An approximation technique for first Painlevé equation

Abstract

In this study, we introduce a new approximative algorithm to get numerical solutions of the nonlinear first Painlevé equation. Indeed, to obtain an approximate solution, a combination of exponential matrix method based on collocation points and quasilinearization technique is used. The quasilinearization method is used to transform the original non-linear problem to a sequence of linear equations while the exponential collocation method is employed to solve the resulting linear equations iteratively. Furthermore, since the exact solution of the model problem is not known, an error estimation based on the residual functions is presented to check the accuracy of the proposed method. Finally, the benefits of the method are illustrated with the aid of numerical calculations. Comparisons with other well-known schemes show that the combined technique is easy to implement while capable of giving results of very high accuracy with a relatively low number of exponential functions.