Numerical method for obtaining periodic steady-state solutions of nonlinear fractional differential-algebraic equations

This paper concerns the design of a new numerical method for determining periodic steady-state solutions in time-dependent nonlinear fractional problems. This method essentially avoids the typical approach of applying a harmonic balance. The new method’s uncommon approach is to reconstruct the steady-state waveforms through their values at selected solution nodes (similar to what is observed in a time-stepping solver). This approach allows to take into account larger distortions in the considered periodic waveforms, while also avoiding potentially long solver computation times when transients with long settling times are considered (as is the case when applying time-stepping solvers). The general form of the problems under consideration takes the form of a system of nonlinear fractional differential–algebraic equations. This form was derived for circuit problems, where there are both classical elements and nonlinear fractional elements (like nonlinear fractional coils). The analysis in the paper may also be useful for other fields, where the problem can be described through the considered general form of nonlinear fractional differential–algebraic equations. The fractional derivative approximation has been described in detail so that interested researchers can also apply it in their implementations. The fractional derivative approximation has been tested on a selected example of a periodic waveform, where an analytical solution can be obtained. As for solving the system of nonlinear fractional differential–algebraic equations, the proposed method was tested on an example of a nonlinear circuit with fractional elements and nonsinusoidal sources, where the solution waveforms contain a significant contribution of higher harmonics. The method was tested not only in terms of accuracy, but also in terms of computation time.