Polynomial Hamiltonians associated with Painlevé equations, II. Differential equations satisfied by polynomial Hamiltonians

Computability with Polynomial Differential Equations

In this paper, we show that there are Initial Value Problems defined with polynomial ordinary differential equations that can simulate universal Turing machines in the presence of bounded noise. The polynomial ODE defining the IVP is explicitly obtained and the simulation is performed in real time.

Polynomial solutions of differential equations

A new approach for investigating polynomial solutions of differential equations is proposed. It is based on elementary linear algebra. Any differential operator of the form L(y) = k=N ∑ k=0 ak(x)y, where ak is a polynomial of degree ≤ k, over an infinite ground field F has all eigenvalues in F in the space of polynomials of degree at most n, for all n. If these eigenvalues are distinct, then th...

Computing with Polynomial Ordinary Differential Equations

In 1941, Claude Shannon introduced the General Purpose Analog Computer (GPAC) as a mathematical model of DiUerential Analysers, that is to say as a model of continuoustime analog (mechanical, and later on electronic) machines of that time. Following Shannon’s arguments, functions generated by the GPAC must satisfy a polynomial diUerential algebraic equation (DAE). As it is known that some compu...

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

Coupled systems of equations with entire and polynomial functions

We consider the coupled system$F(x,y)=G(x,y)=0$,where$$F(x, y)=bs 0 {m_1}   A_k(y)x^{m_1-k}mbox{ and } G(x, y)=bs 0 {m_2} B_k(y)x^{m_2-k}$$with entire functions $A_k(y), B_k(y)$.We    derive a priory estimates  for the sums of the rootsof the considered system andfor the counting function of  roots.