The Cauchy Problem for Partial Difference Equations

摘要

We want to discuss partial difference equations, first of all with respect to the existence and uniqueness of their solution. These equations are considered with solutions on arbitrary subsets of the n-dimensional grid Z~n. The basic theorem enables one to formulate the Cauchy problem for such equations. The solution is proven to be recursively computable for partial difference equations under very mild restrictions. (Variable coefficients for linear equations, systems of equations as well is nonlinear equations are not excluded.) The construction of solutions presented here also allows for some qualitative conclusions, such as boundedness of solutions.