The Constructive Solution of Linear Systems of Partial Difference and Differential Equations with Constant Coefficients

This paper gives a survey of past work in the treated subject and also contains several new results. We solve the Cauchy problem for linear systems of partial difference equations on general integral lattices by means of suitable transfer operators and show that these can be easily computed with the help of standard implementations of Gröbner basis algorithms. The Borel isomorphism permits to transfer these results to systems of partial differential equations. We also solve the Cauchy problem for the function spaces of convergent power series and for entire functions of exponential type. The unique solvability of the Cauchy problem implies that the considered function spaces are large injective cogenerators for which the duality between finitely generated modules and behaviours holds. Already in the beginning of the last century C. Riquier considered and solved problems of the type discussed here.