Path Integral Solution of Linear Second Order Partial Differential Equations II. Elliptic, Parabolic and Hyperbolic Cases

TL;DR

This paper develops a unified path integral approach to solve various types of second order linear PDEs, including elliptic, parabolic, and hyperbolic equations, with boundary conditions, introducing new computational methods.

Contribution

It specializes a general path integral theorem to specific PDE classes and verifies solutions by evaluating known kernels, also presenting new calculation techniques.

Findings

Validated path integral solutions for elliptic, parabolic, and hyperbolic PDEs.

Derived solutions for regions with planar and spherical boundaries.

Introduced novel calculational techniques for PDE kernel evaluation.

Abstract

A theorem that constructs a path integral solution for general second order partial differential equations is specialized to obtain path integrals that are solutions of elliptic, parabolic, and hyperbolic linear second order partial differential equations with Dirichlet/Neumann boundary conditions. The construction is checked by evaluating several known kernels for regions with planar and spherical boundaries. Some new calculational techniques are introduced.