Folks,

trying 2 methods, neither working out for me

1)Method of Characteristics

$\displaystyle \displaystyle \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$ given u(x,0)=x

$\displaystyle \frac{dy}{dx}=1 \implies c=y-x$, $\displaystyle \frac{du}{dx}=0 \implies u=f(c)$

Using IC's, $\displaystyle x=f(-x)$, let$\displaystyle t=-x \implies x=-t=f(t) \therefore u(x,y)=y-x$

but the correct is $\displaystyle u(x,y)=-y+x$

2) Parameterise x and y with s with the given IC's

$\displaystyle x(s); x(0)=x, y(s);y(0)=0$

$\displaystyle \frac{d}{ds} u(x(s),y(s))= \frac{dx}{ds}\frac{\partial u}{\partial x}+\frac{dy}{ds}\frac{\partial u}{\partial y} = 0$

$\displaystyle \frac{dx}{ds}=1 \implies x=s+A, \frac{dy}{ds}=1 \implies y=s+B, \frac{du}{ds}=0 \implies u=f(c)$

Not sure what is next, when I use the IC's, it doesnt look right.

THanks