Linear constant coefficient differential (or difference) equations

Polynomial Solutions to Constant Coefficient Differential Equations

Let Dx, ... , Dr e C[d/dxx, ... , d/dxn) be constant coefficient differential operators with zero constant term. Let S = {fe C[xx,... , x„]\Dj(f) = 0 for all 1 < j < r) be the space of polynomial solutions to the system of simultaneous differential equations Dj(f) = 0. It is proved that S is a module over 3¡(V), the ring of differential operators on the affine scheme V with coordinate ring C[d/...

Linear difference and differential equations

Linear Differential Algebraic Equations with Constant Coefficients

Differential-algebraic equations (DAEs) arise in a variety of applications. Their analysis and numerical treatment, therefore, plays an important role in modern mathematics. The paper gives an introduction to the topics of DAEs. Examples of DAEs are considered showing their importance for practical problems. Some essential concepts that are really essential for understanding the DAE systems are...

Liouvillian solutions of linear difference-differential equations

For a field k with an automorphism σ and a derivation δ, we introduce the notion of liouvillian solutions of linear difference-differential systems {σ(Y ) = AY, δ(Y ) = BY } over k and characterize the existence of liouvillian solutions in terms of the Galois group of the systems. We will give an algorithm to decide whether such a system has liouvillian solutions when k = C(x, t), σ(x) = x + 1,...

Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems

In this paper we present an approach for the numerical solution of Delay Di erential Equations y 0 (t) = Ly (t) +My (t ) t 0 y (t) = ' (t) t 0; (1) where > 0, L;M 2 C I m m and ' 2 C ([ ; 0] ; C I m ), di erent from the classical step-by-step method. We restate (1) as an abstract Cauchy problem and then we discretize it in a system of Ordinary Differential Equations. The scheme of discretizatio...