We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals K and E. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations ar...

Several nonlinear systems such as the Korteweg-de Vries (KdV) and modified KdV equations and lambda phi(4) theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of so...

In this article, a new version of the generalized F-expansion method is proposed enabling to obtain exact solutions Biswas-Arshed equation and Boussinesq defined by Atangana’s beta-derivative. First, introduced, then, nonlinear fractional differential equations expressed with beta-derivative are given. When results examined, it seen that single, combined, mixed Jacobi elliptic function obtained...

In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations of many of Gudermann’s formulas in Carlson’s modern notation. The transformations between squares of ellipt...

A rational elliptic balance method is introduced to obtain exact and approximate solutions of nonlinear oscillators by using Jacobi elliptic functions. To illustrate the applicability of the proposed rational elliptic forms in the solution of nonlinear oscillators, we first investigate the exact solution of the non-homogenous, undamped Duffing equation. Then, we introduce first and second order...

We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a new type of elliptic hypergeometric function. We show that obtained polynomials are orthogonal on the unit circle with respect to a positive measure which is s...

We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a new type of elliptic hypergeometric function. We show that obtained polynomials are orthogonal on the unit circle with respect to a dense point meausure, i.e. ...

We propose a new iterative scheme to compute the numerical solution an over-determined boundary value problem for general quasilinear elliptic PDE. The main idea is repeatedly solve its linearization by using quasi-reversibility method with suitable Carleman weight function. presence of function allows us employ estimate prove convergence sequence generated above desired solution. iteration fas...

We will show the utility of the classical Jacobi Thetanullwerte for the description of certain period lattices of elliptic curves, providing equations with good arithmetical properties. These equations will be the starting point for the construction of families of elliptic curves with everywhere good reduction.