An efficient method for solving Troesch's problem

Troesch's problem is an inherently unstable two-point boundary value problem. This paper describes an efficient method, in which the hyperbolic nonlinearity in Troesch's problem is first converted into the polynomial nonlinearity by variable transformation and the differential transform method is then used directly to solve this transformed problem. The approximate solution obtained is in the form of a rapid convergent power series and highly accurate in comparison with those obtained by other analytical and numerical methods. As compared with the differential transform method, this approach is more efficient and powerful since it provides a much faster convergent series solution with higher accuracy and considerably less computational effort. Finally, this method is successfully applied to other two-point boundary value problems with exponential nonlinearity and the obtained results demonstrate the reliability of the proposed approach.