A method of continuation is a cheap trick to reduce certain BVP to those we already know how to solve. In its easiest form we looked at it in the lectures.
Consider a BVP with one "special" variable $x$ (there could be other variables). This $x$ runs from $0$ to $+\infty$ (there could be other cases). Consider the same problem but with $x$ running from $-\infty$ to $\infty$, thus dropping boundary condition(s) at $x=0$.
Assume that
1) plugging $-x$ instead of $x$ leaves this new boundary problem unchanged. F.e. it happens when we consider equations with the constant coefficients and containing only even order derivatives by $x$;
Good: $u_{t}+u_{xx}$, $u_{yxx}+ u_{y}-u_{xx}$
Bad: $u_{tx}+u_{xx}$, $u_t+u_{xxx}$
Variable coefficients can affect this situation:
Also good: $u_{tt}- xu_{xxx}$
Bad: $u_t + xu_{xx}$
So far we applied method of continuation to wave and heat equations:
$$u_{tt}-c^2u_{xx}=f, \qquad u|_{t=0}=g, \qquad u_t|_{t=0}=h$$
and
$$u_{t}-ku_{xx}=f, \qquad u|_{t=0}=g.$$
2) Assume that boundary conditions contains only terms with all odd order derivatives with respect to $x$ and are homogeneous:
$u_x|_{x=0}=0$ or $(u_x-u_ {xxx})|_{x=0}=0$ fit the bill.
Note that even functions satisfy these boundary conditions automatically. Then:
We continue all known functions to $x<0$ as even functions and solve extended problem (ignoring boundary condition(s) at $x=0$.
2*) Alternatively, assume that boundary conditions contains only terms with all even order derivatives with respect to $x$ and are homogeneous:
$u|_{x=0}=0$ or $(u-u_ {xx})|_{x=0}=0$ fit the bill.
Note that odd functions satisfy these boundary conditions automatically. Then:
We continue all known functions to $x<0$ as odd functions and solve extended problem (ignoring boundary condition(s) at $x=0$.
