Optimization & Operations Research
Mahdi Dehghani Darmian
Abstract
This paper analyzes systems of linear first-order ordinary differential equations (ODEs) with parametric coefficients, a class of problems that arises in control theory, optimization, and applied mathematics. We introduce the notion of a comprehensive solution ...
Read More
This paper analyzes systems of linear first-order ordinary differential equations (ODEs) with parametric coefficients, a class of problems that arises in control theory, optimization, and applied mathematics. We introduce the notion of a comprehensive solution system for such parametric ODEs, constructed using Gröbner systems from computer algebra. Our approach partitions the parameter space into finitely many cells and associates an explicit solution with each cell. Furthermore, we present an algorithm that computes a comprehensive solution system for any given parametric system. To address the computational challenges inherent in Gröbner systems, we adopt the GES algorithm, a parametric variant of Gaussian elimination, which eliminates the need for Gröbner bases. This method builds upon the LDS algorithm proposed in 2017. Both algorithms have been implemented in Maple, and we illustrate the structural framework of the main algorithm with a straightforward example. The results highlight the practicality and effectiveness of the proposed methods for solving parametric linear first-order ODE systems.
