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.