Partial Differential Equations with MATLAB

174 An Introduction to Partial Differential Equations with MATLAB©R

b) Solve the problem

ux−uy=u, x > 0 ,y> 0 ,

u(x,0) = sin 2x, x≥ 0 ,

u(0,y)=sin3y, y≥ 0.

5.2 First-Order PDEs with VariableCoefficients ..........

In Exercise 18 of the previous section, we derived the more general version of
the convection equation

ut+v(x)ux=f(x, t).

Can we extend the method of Section 5.1 to deal with equations where the
coefficients are not all constant? Let’s see.

Example 1 Try to solveux+yuy=0.
We let

ξ=x

η=Ax+By

and theuηcoefficient becomesA+By. However, it is impossible to choose
constantsAandBwhich will makeA+By=0forally(why?). We meet
with a similar fate if we try

ξ=Ax+By

η=y.

What can we do?
In the previous section, our transformation was chosen so that the charac-
teristics were the curvesη=constant. This suggests that we try the same
thing for equations with variable coefficients. The obvious question, then, is,
“What are the characteristics?”
Remember that the characteristics were curves along which the PDE could
be treated as an ODE. Let’s go back to Example 1 of Section 5.1 and see if
we can look at that problem in a different way.

Example 2 2 ux+3uy=0.
Notice that we can write this equation as

(2ˆı+3ˆj)·(uxˆı+uyjˆ)=0