Numerical Study of Nonlinear Equations with Infinite Number of Derivatives

TL;DR

This paper numerically investigates nonlinear equations with infinitely many derivatives, revealing two solution regimes and establishing convergence of an iterative method, with implications for mathematical physics and cosmology.

Contribution

It introduces a numerical approach to solve equations with infinite derivatives, identifying solution regimes and critical parameters, advancing understanding in mathematical physics.

Findings

Two solution regimes: interpolating and periodic

Critical parameter q^2=1.37 separates regimes

Proved convergence of the iterative solution method

Abstract

We study equations with infinitely many derivatives. Equations of this type form a new class of equations in mathematical physics. These equations originally appeared in p-adic and later in fermionic string theories and their investigation is of much interest in mathematical physics and applications, in particular in cosmology. Differential equation with infinite number of derivatives could be written as nonlinear integral equations. We perform numerical investigation of solutions of the equations. It is established that these equations have two different regimes of the solutions: interpolating and periodic. The critical value of the parameter q separating these regimes is found to be q^2=1.37. Convergence of iterative procedure for these equations is proved.