A non-iterative shooting method for a non-linear diffusion problem using automatic differentiation

Abstract

We present a non-iterative shooting method for the solution of a non-linear two-point boundary value problem that models the steady-state temperature distribution in a cylinder of unit radius. By non-iterative it is meant that there is no need to solve one or more initial value problems repeatedly. The method of this article avoids the need for such repetition by numerically obtaining a single algebraic non-linear equation involving only the initial condition. Thus, the appropriate initial condition for the final solution is available just after one solution of the initial value problem corresponding to an arbitrary initial value. In addition, the solution of the initial value problem is obtained as a Taylor series expansion of arbitrary order, using a technique known as automatic differentiation, which is the process of obtaining the coefficients in the Taylor series expansion using recursive formulas. Thus, the method does not face the need to deal with step size issues or the need to carry out lengthy algebraic manipulations. The method successfully reproduces the solutions obtained previously by other researchers.

Keywords:

• Two-point boundary value problems
• Taylor series
• Taylor coefficients
• Automatic differentiation
• Shooting