On Integro-Differential Algebras

Abstract

The concept of integro-differential algebra has been introduced recently in the study of boundary problems of differential equations. We generalize this concept to that of integro-differential algebra with a weight, in analogy to the differential Rota-Baxter algebra. We construct free commutative integro-differential algebras with weight generated by a base differential algebra. This in particular gives an explicit construction of the integro-differential algebra on one generator. Properties of these free objects are studied.

26 pages

References in corpus (6)

• A Unified Algebraic Approach to Classical Yang-Baxter Equation
• Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT
• Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases
• Exact and asymptotic results for insurance risk models with surplus-dependent premiums
• Construction of free commutative integro-differential algebras by the method of Gröbner-Shirshov bases
• Structure theorems of mixable shuffle algebras and free commutative Rota-Baxter algebras

Cited by in corpus (21)

• The Polarized Two-Loop Massive Pure Singlet Wilson Coefficient for Deep-Inelastic Scattering
• The unpolarized two-loop massive pure singlet Wilson coefficients for deep-inelastic scattering
• Rota-Baxter operators on the polynomial algebras, integration and averaging operators
• Construction of free differential algebras by extending Gröbner-Shirshov bases
• Cohomologies of modified $\lambda$-differential Lie triple systems and applications
• Construction of free commutative integro-differential algebras by the method of Gröbner-Shirshov bases
• Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems
• Deformations and extensions of modified $\lambda$-differential $3$-Lie Algebras
• An Integro-Differential Structure for Dirac Distributions
• Constructions of free commutative integro-differential algebras
• Extensions of Operators, Liftings of Monads and Distributive Laws
• The fundamental theorem of calculus in differential rings
• Classification of operator extensions, monad liftings and distributive laws for differential algebras and Rota-Baxter algebras
• Fundamental Theorems of Calculus and Zinbiel Algebras
• An Algebraic Approach to Fourier Transformation
• The global dimension of the algebras of polynomial integro-differential operators ${\mathbb{I}}_n$ and the Jacobian algebras ${\mathbb{A}}_n$
• Finite PDEs and finite ODEs are isomorphic
• On Rings of Differential Rota-Baxter Operators
• Matrix representations of multidimensional integral and ergodic operators
• Stability Problems on D-finite Functions
• Explicit description of generalized weight modules of the algebra of polynomial integro-differential operators $I_n$