Solving Nonlinear Systems of First Order Ordinary Differential Equations Using a Galerkin Finite Element Method

A new numerical technique to solve nonlinear systems of initial value problems for nonlinear first-order differential equations (ODEs) that model genetic networks in systems biology is developed. This technique is based on finding local Galerkin approximations on each sub-interval at a given time grid of points using piecewise hat functions. Comparing the numerical solution of the new method for a single nonlinear ODE with an exact solution shows that this method gives accurate solutions with relative error
$1.88\times{10}^{-11}$
for a time step
$1\times{10}^{-6}$
. This new method is compared with the adaptive Runge Kutta (ARK) method for solving systems of ODEs, and the results are comparable for a time step
$2\times{10}^{-4}$
. It is shown that the relative error of the Galerkin method decreases approximately linearly with the log of the number of hat functions used. Unlike the ARK method, this new method has the potential to be parallelizable and to be useful for solving biological problems involving large genetic networks. An NSF commissioned video illustrating how systems biology helps us understand that a fundamental process in cells is included.