381,030 research outputs found
On polynomial solutions of differential equations
A general method of obtaining linear differential equations having polynomial
solutions is proposed. The method is based on an equivalence of the spectral
problem for an element of the universal enveloping algebra of some Lie algebra
in the "projectivized" representation possessing an invariant subspace and the
spectral problem for a certain linear differential operator with variable
coefficients. It is shown in general that polynomial solutions of partial
differential equations occur; in the case of Lie superalgebras there are
polynomial solutions of some matrix differential equations, quantum algebras
give rise to polynomial solutions of finite--difference equations.
Particularly, known classical orthogonal polynomials will appear when
considering acting on . As examples, some
polynomials connected to projectivized representations of ,
, and are briefly discussed.Comment: 12p
Solving simple quaternionic differential equations
The renewed interest in investigating quaternionic quantum mechanics, in
particular tunneling effects, and the recent results on quaternionic
differential operators motivate the study of resolution methods for
quaternionic differential equations. In this paper, by using the real matrix
representation of left/right acting quaternionic operators, we prove existence
and uniqueness for quaternionic initial value problems, discuss the reduction
of order for quaternionic homogeneous differential equations and extend to the
non-commutative case the method of variation of parameters. We also show that
the standard Wronskian cannot uniquely be extended to the quaternionic case.
Nevertheless, the absolute value of the complex Wronskian admits a
non-commutative extension for quaternionic functions of one real variable.
Linear dependence and independence of solutions of homogeneous (right) H-linear
differential equations is then related to this new functional. Our discussion
is, for simplicity, presented for quaternionic second order differential
equations. This involves no loss of generality. Definitions and results can be
readily extended to the n-order case.Comment: 9 pages, AMS-Te
Dynamics with Infinitely Many Derivatives: Variable Coefficient Equations
Infinite order differential equations have come to play an increasingly
significant role in theoretical physics. Field theories with infinitely many
derivatives are ubiquitous in string field theory and have attracted interest
recently also from cosmologists. Crucial to any application is a firm
understanding of the mathematical structure of infinite order partial
differential equations. In our previous work we developed a formalism to study
the initial value problem for linear infinite order equations with constant
coefficients. Our approach relied on the use of a contour integral
representation for the functions under consideration. In many applications,
including the study of cosmological perturbations in nonlocal inflation, one
must solve linearized partial differential equations about some time-dependent
background. This typically leads to variable coefficient equations, in which
case the contour integral methods employed previously become inappropriate. In
this paper we develop the theory of a particular class of linear infinite order
partial differential equations with variable coefficients. Our formalism is
particularly well suited to the types of equations that arise in nonlocal
cosmological perturbation theory. As an example to illustrate our formalism we
compute the leading corrections to the scalar field perturbations in p-adic
inflation and show explicitly that these are small on large scales.Comment: 26 pages, 2 figure
Solving linear parabolic rough partial differential equations
We study linear rough partial differential equations in the setting of [Friz
and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear
parabolic partial differential equation driven by a deterministic rough path
of H\"older regularity with . Based on a stochastic representation of the solution of the rough
partial differential equation, we propose a regression Monte Carlo algorithm
for spatio-temporal approximation of the solution. We provide a full
convergence analysis of the proposed approximation method which essentially
relies on the new bounds for the higher order derivatives of the solution in
space. Finally, a comprehensive simulation study showing the applicability of
the proposed algorithm is presented
- …