Elliptic really does have a connection with literal geometric ellipses, removed by a few steps. Integrals expressing arc length of ellipses are less elementary than integrals for inverse trigonometric functions, and were named elliptic because of this. Abel and Jacobi discovered in 1827 that the inverse functions are doubly periodic (explained below). This chain of events caused these functions...

In this paper, an algorithm is presented to find exact polynomial solutions of nonlinear differential-difference equations(DDEs) in terms of the Jacobi elliptic functions. The key steps of the algorithm are illustrated by the discretized mKdV lattice. A Maple package JACOBI is developed based on the algorithm to automatically compute special solutions of nonlinear DDEs. The effectiveness of the...

We present two complementary methods, each applicable in a different range, to evaluate the distribution of the lowest eigenvalue of random matrices in a Jacobi ensemble. The first method solves an associated Painlevé VI nonlinear differential equation numerically, with suitable initial conditions that we determine. The second method proceeds via constructing the power-series expansion of the P...

This paper analyzes a class of impulse control problems for multidimensional jump diffusions in the finite time horizon. Following the basic mathematical setup from Stroock and Varadhan [Multidimensional Diffusion Processes, Springer-Verlag, Heidelberg, 2006], this paper first establishes rigorously an appropriate form of the dynamic programming principle. It then shows that the value function ...

In the present study, the nonlinear Boussinesq type equation describe the bi-directional propagation of small amplitude long capillary–gravity waves on the surface of shallow water. By using the extended auxiliary equation method, we obtained some new soliton like solutions for the two-dimensional fourth-order nonlinear Boussinesq equation with constant coefficient. These solutions include symm...

The Scheduled Relaxation Jacobi (SRJ) method is an extension of the classical Jacobi iterative method to solve linear systems of equations (Au = b) associated with elliptic problems. It inherits its robustness and accelerates its convergence rate computing a set of P relaxation factors that result from a minimization problem. This set of factors is employed in cycles of M consecutive iterations...