Markus Rosenkranz

Theorema: Towards computer-aided mathematical theory exploration

Theorema is a project that aims at supporting the entire process of mathematical theory exploration within one coherent logic and software system. This survey paper illustrates the style of Theorema-supported mathematical theory exploration by a

Solving and factoring boundary problems for linear ordinary differential equations in differential algebras

We present a new approach for expressing and solving boundary problems for linear ordinary differential equations in the language of differential algebras. Starting from an algebra with a derivation and integration operator, we construct an algebra of linear integro-differential operators that is expressive enough for specifying regular boundary problems with arbitrary Stieltjes boundary conditions as well as their solution operators. On the basis of these structures, we define a new multiplication on regular boundary problems in such a way that the resulting Greens operator is the reverse composition of the constituent Greens operators. We provide also a method for lifting any factorization of the underlying differential operator to the level of boundary problems. Since this method only needs the computation of initial value problems, it can be used as an effective alternative for computing Greens operators in the case where one knows how to factor the given differential operators.

Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases

We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integro-differential operators, is used for both stating and solving linear boundary problems. The other structure, called integro-differential polynomials, is the key tool for describing extensions of integro-differential algebras. We use the canonical simplifier for integro-differential polynomials for generating an automated proof establishing a canonical simplifier for integro-differential operators. Our approach is fully implemented in the Theorema system; some code fragments and sample computations are included.

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 differential algebra. This gives in particular an explicit construction of the integro-differential algebra on one generator. Properties of the free objects are studied.

Integro-differential polynomials and operators

We propose two algebraic structures for treating integral operators in conjunction with derivations: The algebra of integro-differential polynomials describes nonlinear integral and differential operators together with initial values. The algebra of integro-differential operators can be used to solve boundary problems for linear ordinary differential equations. In both cases, we describe canonical/normal forms with algorithmic simplifiers.

A skew polynomial approach to integro-differential operators

We construct the algebra of integro-differential operators over an ordinary integro-differential algebra directly in terms of normal forms. In the case of polynomial coefficients, we use skew polynomials for defining the integro-differential Weyl algebra as a natural extension of the classical Weyl algebra in one variable. Its normal forms, algebraic properties and its relation to the localization of differential operators are studied. Fixing the integration constant, we regain the integro-differential operators with polynomial coefficients.

A Symbolic Framework for Operations on Linear Boundary Problems

We describe a symbolic framework for treating linear boundary problems with a generic implementation in the Theorema system. For ordinary differential equations, the operations implemented include computing Greens operators, composing boundary problems and integro-differential operators, and factoring boundary problems. Based on our factorization approach, we also present some first steps for symbolically computing Greens operators of simple boundary problems for partial differential equations with constant coefficients. After summarizing the theoretical background on abstract boundary problems, we outline an algebraic structure for partial integro-differential operators. Finally, we describe the implementation in Theorema, which relies on functors for building up the computational domains, and we illustrate it with some sample computations including the unbounded wave equation.

Regular and singular boundary problems in maple

We describe a new Maple package for treating boundary problems for linear ordinary differential equations, allowing two-/multipoint as well as Stieltjes boundary conditions. For expressing differential operators, boundary conditions, and Greens operators, we employ the algebra of integro-differential operators. The operations implemented for regular boundary problems include computing Greens operators as well as composing and factoring boundary problems. Our symbolic approach to singular boundary problems is new; it provides algorithms for computing compatibility conditions and generalized Greens operators.

A maple package for integro-differential operators and boundary problems

We present a Maple package for computing in algebras of integro-differential operators. This provides the appropriate algebraic setting for treating boundary problems [7] for linear ordinary differential equations symbolically. They allow to formulate a boundary problem - a differential equation and boundary conditions - but they are also expressive enough for describing its solution via an integral operator, which is called Greens operator. Our package provides procedures for algebraic operations on integro-differential operators as well as for solving LODEs with general boundary conditions, given a fundamental system. The implementation was tested in Maple 11, 12 and 13. It is available with an example worksheet at http://www.risc.jku.at/people/akorpora/index.html.

Exact and Asymptotic Results for Insurance Risk Models with Surplus-dependent Premiums

In this paper we develop a symbolic technique to obtain asymptotic expressions for ruin probabilities and discounted penalty functions in renewal insurance risk models when the premium income depends on the present surplus of the insurance portfolio. The analysis is based on boundary problems for linear ordinary differential equations with variable coefficients. The algebraic structure of the Greens operators allows us to develop an intuitive way of tackling the asymptotic behavior of the solutions, leading to exponential-type expansions and Cramer-type asymptotics. Furthermore, we obtain closed-form solutions for more specific cases of premium functions in the compound Poisson risk model.