2
$\begingroup$

I wanna solve the equation

$$u_x+u_y+u=\exp(x+2y), \quad u(x,0) = 0$$

I have just learned method of characteristics. But I don't know how to deal with $u$ term and inhomogeneous term simultaneously. Can you help me?

$\endgroup$
0
$\begingroup$

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dy}{dt}=1$ , letting $y(0)=0$ , we have $y=t$

$\dfrac{dx}{dt}=1$ , letting $x(0)=x_0$ , we have $x=t+x_0=y+x_0$

$\dfrac{du}{dt}=e^{x+2y}-u=e^{3t+x_0}-u$ , we have $u(x,y)=\dfrac{e^{3t+x_0}}{4}+f(x_0)e^{-t}=\dfrac{e^{x+2y}}{4}+f(x-y)e^{-y}$

$u(x,0)=0$ :

$f(x)=-\dfrac{e^x}{4}$

$\therefore u(x,y)=\dfrac{e^{x+2y}-e^{x-2y}}{4}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.