Symbolic Computation of Exact Solutions of Nonlinear Mathematical Physics Equations Based on a Geometric Grid Factorization Algorithm

Exploring exact solutions of non-linear mathematical physics equations can help solve problems in more engineering fields. In this paper, the exact solutions of the periodic exact traveling wave solution of the rod equation, the regular long-wave equation, and the KdV-Burgers equation are solved by a geometric lattice factorization algorithm based on isolator theory and conformal computing systems. From the exact solutions of the rod equation, the periodic wave becomes an anticlockwise isolated wave when
\( c < 0 \), \( g_1(c) < g < g_2(c) \), \( \varphi_0 \) satisfies \( \varphi_- < \varphi_0 < \varphi_+ \), and \( \varphi_0 \) converges to \( \varphi_- \). From the exact solutions of the regular long-wave equation and the KdV-Burgers equation, different parameter forms can be obtained by the factorization algorithm. It is thus shown that the use of geometric grid factorization algorithms enables the exact solution of non-linear mathematical physics equations and provides a new research direction for the development of the engineering field.