Analysis and application of initial solutions from the function interpolation method

Abstract The function interpolation method (FIM) is a continuous solution-tracking technique that combines the triangular collocation method with the Runge–Kutta method. Previously, the incremental harmonic balance (IHB) method was utilized to generate initial solutions for the FIM but the IHB method may encounter convergence limitations. In this work, three strategies are introduced to provide initial solutions of the FIM. Strategy 1 eliminates secular terms to derive solutions for linear differential systems, thereby establishing a connection between the Lindstedt-Poincaré (LP) method and the IHB method. Strategy 2 controls a certain system parameter so that one equation in the nonlinear system degenerates into a solvable linear differential equation, from which solutions for the entire system can be constructed. Strategy 3 transforms nonlinear differential systems into algebraic equations by making the excitation frequency equal to zero, making initial solutions straightforward to obtain. After obtaining initial solutions, the FIM can extend them to any place. Four examples are presented to verify the effectiveness of the strategies: the coupled van der Pol oscillator, wave propagation in the nonlinear diatomic chain, the coupled van der Pol oscillator with external excitation, and the van der Pol-Mathieu oscillator with external excitation. The results show that the FIM provides more accurate initial values for semi-numerical and semi-analytical methods compared to the LP method. Generally, solutions from the FIM can satisfy computational requirements. Moreover, solutions from the FIM and the IHB method agree well with numerical integration, which further verifies the reliability of initial solution strategies for the FIM.