# Toronto Math Forum

## APM346-2012 => APM346 Math => Misc Math => Topic started by: Dana Kayes on October 05, 2012, 11:09:44 AM

Title: Method of Continuation
Post by: Dana Kayes on October 05, 2012, 11:09:44 AM
Could anyone help me understand the method of continuation? The notes for Lecture 8 are just a little too complex for me to follow. What are the steps, and what is the aim? I think that if I can better understand what we're trying to achieve by using it, I'll be able to follow the notes better.

Thanks :)
Title: Re: Method of Continuation
Post by: Vitaly Shemet on October 06, 2012, 09:29:29 AM
I understood the even/odd "trick" as a classification, what to do in each case, but I can't understand the reason and idea of applying, what results in disability to apply method to another types of equations (not 1dwe)
Title: Re: Method of Continuation
Post by: Victor Ivrii on October 06, 2012, 10:16:32 AM
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 $x=0$.

Title: Re: Method of Continuation
Post by: Aida Razi on October 14, 2012, 05:39:40 PM
The best explanation for method of continuation  :)
Thank you professor,