A Shooting Method to Solve Singular Heat Transfer Equation Using Automatic Differentiation
Keywords:Shooting Method, Taylor Series Expansion, Automatic Differentiation
In this study, Taylor series expansion and shooting method have been applied to find a solution of the nonlinear two-point boundary value problem that models the steady-state temperature distribution in a cylinder with an endpoint singularity. To obtain the coefficients in the Taylor series expansion, this study uses automatic differentiation the aid of recursive formulas derived from the governing differential equation itself. In addition, this method does not have to carry out lengthy algebraic manipulations for obtaining higher-order derivatives because it is only needs to substitute the value in order to find coefficients in the Taylor series expansion. Shooting method is the techniques to solve initial-value problems repeatedly until the boundary conditions is satisfied using Maple18 and are presented in the form of table. The solution of the initial value problem is obtained as a Taylor series expansion of arbitrary order. A solution of the result of Taylor series expansion are depends on heat generation constants (alpha). The heat generation constants (alpha) in Taylor series expansion are analyzed and discussed in the form of table and graph. It is found that the difference between alpha equal to 0.1 until alpha equal to 0.8 in Taylor series expansion of the equation are too small.