"Elisa" <elisa.d
...@gmail.com> wrote in message
> Hi,
> I have to solve this pde:
> dW/dz + a(dW/dt)^2+g(t)(dW/dt)=f(t)
> with
> W(0,t)=0
> any idea about possible solutions?
Yes, but you may need to be more specific with g(t) and f(t). For example,
write the equation as follows
w_z+a(w_t)^2 +g(t)w_t-f(t)=0 such that w(0,t)=0 and a=constant
you then have the five Lagrange-Charpit equations to solve, namley
dz/ds = 1
dt/ds=2aw_t + g(t)
dw_z/ds=0
dw_t/ds=-g'(t)w_t + f'(t)
dw/ds=w_z+2a(w_t)^2+g(t)w_t
(i am omiting * signs for multiplication here becuase everything should be
obvious in this context and for what follows)
parametrise the initial data by choosing for s=0, z(0,r)=0 and t(0,r)=r, so
the initial condition is w(0,r)=0
by doing so I get w_t(0,r)=0 and w_z(0,r)=f(r)
now the first equation --> z=s, and the second that w_z(s,r) = w_z(0,r) =
f(r).
the fourth equation --> w_t(s,r) = f'(t)/g'(t) - f'(r)/g'(r) exp( -sg'(t) )
the second equation then -->
dt/ds = 2a [ f'(t)/g'(t) - f'(r)/g'(r) exp( -sg'(t) ) ] +g(t)
and this is where i run into trouble, because we need to solve this for
t(s,r) in order to successfully solve the last remaining eqauation. i can
find no symmetries of this general equation and can see no obvious way it
can be integrated without specifying some sort of functional relationship on
f and g.
So we can keep going and illustrate the idea, let's assume, as an example,
that g(t)=f(t)=t then
w_t = 1 - exp(-s) and
dt/ds = 2a(1-exp(-s)) + t
solving this we find t(s,r) = r exp(s) + 2a(cosh(s)-1)
From the very last equation...
dw/ds = r + 2a(1-exp(-s))^2 + (1-exp(-s)) (r exp(s) + 2a(cosh(s)-1))
solving this we have
w(s,r) = r (1+2s) - (2+s)a +exp(s) (a-r) +a exp(-s) ( 1+sinh(s) )
we know s=z and can find r from the equation for t(s,r), i.e. r = -exp(-z)
(-2a-t+2acosh(z))
putting it all into w we get after a bit of manipulation
w(z,t) = t + (sinh(z)-cosh(z)) ( 2a+t+a(z-2)cosh(z)+a(z-1)sinh(z) )
and this can indeed be verified to solve
w_z+a(w_t)^2 +g(t)w_t - f(t)=0 where g(t)=f(t)=t and w(0,t)=0
If you want to understand where the Lagrange-Charpit equations come from,
and why all this works - look at Copson's "Partial Differential Equations" -
i just found my dusty copy last night after a three day search. it's a good
read.
cheers
moth
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