Symbolic-numeric algorithms for simplifying differential systems and their application to the determination of approximate Lie symmetry algebras
Abstract
PDE models arising in applications can be over-determined, and usually have non-trivial integrability conditions. Differential elimination methods can be applied to these systems, which involve a finite number of differentiations and eliminations to reveal hidden constraints, and form the determining equations of symmetries. Normally, such PDE systems have approximate parameters, and the direct application of well-defined symbolic differential elimination algorithms on such systems is prone to instability since these algorithms strongly depend on the ordering of the variables involved (similar to the instability of naïve Gaussian elimination).
To avoid such instability, numerical differential elimination methods are necessary. In this thesis, the Geometric Involutive Form is mainly used to perform numerical differential elimination of linear homogeneous DE systems. However, the application of GIF to the whole system leads to singular value decomposition on large matrices. To reduce the size of matrices and increase efficiency, this thesis presents an important idea of partitioning the whole differential equation system into exact and approximate subsystems. After that, applying symbolic methods to the exact subsystem and numerical techniques to the approximate subsystem ensures stability. Finally, checking that the jointly prolonged system is geometrically involutive is done by computing dimension criteria of the approximate subsystem and the differential Hilbert function of the exact subsystem. This approach enhances the efficiency in simplifying PDE systems involving both exact and approximate parameters.
Another contribution of this thesis is the establishment of a foundation for constructing and analyzing local approximate Lie symmetry algebras of differential equations. This thesis includes the definition of local approximate Lie symmetry of DEs, the computation of structure constants of the approximate Lie symmetry algebra of DEs, and the evaluation of the reliability of computed results.