An abstract robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed. The term “robust” refers to the convergence rate of the method being independent of discretization parameters, i.e., the problem size, and problem parameters. Important instances of such problem parameters are in particular (highl...

A method is proposed for finding exact solutions of the nonlinear Schr~dinger equation. It uses an ansatz in which the real and imaginary parts of the unknown function are connected by a linear relation with coefficients that depend only on the time. The method consists of constructing a system of ordinary differential equations whose solutions determine solutions of the nonlinear Schr~dinger e...

The discovery of elliptic functions emerged from investigations of integral addition theorems. An addition theorem for a function f is a formula expressing f(u+ v) in terms of f(u) and f(v). For a function defined as a definite integral with a variable upper limit, an addition theorem takes the form of an equation between the sum of two such integrals, with upper limits u and v, and an integral...

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, standard two-level schemes are used. The approximate solution at a new time-level is obtained as a solution of a discr...

In this paper, we find all the solutions of the Diophantine equation x + 2 513 = y in nonnegative integers x, y, α, β, γ, n ≥ 3 with x and y coprime. In fact, for n = 3, 4, 6, 8, 12, we transform the above equation into several elliptic equations written in cubic or quartic models for which we determine all their {2, 5, 13}-integer points. For n ≥ 5, we apply a method that uses primitive diviso...

Abstract: In this paper, the generalized Jacobi elliptic functions expansion method with computerized symbolic computation are employed to investigate explicitly analytic solutions of the (N + 1)-dimensional generalized Boussinesq equation. The exact solutions to the equation are constructed analytically under certain circumstances, some of these solutions are degenerated to soliton-like soluti...

where A is elliptic and B monotone (not strictly). It is the case, for example, in porous medium, semiconductor equations, ... In our application, we considered the injection moulding of a thermoplastic, with a mold of small thickness with respect to its other dimensions. By averaging Navier-Stokes equations across the thickness of the mold, and under an assumption (of Hele-Shaw) stating that t...

A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz, this providing an elliptic version of the idea, in...

‎In this paper‎, ‎we propose some Diffie-Hellman type key exchange protocols using isogenies of elliptic curves‎. ‎The first method which uses the endomorphism ring of an ordinary elliptic curve $ E $‎, ‎is a straightforward generalization of elliptic curve Diffie-Hellman key exchange‎. ‎The method uses commutativity of the endomorphism ring $ End(E) $‎. ‎Then using dual isogenies‎, ‎we propose...

The Hirota equation is a higher order extension of the nonlinear Schrödinger equation by incorporating third order dispersion and one form of self steepening effect. New periodic waves for the discrete Hirota equation are given in terms of elliptic functions. The continuum limit converges to the analogous result for the continuous Hirota equation, while the long wave limit of these discrete per...