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Exact Solutions > Interesting Papers > N.A. Kudryashov. Seven common errors in finding exact solutions of nonlinear differential equations > Conclusion

Seven common errors in finding exact solutions of nonlinear differential equations
© N.A. Kudryashov
Communications in Nonlinear Science and Numerical Simulation (to be pubslished in 2009)

Contents

  1. Abstract
  2. Introduction
  3. First error: some authors use equivalent methods to find exact solutions
  4. Second error: some authors do not use the known general solutions of ordinary differential equations
  5. Third error: some authors omit arbitrary constants after integration of equation
  6. Fourth error: using some functions in finding exact solutions some authors lose arbitrary constants
  7. Fifth error: some authors do not simplify the solutions of differential equations
  8. Sixth error: some authors do not check solutions of differential equations
  9. Seventh error: some authors include additional arbitrary constants into solutions
  10. Conclusion
  11. References

9. Conclusion

Let us shortly formulate the results of this paper. First of all we have tried to give some classification of the common errors and mistakes which occur in finding the solitary wave solutions of nonlinear differential equations. We have illustrated these errors using the examples from the recent publications in which the authors presented the exact solutions of nonlinear differential equations.

We have tried to classify the wide - spread errors in finding the exact solutions of nonlinear differential equations. Unfortunately we have no possibility to consider all errors and all problems of computer methods in finding the exact solutions. However we can note that the authors of many papers do not analyze the solutions of nonlinear differential equations that they obtain. Some errors are mentioned above due to this reason. It is important to discuss the solutions of nonlinear differential equations. We need to study how a solution depends on the parameters of nonlinear differential equation. It is useful to study the plots for the solutions of differential equation. We are interested in the stable exact solutions of nonlinear differential equations. These solutions are useful for the investigation of physical processes.

In our paper we have discussed the solitary wave solutions of nonlinear differential equations, but certainly we can find the same errors in looking for the other exact solutions.

We believe the results of this paper will be useful for young people who are going to study the exact solutions of nonlinear differential equations. We also hope that the material of this paper will be interesting for some referees.

10. Acknowledgements

This work was supported by the International Science and Technology Center under Project B 1213.

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