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APM346-2012 => APM346 Math => Home Assignment 1 => Topic started by: Djirar on September 25, 2012, 12:06:23 PM

Title: Problem 1
Post by: Djirar on September 25, 2012, 12:06:23 PM
my solution to problem 1.
Title: Re: Problem 1
Post by: Victor Ivrii on September 25, 2012, 03:08:21 PM
So, what are the answers for (c), (d)? Especially (d). Formulate them explicitly

Scanning is passable but not spectacular
Title: Re: Problem 1
Post by: Rouhollah Ramezani on September 25, 2012, 04:28:16 PM
a,b) This is a first order linear PDE with constant coefficient. We begin by writing  equation of characteristic lines:
\Rightarrow y=-3x+C
This implies $u(x,y)=\phi(3x+y)$, for some arbitrary $\phi$. Letting $u|_{x=0}=e^{-y^2}$ we get:
Hence solution to IVP is$$
c) With initial condition $u|_{x=0}=y$ we have $\phi(y)=y$, $y>0$. Hence the solution is $$u(x,y)=3x+y$$
Since $3x+y>0$ $\forall x,y>0$, this solution is valid everywhere in the domain of $u(x,y)$.
d) In this case, the solution is defined only where $3x+y>0$ i.e. where $y>|3x|$. To find solution on the whole domain, we need to impose another condition e.g. at $y=0$. Letting $u|_{y=0}=x$ we get:
$$\phi(3x)=x, (x<0) \rightarrow \phi(x)=\frac{x}{3}, (x<0)$$ $$\Rightarrow u(x,y)=x+ \frac{y}{3}$$
Above equation is constrained to $x+ \frac{y}{3}<0$.
Final solution would be:
u(x,y)= \left\{
&x+ \frac{y}{3}, &y<|3x|