Numerical solution of a third-order nonlinear boundary-value problem by automatic differentiation

We develop a simple numerical method for obtaining Taylor series approximation to the solution of a nonlinear third-order boundary-value problem. We use recursive formulas derived from the governing differential equation itself to calculate exact values of the derivatives needed in the Taylor series. Since we do not use difference formulas or symbolic manipulation for calculating the derivatives, our method requires much less computational effort when compared with the techniques previously reported in the literature. We will illustrate the effectiveness of our method with several test problems.