Approximate analytical solutions for a nonlinear differential equation of the corneal geometry

A precise determination of the corneal shape and predicting the changes in the corneal geometry can be of paramount importance for early detection of certain eye diseases and for assessing the postoperative outcomes of refractive surgery. In this article, a simple approach is proposed to solve a nonlinear differential equation that models the human corneal shape. Based on this approach, three approximate analytical solutions are developed using linear solution and Taylor series method. The accuracy and the convergence rate of the solutions are evaluated for different values of constant parameters of the equation that can arise in various physical situations. Our results show that the proposed solutions are more accurate than the other approximate analytical solutions available in the literature. In comparison with the homotopy perturbation method and perturbation method, our approach has less computational complexity.