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##1.2. Initial and Boundary Value Problems
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> 1. [Problems for PDEs](#sect-1.2.1)
> 2. [Notion of 'well-posedness'](#sect-1.2.2)
###Problems for PDEs
We know that solutions of ODEs typically depend on one or several constants. For PDEs situation is more complicated. Consider simplest equations
\begin{gather}
u\_x =0,
\label{eq-1.2.1}\\\\
v\_{xy}=0
\label{eq-1.2.2}
\end{gather}
with $u=u(x,y)$ and $v=v(x,y)$. Equation (\ref{eq-1.2.1}) could be treaded as an ODE with respect to $x$ and its solution is a constant but this is not a genuine constant as *it is constant only with respect to $x$ and can depend on other variables*; so $u(x,y)=\phi(y)$.
Then for solution of (\ref{eq-1.2.2}) we have $v_y= \phi(y)$ where $\phi$ is an arbitrary function of one variable and it could be considered as ODE with respect to $y$; then $(v-g(y))_y=0$ where $g(y)=\int \phi(y)\,dy$, and therefore $v-g(y)=f(x)\implies v(x,y)=f(x)+g(y)$ where $f,g$ are arbitrary functions of one variable.
Considering these equations again but assuming that $u=u(x,y,z)$, $v=v(x,y,z)$ we arrive to $u=\phi(y,z)$ and $v=f(x,z)+g(y,z)$ where $f,g$ are arbitrary functions of two variables.
Solutions to PDEs typically depend not on several arbitrary constants but on one or several arbitrary functions of $n-1$ variables. For more complicated equations this dependance could be much more complicated and implicit. To select a right solutions we need to use some extra conditions.
The sets of such conditions are called *Problems*. Typical problems are
- IVP (*Initial Value Problem*): one of variables is interpreted as *time* $t$ and conditions are imposed at some moment; f.e. $u|_{t=t_0}=u_0$;
- BVP (*Boundary Value Problem*) conditions are imposed on the boundary of the spatial domain $\Omega$: f.e. $u|_{\partial\Omega}=\phi$ where $\partial\Omega$ is a boundary of $Omega$;
- IVBP (*Initial-Boundary Value Problem* aka *mixed problem*): one of variables is interpreted as *time* $t$ and some conditions are imposed at some moment but other conditions are imposed on the boundary of the spatial domain.
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