Solutions of the Van der Pol Equation

Summary

Presented in this paper are various solutions to the Van der Pol equation. Numerical solutions are utilized as an independent means of validating the various solutions discussed. A new solution in the form of a power series has been found. Although this solution is exact, its interval of convergence can only be estimated for a special case. Numerical experiments reveal that the power series solution can provide an exact solution over intervals where other approximate solutions are not valid. Thus, the new solution represents an additional solution that can complement other existing solutions. This work also emphasizes the importance and the role of computation. Although the power series solution along with the other approximate solutions mentioned are of theoretical interest, their restrictions limit their usefulness in real applications, and therefore numerical methods should also be considered.

Additional information

Notes on contributors

Serge D’Alessio

Serge D’Alessio ([email protected], ORCID 0000-0002-0350-4672) completed his undergraduate studies in Engineering Physics at McMaster University and received his Ph.D. in Applied Mathematics from Western University. He is a professor in the Centre for Education in Mathematics and Computing housed in the Faculty of Mathematics at the University of Waterloo, and actively visits high schools across Canada and abroad to promote and motivate the application of mathematics in the sciences and engineering. He is also a licensed professional engineer with research interests in fluid mechanics, and an avid cyclist.