Computability with Polynomial Differential Equations

Tau Numerical Solution of Volterra Integro-Differential Equations With Arbitrary Polynomial Bases

Computational bounds on polynomial differential equations

In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations. We consider elementary (in the sense of Analysis) discrete-time dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into full...

A Universal Ordinary Differential Equation

An astonishing fact was established by Lee A. Rubel in 81: there exists a fixed non-trivial fourthorder polynomial differential algebraic equation (DAE) such that for any positive continuous function φ on the reals, and for any positive continuous function (t), it has a C∞ solution with |y(t) − φ(t)| < (t) for all t. Lee A. Rubel provided an explicit example of such a polynomial DAE. Other exam...

Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length: The General Purpose Analog Computer and Computable Analysis Are Two Efficiently Equivalent Models of Computations

The outcomes of this paper are twofold. Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous (time and space) elegant and simple charact...

NUMERICAL SOLUTION OF THE MOST GENERAL NONLINEAR FREDHOLM INTEGRO-DIFFERENTIAL-DIFFERENCE EQUATIONS BY USING TAYLOR POLYNOMIAL APPROACH

In this study, a Taylor method is developed for numerically solving the high-order most general nonlinear Fredholm integro-differential-difference equations in terms of Taylor expansions. The method is based on transferring the equation and conditions into the matrix equations which leads to solve a system of nonlinear algebraic equations with the unknown Taylor coefficients. Also, we test the ...