Seven common errors in finding exact solutions of nonlinear differential equations

By N. Kudryashov. 2009

285 citations recorded by the source corpus.

Abstract We analyze the common errors of the recent papers in which the solitary wave solutions of nonlinear differential equations are presented. Seven common errors are formulated and classified. These errors are illustrated by using multiple examples of the common errors from the recent publications. We show that many popular methods in finding the exact solutions are equivalent each other. We demonstrate that some authors look for the solitary wave solutions of nonlinear ordinary differential equations and do not take into account the well - known general solutions of these equations. We illustrate several cases when authors present some functions for describing solutions but do not use arbitrary constants. As this fact takes place the redundant solutions of differential equations are found. A few examples of incorrect solutions by some authors are presented. Several other errors in finding the exact solutions of nonlinear differential equations are also discussed.

Subfields: Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f52e1669aa1e9bf7d5bb8714fe1c7348576c6a6e

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Citations in the corpus

Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm (2012, methodology): “The arbitrariness of the constants ci; 1 6 i 6 n, brings more choices for solutions than we used to [27,28].”
Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity (2010, methodology): “Finally, we would like to note two of the papers of Kudryashov [39,40] who advises us to be careful when using the methods for obtaining of exact solutions of the nonlinear differential equations.”
Modified Kudryashov method for finding exact solitary wave solutions of higher‐order nonlinear equations (2011, methodology): “Kudryashov [32, 36] showed that all solitary wave solutions by Wazwaz [13] and Chen et al.”
Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics (2010, methodology): “Finally, we would like to note two of the papers of Kudryashov [59,60] who advises us to be careful when use the methods for obtaining of exact solutions of the nonlinear differential equations.”
Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs (2011, methodology): “Finally, we would like to note two of the papers of Kudryashov [64,65] who advises us to be careful when use the methods for obtaining of exact solutions of the nonlinear differential equations.”