Hello,
show your working and we'll tell you what's wrong (although I don't think that'll be me ^^)
Solve the equation u_xx - 3u_xt - 4u_tt = 0 subject to the initial conditions u(x,0) = x^2 and u_t(x,t) = e^x.
Hint: consider a change of coordinates.
Having spent a large amount of time getting wrong solutions to the above question, any help that can be offered would be very much appreciated!
Thanks!
Ok no problem!
I have seen something similar where you factor out the operator: (d_xx - 3d_xt - 4d_tt) = 0
This gives us: (d_t + d_x)(4d_t + d_x)u = 0
I then have attempted to define new coordinates. I was led to believe that I should try, for example: a = x +t and b = -4x + t (though this does not appear to work so perhaps this is where I have gone wrong?).
We then want to find a v(a,b) such that it solves the equation.
We then get;
u_x = v_a - 4v_b
u_t = v_a + v_b
I then continue...but will avoid writing out the rest of it now in case my coordinates are wrong! If anybody can advise me if they are or are not correct then I will post the rest of my solution on that basis!
Many thanks!
Keep going, you're on the right lines. Now use the chain rule to find and in terms of and . For example,
You should then find that is some constant times (the terms in and all cancel out). So your equation is equivalent to , whose general solution is for arbitrary (twice-differentiable) functions f, g. So the solution for u(x,t) is .
That is a brilliant help thanks very much! I have now got to this stage...but am now a little confused as to how I take into the account the initial conditions.
I have attempted to somehow adapt d'Alambert's solution to the wave equation (or something along these lines) but the solution I attain does not seem to satisfy the original equation!
In particular I obtained a solution along the lines of:
(1/2) (x-t)^2 + (1/2) (4x+t)^2 + the integral of e^y dy (between the limits (x-t) and (4x+t))
Again any help in finishing this question would be much appreciated! Thanks!
In that case, if the general solution is then the initial conditions are
Integrate the second one to get . (There should be a constant of integration somewhere there, but it will cancel out in the eventual solution, so let's ignore it.)
Solve those two simultaneous equations for f(x) and g(4x) to get
Therefore
Add those together to get the solution for u(x,t).