Numerical study of nonlinear equations with an infinite number of derivatives

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 equations with an infinite number of derivatives can be written as nonlinear integral equations. We perform a numerical investigation of the solutions of these equations. It is established that these equations have two different regimes of solutions: interpolating and periodic. The critical value of the parameter q separating these regimes is found to be q(cr)(2) approximate to 1.37. The convergence of the iterative procedure for these equations is proven.