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.
Keywords
perturbation methods , Jacobi elliptic function , Lindstedt-Poincare technique
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