A Computer Program for the Nonnumerical Testing and Reduction of Sets of Algebraic Partial Differential Equations

ABSTRACT

A program is described which enables a digital computer to perform the formal algebraic manipulations and differentiations required to test a set of algebraic partial differential equations for consistency, and, if the set is consistent, to reduce the equations in it to such an extent that the nature of the arbitrary functions uniquely generating all local, analytic solutions is apparent upon inspection. The computer performs these operations on polynomials in a nonnumerical sense, treating variables and derivatives of variables precisely as they are treated in the symbolic operations of algebra and calculus. The language which permits the computer to deal with variables and polynomials in this fashion is described. The general mathematical problem of testing and reduction of sets of algebraic partial differential equations is briefly described together with the techniques available for resolving it as adapted to this computer language. A brief description of the FORTRAN program itself is also given. The motivation for the development of this program was the study of the Einstein gravitational equations in general relativity. It has been shown that these equations can be invariantly reduced to a finite number of sets of algebraic partial differential equations, and the problem of testing and reducing these sets led to this program. The program reported on in this paper has been successfully tested for comparatively simple problems, some of which are described here, but must be further developed for the relativity applications.

REFERENCES

• 1: BRANS, CARL H. Invariant approach to tim geometry of spaces in general relative J. Math. Phys. 6 (1965), 94--102.
• 2: RITT, J. Differengal Equations from the Algebraic Standpoint. Amer. Math. Soc. Co Publications, Vol. 14, New York, N. Y., 1932.
• 3: BRowN, W. The ALPAK system for nonnumerical algebra on the digital compute Polynomials in several variables and truncated power series with polynomial coeflici Bell Syst. Tech. J. 2 (1963), 2081-2119.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations
          Subjects: Error analysis

General Terms:
Performance, Reliability, Theory