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 $x=0$.