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Symbolic Solution of Systems of Polynomial Differential Equations Via The Cauchy-Riemann Equations. Applications to Kinetic Differential Equations

Kelvin Kiprono, János Tóth

Abstract

The differential equations of chemical kinetics are systems of nonlinear (polynomial) differential equations, therefore their solutions cannot usually be found in symbolic form. Here we offer a method to solve classes of kinetic differential equations based on the Cauchy--Riemann equations. It turns out that the method can be used to symbolically solve some polynomial differential equations that are not necessarily kinetic, as well.

Symbolic Solution of Systems of Polynomial Differential Equations Via The Cauchy-Riemann Equations. Applications to Kinetic Differential Equations

Abstract

The differential equations of chemical kinetics are systems of nonlinear (polynomial) differential equations, therefore their solutions cannot usually be found in symbolic form. Here we offer a method to solve classes of kinetic differential equations based on the Cauchy--Riemann equations. It turns out that the method can be used to symbolically solve some polynomial differential equations that are not necessarily kinetic, as well.
Paper Structure (19 sections, 6 theorems, 60 equations, 3 figures)

This paper contains 19 sections, 6 theorems, 60 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The method
  3. Special cases
  4. Linear systems, first-order reactions
  5. Second-degree systems, second-order reactions
  6. Third-degree systems, third-order reactions
  7. Rth-degree systems
  8. Application of multivariate complex functions
  9. Discussion and Outlook
  10. Further directions
  11. Partial differential equations
  12. Approximations
  13. Final remarks
  14. Appendix
  15. Useful formulas
  16. ...and 4 more sections

Key Result

Theorem 1

Let $R\in\mathbb{N}_0,$ and suppose that the polynomials $u,v$ defined as satisfy the Cauchy--Riemann equations eq:cauchy. Then, the system of differential equations can be simplified to Furthermore, for the complex-valued function $z:=x+i y$ one has

Figures (3)

  • Figure 5: FHJ graph of the reaction \ref{['eq:secondex']}
  • Figure 6: Differences between the first coordinates of the solution of \ref{['eq:epsilon']} with $\varepsilon=1$ and $\varepsilon=0$.
  • Figure 7: Differences between the second coordinates of the solution of \ref{['eq:epsilon']} with $\varepsilon=1$ and $\varepsilon=0$.

Theorems & Definitions (8)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Remark 2