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Generalization of the simplest equation method for nonlinear non-autonomous differential equations |
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| Institution: | 1. Mathematical Department, Zhejiang Normal University, Zhejiang 321004, PR China;2. School of Mathematical Science, Peking University, Beijing 100871, PR China;1. Department of Mathematics, Shanghai University, Shanghai 200444, PR China;2. Department of Mathematics, Shanghai Normal University, Shanghai 200234, PR China;3. Center for Applied Mathematics and Theoretical Physics, University of Maribor, SI-2000 Maribor, Slovenia;4. Faculty of Natural Science and Mathematics, University of Maribor, SI-2000 Maribor, Slovenia |
| Abstract: | It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order. |
| Keywords: | Simplest equation method Painlevé equation Special solution Riccati equation |
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