Comparing a Three-Term Perturbation Solution of the Nonlinear ODE of the Jacobi Elliptic SN Function to Its Approximation into Circular Functions

Abstract

In this paper, the nonlinear differential equation of the elliptic snfunction is solved analytically using the Lindstedt-Poincare perturbationmethod. This differential equation has a cubic nonlinearity and a con-stant known as the modulus of elliptic integral. This constant takes anyvalue from zero to one and the square of its value is used as a smallparameter to expand the dependent variable in series and start analyti-cal iterations. Fortunately, there is an exact solution to this differentialequation known as the Jacobi sn elliptic function. When the modulusapproaches zero the elliptic differential equation becomes linear with thecircular sine function as exact solution. Thus, the sine function is con-sidered as the unperturbed solution and is used as the basis to add morecorrection terms through analytical iterations. The Lindstedt-Poincaretechnique is used to render the perturbation solution uniformly valid atlarger values of the independent variable. A three-term perturbation so-lution is obtained and shows good convergence and boundedness. Thissolution is compared with the exact, numerically calculated, sn ellipticfunction. It is also compared analytically with the approximate expan-sion of the elliptic function into circular functions in case of a smallmodulus. The relative percentage error between the perturbation solutionand the exact one is calculated at certain values of the modulus and forall values of the independent variable. The relative error is reasonablysmall but increases at larger values of the modulus. In addition, the ap-proximation of the exact solution gives smaller relative error than thatof the perturbation solution including the same order of the modulus.