A shooting method for nonlinear heat transfer using automatic differentiation

Abstract

The steady-state temperature distribution in a cylinder of unit radius is modeled by a nonlinear two-point boundary value problem with an endpoint singularity. We present a simple shooting method for the solution of this problem. It is well known that shooting methods solve initial-value problems repeatedly until the boundary conditions are satisfied. For the solution of the initial value problems, the method of this paper uses a technique known as automatic differentiation and obtains a Taylor series expansion for the solution. Automatic differentiation is the process of obtaining the exact values of derivatives needed in the Taylor series expansion using recursive formulas derived from the governing differential equation itself. Thus, the method does not face the need to deal with step-size issues or the need to carry out lengthy algebraic manipulations for obtaining higher-order derivatives. The method successfully reproduces the solutions obtained by previous researchers.