solve Uxx-3Uxt-4Utt=0 with U(x,0)=x^2 and Ut(x,0)=e^x
I know that this is hyperbolic since D=(-1.5)^2+4 >0 so I have to
transform the variables x and t linearly to obtain the wave equation
of the form
(Utt-c^2Uxx=0). The above equation is equivalent to (My work):
(d/dx - 1.5 d/dt)*(d/dx - 1.5 d/dt)u - 6.25 d^2u/dt^2 = 0
let x=b
let t=-1.5b + 2.5a
Thus,
Ub=Ux - (1.5) Ut
Ua=2.5 Ut
thus Ubb-Uaa=0. This is where I am stuck..
I know the general solution is U(a,b)=f(a+b)+g(a-b)
(not sure if next part is needed but here it is).
Also the explicit solution is U(a,b)=(1/2)*[h(a+b)+h(a-b)]*(1/2c)*(integral
k(s)ds from a-b to a+b). according to d'Alamber's solution
where U(a,0)=h(a) and Ub(a,0)=k(a).
The solution is (4/5)*[e^(9x+t/4)-e^(x-t)]+x^2+(1/4)*t^2
but how to obtain it? |
