Differential equations with distributional sources---in particular, involving delta distributions and/or derivatives thereof---have become increasingly ubiquitous in numerous areas of physics and applied mathematics. It is often of considerable interest to obtain numerical solutions for such equations, but the singular (" point-like " ) modeling of the sources in these problems typically introduces nontrivial obstacles for devising a satisfactory numerical implementation . A common method to circumvent these is through some form of delta function approximation procedure on the computational grid , yet this strategy often carries significant limitations  . In this paper, we present an alternative technique for tackling such equations  : the "Particle-without-Particle" method. Previously introduced in the context of the self-force problem in gravitational physics, the idea is to discretize the computational domain into two (or more) disjoint pseudospectral (Chebyshev-Lobatto) grids in such a way that the "particle" (the singular source location) is always at the interface between them; in this way , one only needs to solve homogeneous equations in each domain, with the source effectively replaced by jump (boundary) conditions thereon. 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 .
Partial differential equations with distributional sources---in particular, involving (derivatives of) delta distributions---have become increasingly ubiquitous in numerous areas of physics and applied mathematics. It is often of considerable interest to obtain numerical solutions for such equations, but any singular (" particle " -like) source modeling invariably introduces nontrivial computational obstacles . A common method to circumvent these is through some form of delta function approximation procedure on the computational grid ; however, this often carries significant limitations on the efficiency of the numerical convergence rates, or sometimes even the resolvability of the problem at all . In this paper, we present an alternative technique for tackling such equations which avoids the singular behavior entirely : the "Particle-without-Particle" method. Previously introduced in the context of the self-force problem in gravitational physics, the idea is to discretize the computational domain into two (or more) disjoint pseudospectral (Chebyshev-Lobatto) grids such that the "particle" is always at the interface between them; thus , one only needs to solve homogeneous equations in each domain, with the source effectively replaced by jump (boundary) conditions thereon. We prove here that this method yields solutions to any linear PDE the source of which is any linear combination of 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 and acoustics ), parabolic ( with applications to finance), and elliptic. We generically obtain improved convergence rates relative to typical past implementations relying on delta function approximations .
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