derivatives on set $t\in (0; 1), x\in (0; \pi),$\\
3. $u(x,t)$ satisfies the differential equation (\ref{eqn1-1}) on set $t\in (0; 1), x\in (0; \pi),$\\
4. $u(x, t)$ satisfies all the boundary conditions of the problem (\ref{eqn1-1}).
\end{definition}
The complexity of the problem (\ref{eqn1-1}) is that its solution for certain lengths of the time interval is not unique \cite{Ivanov} or may not exist at all (we choose the case corresponding to value $\alpha=1$ in the notation of \cite{Ivanov}). Moreover, the problem is unstable with respect to the initial data \cite{Ivanov}. Therefore, we construct and consider an auxiliary Cauchy problem, whose classical solution is well known (for example, \cite{Sveshnikov}). It is necessary to answer the question of whether the solution to the auxiliary problem also coincides with the solution to the problem (\ref{eqn1-1}).

Let us consider the question of eigenfunctions, since we construct a solution by expanding it in terms of the eigenfunctions of the corresponding Sturm-Liouville problem.

We also need to consider the auxiliary Cauchy problem (the direct problem), since we use it later.
\begin{eqnarray}\label{eqn1-1b}
\left\{
\begin{array}{lll}
\displaystyle\frac{\partial^2 u(x, t)}{\partial t^2}=\frac{\partial^2u(x, t)}{\partial x^2}, \quad x\in (0; \pi), \quad t \in (0; +\infty),\\
\displaystyle u(0, t)=0, \quad t\in [0; +\infty), \quad u(1, t)=0, \quad t\in [0; +\infty);\\
\displaystyle u(x, 0) = f(x), \quad x\in [0; \pi], \quad \frac{\partial u(x, 0)}{\partial t}=g(x), \quad x\in [0; \pi],\\
\end{array}\right.
\end{eqnarray}
We do not specify the spaces in which we construct the solution, because their clarification is preceded by the properties of the constructed solution.