Before revision:  We prove here that this method is applicable to any linear PDE (of arbitrary order) the source of which is a linear combination of one-dimensional delta distributions and derivatives thereof supported at an arbitrary number of particles. We furthermore apply this method to obtain numerical solutions for various types of distributionally-sourced PDEs: we consider first-order hyperbolic equations with applications to neuroscience models (describing neural populations ), parabolic equations with applications to financial models (describing price formation), second-order hyperbolic equations with applications to wave acoustics, and finally elliptic(Poisson) equations .
After revision:  We prove here that this method is applicable to any linear PDE (of arbitrary order) the source of which is a linear combination of one-dimensional delta distributions and derivatives thereof supported on a one-dimensional subspace of the problem domain. We then implement it to numerically solve a variety of relevant PDEs: hyperbolic ( with applications to neuroscience models (describing neural populations ), parabolic equations with applications to financial models (describing price formation), second-order hyperbolic equations with applications to wave acoustics, and finally elliptic(Poisson) equations .