So I have a question asking me to solve the PDE Δu = $u_{tt}$ in $R^3$ subject to the initial conditions u(0,x,y,z) = $\sqrt{(x^2 + y^2 + z^2)}$ and $u_t(0,x,y,z) = x^2 + y^2 + z^2$. There is a hint to use the substitution w = ru, but I'm just not sure how to start solving this.

My thoughts are to first convert the PDE to spherical coordinates and then do the substitution, but I'm not sure what it means by find the "radial" solutions. Can someone help me figure out the solution steps?


1 Answer 1



In spherical coordinates the Laplacian is :

enter image description here


In the present case, the initial conditions are functions of $r$ only. Thus the solution is not function of the angles $\phi$ and $\theta$, but function of $r$ only for the geometric parameters and of course also function of $t$, so that the sought solution be on the form $u(t,r)$ .

That is why in the wording of the problem it is asked for a "radial" solution.

In this context the PDE and initial conditions are transformed into : $$\begin{cases} \frac{1}{r^2} \frac{\partial }{\partial r}\left(r^2 \frac{\partial u(t,r)}{\partial r}\right)=u_{tt} \\ u(0,r)=r \\ u_t(0,r)=r^2 \end{cases}$$

Note : The hint given in the wording of the problem $W=ru$ simplifies again the PDE to $W_{rr}=W_{tt}$

I suppose that you can take it from here.


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