Simple Nonlinear 2

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Simple Nonlinear 2

Equations:

$$\begin{align}\dot{x}_{0}&=1.0 - x_{0}^{2}\end{align}$$

Description:

A non-linear separable ODE.

Download file

Plots:

C5 WPD

Initial value problems:

22 (PINLAU)
23 (PINLANU)

IVP 22

Class: PINLAU

Initial conditions:

$$\begin{align}x_{0}(0)&\subset~\left[ 3.9, 4.1\right]\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&\subset~1.0 - \frac{2.0}{e^{2.0 \cdot t} \cdot \left[ -1.68965517241379, -1.64516129032258\right] + 1.0}\end{align}$$

View test data:

Solver
Settings
Results

VNODE-LP

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.00510
c5: t_us=0.00490 , e_us=2.084489528779709e-04
c6: t_bd=10.000000 , e_bd=2.866373804977229e-12

VNODE-LP

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.01010
c5: t_us=0.00880 , e_us=2.443810355188258e-04
c6: t_bd=10.000000 , e_bd=3.409716953228781e-12

VNODE-LP

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.01510
c5: t_us=0.01280 , e_us=2.763395779545874e-04
c6: t_bd=10.000000 , e_bd=3.926858838099179e-12

VNODE-LP

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.04160
c5: t_us=0.03880 , e_us=1.990223730716778e-04
c6: t_bd=10.000000 , e_bd=2.745581539898012e-12

Generate graphics:

VNODE-LP

WPD for Problem 22

IVP 23

Class: PINLANU

Initial conditions:

$$\begin{align}x_{0}(0)&=4.0\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&=1.0 - \frac{2.0}{1.0 - 1.66666666666667 \cdot e^{2.0 \cdot t}}\end{align}$$

View test data:

Solver
Settings
Results

VNODE-LP

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.00500
c5: t_us=0.00380 , e_us=8.437694987151190e-15
c6: t_bd=10.000000 , e_bd=1.776356839400250e-15

VNODE-LP

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.00990
c5: t_us=0.00730 , e_us=3.108624468950438e-15
c6: t_bd=10.000000 , e_bd=4.440892098500626e-16

VNODE-LP

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.01490
c5: t_us=0.00990 , e_us=1.998401444325282e-15
c6: t_bd=10.000000 , e_bd=4.440892098500626e-16

VNODE-LP

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.04140
c5: t_us=0.03400 , e_us=3.075317778211684e-13
c6: t_bd=10.000000 , e_bd=1.798561299892754e-14

Generate graphics:

VNODE-LP

WPD for Problem 23

^* rounded (non-rigorously) to 5 decimal places to preserve screen real estate.

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