Closed-form numerical formulae for solutions of strongly nonlinear oscillators

Abstract

Based on Gauss–Kronrod quadrature rule, this paper provides closed-form numerical formulae of the period, periodic solution and Fourier expansion coefficients for a class of strongly nonlinear oscillators. Firstly, the period derived in the form of definite integral is addressed by a new equation constructed according to the fundamental theorem of calculus. Then, an approximate closed-form expression of the period can be established by employing only a low-order Gauss–Kronrod quadrature formula. Changing the lower limit of the integral, the closed-form expression becomes a numerical formula that can give the periodic solution of the system. After this, according to the partial integration rule, the calculation of the Fourier coefficients is derived in a very concise form. In general, the relative error of the approximate period can be reduced to 1e−6 only by employing a 31-point Kronrod rule. Error magnitude of the period indicates the maximum error level of the periodic solution and Fourier coefficients. In addition, the proposed formulae are stable convergent and the exact solutions being their convergence limits. Three very typical examples are given to illustrate the usefulness and effectiveness of the proposed technique.