ABSTRACT
This chapter provides the fundamental algorithms for working with linear ordinary and partial differential equations that are the building blocks for the solution algorithms given later in this book. Systems of linear pde’s are characterized in the first place by the number m of dependent and the number n of independent variables. Additional quantities of interest are the number N of equations, the order of the highest derivatives that may occur and the smallest function field in which the coefficients are contained, called the base field. Without further specification this is assumed to be the field of rational functions in the independent variables with rational number coefficients. The special case m = n = N = 1 corresponds to a single linear ode. The fundamental concepts described in this chapter are the Loewy decomposition, Loewy [127], and the Janet basis, Janet [83]. The latter term is chosen to honour the French mathematician Maurice Janet who described this concept and gave an algorithm to obtain it. After it had been forgotten for about fifty years, it was rediscovered [163] and utilized in various applications as described in this book. A good survey is also given in the article by Plesken and Robertz [148]. Janet bases are the differential counterpart of Gro¨bner bases that have been introduced by Bruno Buchberger and are a well-established tool in polynomial ideal theory and algebraic geometry now. The relevance of a Janet basis for the main subject of this book, i.e., solving ordinary differential equations or ode’s for short, originates from the fact that the symmetries of any such equation are determined by a system of linear homogeneous pde’s. General references for this chapter are the book by Ince [80] or the two volumes on linear equations by Schlesinger [160]. For Chapter 2.1 the first 150 pages of the book by van der Put and Singer [185], or the article by Buium and Cassidy [23] are highly recommended.
