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 sn function is solved analytically using the Lindstedt-Poincare perturbation method. This differential equation has a cubic nonlinearity and a con- stant known as the modulus of elliptic integral. This constant takes any value from zero to one and the square of its value is used as a small parameter to expand the dependent variable in series and start analyti- cal iterations. Fortunately, there is an exact solution to this differential equation known as the Jacobi sn elliptic function. When the modulus approaches zero the elliptic differential equation becomes linear with the circular sine function as exact solution. Thus, the sine function is con- sidered as the unperturbed solution and is used as the basis to add more correction terms through analytical iterations. The Lindstedt-Poincare technique is used to render the perturbation solution uniformly valid at larger values of the independent variable. A three-term perturbation so- lution is obtained and shows good convergence and boundedness. This solution is compared with the exact, numerically calculated, sn elliptic function. It is also compared analytically with the approximate expan- sion of the elliptic function into circular functions in case of a small modulus. The relative percentage error between the perturbation solution and the exact one is calculated at certain values of the modulus and for all values of the independent variable. The relative error is reasonably small but increases at larger values of the modulus. In addition, the ap- proximation of the exact solution gives smaller relative error than that of the perturbation solution including the same order of the modulus.