Solving an initial value problem for a PDE
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Salma fathi 2022년 4월 26일
Having the following initial value problem
with some mathematical computations we reach to an end that an implicit general solution of this pde can have the following form
if we had phi=e^(-x^2) for example,
I have been able to solve a similar problem to this but the genral solution was only a function of x and t, but here we have also u, so how can we possibly do that.
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편집: Torsten 2022년 4월 26일
The method of characteristics gives the equations
dt/ds = 1, t(0) = 0
dx/ds = u, x(0) = x0
du/ds = 0, u(0) = phi(x0)
x = x0 + phi(x0) * t
Thus to get the solution u(x,t) in (x,t), you will have to solve
x - x0 - phi(x0)*t = 0
The solution u(x,t) in (x,t) is then given by u(x,t) = phi(x0).
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