Exact solution of the Heat Equation for initial polynomials or splines

TL;DR

This paper derives exact solutions for the heat equation with initial conditions modeled as polynomials or splines, providing explicit formulas in 1D, 2D, and 3D for temperature evolution.

Contribution

It introduces a method to obtain exact solutions of the heat equation for piecewise polynomial initial distributions in multiple dimensions.

Findings

Solutions expressed as finite combinations of Gaussians and Error Functions.

Applicable to 1D, 2D, and 3D cases with boundary conditions on grid-aligned boundaries.

Provides detailed theoretical development and illustrative examples.

Abstract

The exact evolution in time and space of a distribution of the temperature (or density of diffusing matter) in an isotropic homogeneous medium is determined where the initial distribution is described by a piecewise polynomial. In two dimensions, the boundaries of each polynomial must lie on a grid of lines parallel to the axes, while in three dimensions the boundaries must lie on planes perpendicular to the axes. The distribution at any position and later time is expressed as a finite linear combination of Gaussians and Error Functions. The underlying theory is developed in detail for one, two, and three dimensional space, and illustrative examples are examined.