Solving a PDE by wavelets

Question

I know how to solve a PDE by wavelets. I found a lot of articles about it.

But, if the PDE includes Dirac Delta distribution $( \delta)$, how to solve it by wavelets? (Haar, Legendre, Chebyshev wavelets don' t matter) I haven' t found any articles or books about it.

For example: $$\begin{aligned} \frac{\partial^{2} y}{\partial t^{2}}=\delta(x-v t) +\frac{\partial^{2} y}{\partial x^{2}} v^{2} \delta(x-v t) \end{aligned}$$ with Appropriate Initial and boundary conditions. (in here $v$ is a real number)

Could you give me some hints or suggest some articles which include PDE with Dirac Delta distribution?

Answer 1

Possible recipe for numerical work:

• Generic form of the PDE $$\partial_t^2y-a(x,t)\partial_x^2y=b(x,t)$$
• Weak form in space (you could extend to space-time if needed) with trial function $y$ and test function $v$ $$\forall v(x),\quad \int v (\partial_t^2y-a(x,t)\partial_x^2y-b(x,t))\,\mathrm{d}\Omega=0$$ that is [loose integration by parts but you do not specify the domain of interest and Boundary conditions, if any] $$\forall v,\quad \int v\,\partial_t^2y+A(x,t)\partial_xv\,\partial_xy\,\mathrm{d}\Omega=\int v\, b\,\mathrm{d}\Omega$$ (I am not sure about the expression of $A$ above)
• Wavelet discretization $$v(x)=\sum_i a_i \phi_i(x)\quad\text{and}\quad y(x,t)=\sum_i b_i(t)\psi_i(x)$$ where the $\phi_i$ and $\psi_i$ families are wavelets of your choice.
• Numerically compute the above integrals (note that since there are $\delta$ in the above integrals, the latter should be read in the sense of distributions, ie $\int \delta_a f =f(a)$) and solve in $b_i(t)$