Analytical Solutions of Generalised Emden–Fowler Initial and Boundary Value Problems of Higher Order

In this paper, the analytical solutions of a class of generalised linear and nonlinear Emden–Fowler-type equations of higher order are presented using a power series method. The proposed equations are subject to appropriate initial and boundary conditions. A generalised Cauchy product is applied to reduce the nonlinear terms to power series whose expansion coefficients are expressed recursively as finite sums involving the expansion coefficients of the assumed series solution. In the case of boundary value problems, required initial conditions are assumed and introduced as dummy constants, which are subsequently determined using the corresponding right boundary conditions. Several examples of Emden–Fowler-type initial and boundary value problems of the third, fourth, fifth, and sixth orders are explicitly given to illustrate the proposed method. For comparison purposes, we consider known problems in the literature. Interestingly, in all the examples considered, our series solutions agree excellently with the exact solutions and published results in the literature.