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Dehghani Darmian, Mahdi. (1404). A Computer Algebra Approach to Linear ODE Systems with Parametric Coefficients. فصلنامه علمی زیست فناوری گیاهان زراعی, 10(2), 135-152. doi: 10.30473/coam.2025.74498.1307
Mahdi Dehghani Darmian. "A Computer Algebra Approach to Linear ODE Systems with Parametric Coefficients". فصلنامه علمی زیست فناوری گیاهان زراعی, 10, 2, 1404, 135-152. doi: 10.30473/coam.2025.74498.1307
Dehghani Darmian, Mahdi. (1404). 'A Computer Algebra Approach to Linear ODE Systems with Parametric Coefficients', فصلنامه علمی زیست فناوری گیاهان زراعی, 10(2), pp. 135-152. doi: 10.30473/coam.2025.74498.1307
Dehghani Darmian, Mahdi. A Computer Algebra Approach to Linear ODE Systems with Parametric Coefficients. فصلنامه علمی زیست فناوری گیاهان زراعی, 1404; 10(2): 135-152. doi: 10.30473/coam.2025.74498.1307

A Computer Algebra Approach to Linear ODE Systems with Parametric Coefficients

Control and Optimization in Applied Mathematics
مقاله 5، دوره 10، شماره 2 - شماره پیاپی 20، مهر 2025، صفحه 135-152    اصل مقاله (554.58 K)
نوع مقاله: Research Article
شناسه دیجیتال (DOI): 10.30473/coam.2025.74498.1307
نویسنده
Mahdi Dehghani Darmian*
Department of Basic Sciences‎, ‎Technical and Vocational University (TVU)‎, ‎Tehran‎, ‎Iran‎.
چکیده
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.

تازه های تحقیق
  • Investigation of linear first-order ODE systems with parametric coefficients, motivated by applications in control theory, optimization, and applied mathematics.
  • Definition of a comprehensive solution system (CSS) for parametric linear ODEs, which organizes parameter space into finite cells, with each cell paired to an explicit closed-form solution.
  • Development of the CSS algorithm to compute comprehensive solution systems for any given parametric linear first-order ODE system.
  • Adoption of the GES algorithm (parametric Gaussian elimination) to tackle computational challenges in Gröbner-system-based approaches, reducing reliance on Gröbner bases.
  • The CSS framework builds upon prior LDS methodology and integrates parametric linear algebra techniques to efficiently handle parameter dependencies.
  • All algorithms are implemented in Maple, enabling practical experimentation and reproducibility.
  • The Improved-CSS algorithm (based on GES) outperforms traditional Gröbner-based methods in efficiency across benchmark examples, indicating potential for real-time and large-scale parameter analyses.
  • Potential impact on control theory and optimization through systematic, parameter-aware analysis, including gain scheduling, robust stability analysis, and design of controllers or estimators with stability guarantees across parameter ranges.
کلیدواژه‌ها
Parametric linear ODE systems‎؛ ‎Gröbner system‎؛ ‎Gaussian elimination system algorithm‎؛ ‎Comprehensive solution system‎؛ ‎Parameter space‎
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