If leading linear equations with usable pivots are found in the algebraically simplified set, rifsimp removes them from the set, and treats them using leading linear methods (i.e. This can increase the number of cases in the output, but is required for the proper functioning of the more compact nonlinear equation handling algorithm, so if this splitting is not desired, pure Groebner basis methods can be used instead with the grobonly option.
If there was no condition on u x + u, then rifsimp would split into the cases u x + u ≠ 0 and u x + u = 0. So for the example above, the initial is u x + u. This coefficient is called the initial of the equation. Note: during the course of the algorithm, when nonlinear equations are encountered, case splitting is performed on the coefficient of the highest degree of the leading derivative each equation. This is simply chosen to be pure lexicographical ordering with respect to the ordering imposed on the linear derivatives.įor example, the equation u xx 2 u x + u xx 2 u − u x 2 − u 2 for a system in which u x + u ≠ 0 holds is algebraically solved for u xx 2 giving the nonlinear relation: u xx 2 = u 2 + u x 2 u x + u. In order to perform the algebraic simplification of these nonlinear equations, a monomial ordering must be imposed for all possible nonlinear terms. Use of this set of equations allows recognition of, and elimination of redundant equations, and produces a more compact representation (due to the use of pivots and lack of S-polynomials). This simplification process resembles an algebraic Groebner basis, but takes the inequations (pivots) of the system into account, and does not compute algebraic S-polynomials. Conversely, the equation u tt 2 − u t = 0 is leading nonlinear, and is always handled through the nonlinear methods described here.ĭuring the course of the computation, nonlinear equations are always differentially simplified with respect to the linear equations of the system, but they are also algebraically simplified with respect to each other. This equation is linear in its leading indeterminate, so as long as u can be present in a pivot it will be handled through case splitting rather than nonlinear equation handling (see rifsimp ). This help page explains how rifsimp handles equations which are polynomially nonlinear in their leading indeterminate (called leading nonlinear equations), or equations that are leading linear, but have a coefficient that depends on pivot restricted variables (see nopiv in rifsimp ).Īs an example, consider the equation u tt u − u t 2 = 0.
Information and options specific to nonlinear equations