## New Nonstandard Finite Difference Methodsfor the Heat Equation

##### Abstract

Many physical phenomena around us can be described by mathematical models, which often take the form of a differential equation. Diffusion of heat through a homogenous bar or rod obeys a linear partial differential equations (PDE) called the heat equation. Exact solutions to this equation with homogenous boundary conditions are known for a wide class of initial conditions. There are also various numerical methods for ?solving? differential equations. Conventional numerically stable discretization schemes for this PDE, such as the Backward-time Centered-space (BTCS) method, only approximate these exact solutions. We modified BTCS to obtain a numerical solution that, for many initial conditions, agrees EXACTLY with the analytic solution at the grid points. These modifications, extended ideas of Mickens and Buckmire, involve replacing the usual time and space step-sizes with unknown ?denominator functions?. We worked backwards from known analytic solutions to determine exact formulas for ratios for these denominator functions. Our method has been successfully applied to the one-dimensional heat equation in both rectangular and cylindrical coordinates. This new method is of interest because it may prove more accurate for nonlinear PDEs than standard schemes.