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

Figure 1

Figure 1
Display of several hat functions.

Figure 2

Figure 2
The solutions for the single nonlinear ODE and the effect of averaging on the numerical solution.

Figure 3

Figure 3
Maximum relative error (on a log scale) decreases linearly with the log of the number of hat functions for the single nonlinear ODE compared with the exact solution. Different numbers of hat functions (Formula$1\times{10}^{2}$, Formula$1\times{10}^{3}$, Formula$1\times{10}^{4}$, Formula$1\times{10}^{5}$, Formula$1\times{10}^{6}$, and Formula$1\times{10}^{7}$) over the solution interval Formula$[0~10]$ are used.

Figure 4

Figure 4
Solutions of the exact, galerkin, and the ARK methods using Formula$1\times{10}^{2}$ hat functions over the time period Formula$[0~10]$ ODE.

Figure 5

Figure 5
The solutions of the exact and galerkin methods using relatively few number of hat functions (50 hat functions) over time period Formula$[0~10]$ for the single nonlinear ODE. Such a solution is sufficient for biological problems with acceptable accuracy and high stability using few hat functions.

Figure 6

Figure 6
The maximum global relative error (on a log scale) among all of the species (global error of two species) for ODEs of the genetic network of the toggle switch decreases approximately linearly with the log of the number of hat functions. These results are based on the comparison with the ARK method (with absolute and relative errors for ARK are equal to Formula$1\times{10}^{-13}$). Different numbers of hat functions (Formula$1\times{10}^{2}$, Formula$1\times{10}^{3}$, Formula$1\times{10}^{4}$, Formula$1\times{10}^{5}$, Formula$1\times{10}^{6}$, and Formula$1\times{10}^{7}$) over the solution interval Formula$[0~10]$ are used.

Figure 7

Figure 7
The solutions of u(t) using the Galerkin method with Formula$1\times 10^{2}$ hat functions and the ARK method over the time period Formula$[0~10]$ for the genetic network of the toggle switch.

Figure 8

Figure 8
The solutions of v(t) using the Galerkin method with Formula$1\times{10}^{2}$ hat functions and the ARK method over the time period Formula$[0~10]$ for the genetic network of the toggle switch.

Figure 9

Figure 9
The maximum global relative error (on a log scale) among the whole species (seven species) for ODEs of the genetic network of the biological clock of Neurospora crassa decreases approximately linearly with the log of the number of hat functions compared with the ARK method (with absolute and relative errors for ARK equal to Formula$1\times{10}^{-13}$). Different numbers of hat functions (Formula$2\times{10}^{2}$, Formula$1\times{10}^{3}$, Formula$1\times{10}^{4}$, Formula$1\times{10}^{5}$, Formula$1\times{10}^{6}$, and Formula$1\times{10}^{7}$) over solution interval Formula$[0\,200]$ are used.

Figure 10

Figure 10
The solution of Formula${\mathrm{f}}_{1}$(t) using the Galerkin and the ARK method using Formula$1\times{10}^{3}$ hat functions over the time period Formula$[0\,200]$ for the genetic network of the biological clock of Neurospora crassa.

Figure 11

Figure 11
The solution of Formula${\mathrm{f}}_{r}$(t) using the Galerkin and the ARK methods using Formula$1\times{10}^{3}$ hat functions over the time period Formula$[0\,200]$ for the genetic network of the biological clock of neurospora crassa.

Figure 12

Figure 12
The solution of Formula${\mathrm{f}}_{p}$(t) using the Galerkin and the ARK methods using Formula$1\times{10}^{3}$ hat functions over the time period Formula$[0\,200]$ for the genetic network of the biological clock of neurospora crassa.

Figure 13

Figure 13
The solution of w(t) using the galerkin and the ARK methods using Formula$1\times{10}^{3}$ hat functions over the time period Formula$[0\,200]$ for the genetic network of the biological clock of neurospora crassa.

Figure 14

Figure 14
The solution of up(t) using the galerkin and the ARK methods using Formula$1\times{10}^{3}$ hat functions over the time period Formula$[0\,200]$ for the genetic network of the biological clock of neurospora crassa.

Figure 15

Figure 15
The solution of Formula${\mathrm{u}}_{\mathrm{r1}}$(t) using the Galerkin and the ARK methods using Formula$1\times{10}^{3}$ hat functions over the time period Formula$[0\,200]$ for the genetic network of the biological clock of neurospora crassa.

Figure 16

Figure 16
The solution of Formula${\mathrm{ur}}_{0}$(t) using the galerkin and the ARK methods using Formula$1\times{10}^{3}$ hat functions over the time period Formula$[0\,200]$ for the genetic network of the biological clock of neurospora crassa.

Figure 17

Figure 17
n subinterval with a hat function for each of these subintervals divided into k test grid points used to estimate the quality of the solution. One or more hat function(s) could be assigned to one slave processor.

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