Models and numerical schemes for generalized van der Pol equations

• Three generalizations of the van der Pol equation (VDPE) using three newly proposed generalized operators are presented.
• These operators allow kernels to be selected from a much larger set.
• Numerical schemes are presented to solve the generalized van der Pol equations (GVDPEs).
• The nonlinear including the limit cycles behaviors of the GVDPEs are examined.
• These generalizations may turn out to be useful where integer and fractional order VDPEs do not work well.

This paper presents three generalizations of the van der Pol equation (VDPE) using newly proposed three new generalized K-, A- and B-operators. These operators allow kernel to be arbitrary. As a result, these operators provide a greater generalization of the VDPE than the fractional integral and differential operators do. Like the original VDPE, the generalized van der Pol equations (GVDPEs) are also nonlinear equations, and in most cases, they can not be solved analytically. Numerical algorithms are presented and used to solve the GVDPEs. Results for several examples are presented to demonstrate the effectiveness of the numerical algorithms, and to examine the behavior of the GVDPEs and the limit cycles associated with them. Although the numerical algorithms have been used to solve the GVDPEs only, they can also be used to solve many other generalized oscillators and generalized differential equations. This will be considered in the future.