Title:        Solving SNPE as a New Basic Symbolic-Numerical
                Operation for Modeling PDEs
  Authors:      Valentine D. Borisevich, Valeriy G. Potemkin
  Affiliation:  Moscow State Engineering Physics Institute
                (Technical Univ.) 
  Abstract:
As is known in the case of solving Partial Differential Equations (PDEs) the
initial system is reduced to the nonlinear system of finite difference
equations that are the System of Nonlinear Algebraic Equations (SNAE).  Solving
of the latter is in turn reduced to System of Linear Algebraic Equations (SLAE)
of a large size, which is considered conventionally as a basic operation for
modelling PDEs.  As the result of our recent research we offer to apply a
direct technique of solving SNAE using instead of SLAE a System of Nonlinear
Polynomial Equations (SNPE) as the model.  In fact, it means the replacement of
the linear basic operation to the nonlinear one.  To solve SNPE by means of our
new basic operation we offer (i) to analyze them as an object of the theory of
ideals; (ii) to reduce the SNPE to a Groebner basis; (iii) to transform SNPE in
the Groebner basis to the System of Spectral Problems (SSP) for rectangular
matrix pencils.  So the new basic operation includes two different types of
computations: the first is a symbolic one (reducing SNPE to a Groebner basis)
and the second is a numeric one (solving of SSP).  We come to conclusion that
to get rise a performance of computations it is possible to use two
coprocessors for symbolic, and numeric computations, respectively.
Additionally, it is necessary to note that the algorithm of solving SNPE
offered in contrast to the conventional algorithms of solving SLAE possesses an
important feature of a natural parallelism.  At the moment the algorithm is
accomplished as a solver in the MATLAB environment.  Its application to
solution of various real-life problems demonstrated that the basic operation of
solving SNPE offered allows to refine the nonlinear components of the PDEs
solutions that cannot be found by the traditional basic operation of solving
SLAE.