An Extended Picard Method to solve non-linear systems of ODE with applications

TL;DR

The paper presents the Extended Picard method, a Picard-like iterative scheme enhanced with segmentary integration for solving first-order nonlinear ODE systems with variable coefficients. It establishes a convergence framework and details how to partition the integration interval, approximate nonlinear terms, and propagate solutions across segments to control error. The method is validated on classical equations (Mathieu, quintic Duffing, Bratu) and applied to chemical-reaction models (Glycolysis and Brusselator), showing high accuracy with few iterations and robust performance on long intervals. Overall, Extended Picard offers explicit analytic-like approximations, favorable accuracy compared with Taylor and Runge-Kutta methods, and practical applicability to stiff, nonlinear problems in chemistry and physics.

Abstract

We provide of a method to integrate first order non-linear systems of differential equations with variable coefficients. It determines approximate solutions given initial or boundary conditions or even for Sturm-Liouville problems. This method is a mixture between an iterative process, a la Picard, plus a segmentary integration, which gives explicit approximate solutions in terms of trigonometric functions and polynomials. The segmentary part is particularly important if the integration interval is large. This procedure provide a new tool so as to obtain approximate solutions of systems of interest in the analysis of chemical reactions. We test the method on some classical equations like Mathieu, Duffing quintic equation or Bratu's equation and have applied it on some models of chemical reactions.

Figures (7)

• Figure 1: Graphics for \ref{['29']}. This is the approximate iterative solution for $n=2$. It is noteworthy how good the exact solution matches with ours.
• Figure 2: Comparison of the solution obtained by Extended Picard with $k=2$ (dashed curve) with the solution obatained by eight order Runge-Kutta (continuous curve).
• Figure 3: Approximated solution of the Duffing equation with segmentary Extended Picard. The Runge-Kutta solution matches with our solution. The interval has been extended to $[0,14]$ in order to display the periodicity. This periodicity shows since $a>0$.
• Figure 4: Local solution to the Glycolisis equation for ADP concentration by iteration (dashed line) without making use of segmentary iteration, as compared with the Runge Kutta solution, when using $a=0.4$ and $b=0.6$ in \ref{['48']}. We have made use of two iterations only, yet we have a reasonable fit for at least the half of the interval. Note that both curves fit quite well up to values of $t$ of the order $t=1.5$ or bigger and, then, both curves split. This split is what the segmentary iteration avoids.
• Figure 5: Local solution to the Glycolisis equation for ADP concentration by segmentary iteration up to $t=10$. It completely matches with the Runge Kutta solution. We have taken \ref{['21']}$a=0.4$ and $b=0.6$. Note than in this case the fixed point is asymptotically stable.