Bachman's Example

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Bachman's Example

Equations:

$$\begin{align}\dot{x}_{0}&=x_{0} - 2.0 \cdot x_{1}\\\dot{x}_{1}&=3.0 \cdot x_{0} - 4.0 \cdot x_{1}\end{align}$$

Description:

A linear system with constant coefficients.

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Plots:

C5 WPD

Initial value problems:

26 (PILBU)
27 (PILBNU)
83 (PILBU)

IVP 26

Class: PILBU

Initial conditions:

$$\begin{align}x_{0}(0)&\subset~\left[ 0.5, 1.5\right]\\x_{1}(0)&\subset~\left[ -1.5, -0.5\right]\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&\subset~e^{- 2.0 \cdot t} \cdot \left[ -6.0, -2.0\right] + e^{- t} \cdot \left[ 2.5, 7.5\right]\\x_{1}(t)&\subset~e^{- 2.0 \cdot t} \cdot \left[ -9.0, -3.0\right] + e^{- t} \cdot \left[ 2.5, 7.5\right]\end{align}$$

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=0.00590
c5: t_us=0.00380 , e_us=1.027385506437649e+00
c6: t_bd=10.000000 , e_bd=2.269872819097235e-04

VNODE-LP

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

c4: t_c=0.01020
c5: t_us=0.00790 , e_us=1.027385506437557e+00
c6: t_bd=10.000000 , e_bd=2.269872818913145e-04

VNODE-LP

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

c4: t_c=0.01410
c5: t_us=0.01020 , e_us=1.027385506437547e+00
c6: t_bd=10.000000 , e_bd=2.269872818908145e-04

VNODE-LP

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

c4: t_c=0.07760
c5: t_us=0.07210 , e_us=1.027385506445530e+00
c6: t_bd=10.000000 , e_bd=2.269872823911188e-04

Generate graphics:

VNODE-LP

WPD for Problem 26

IVP 27

Class: PILBNU

Initial conditions:

$$\begin{align}x_{0}(0)&=1.0\\x_{1}(0)&=-1.0\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&=- 4.0 \cdot e^{- 2.0 \cdot t} + 5.0 \cdot e^{- t}\\x_{1}(t)&=- 6.0 \cdot e^{- 2.0 \cdot t} + 5.0 \cdot e^{- t}\end{align}$$

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=0.00510
c5: t_us=0.00230 , e_us=3.286260152890463e-14
c6: t_bd=10.000000 , e_bd=1.729137124283076e-14

VNODE-LP

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

c4: t_c=0.00910
c5: t_us=0.00580 , e_us=2.442490654175344e-15
c6: t_bd=10.000000 , e_bd=4.866712501744308e-16

VNODE-LP

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

c4: t_c=0.01310
c5: t_us=0.00750 , e_us=4.440892098500626e-16
c6: t_bd=10.000000 , e_bd=3.973600962159374e-17

VNODE-LP

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

c4: t_c=0.04370
c5: t_us=0.03620 , e_us=8.237854842718662e-13
c6: t_bd=10.000000 , e_bd=4.364828810887733e-13

Generate graphics:

VNODE-LP

WPD for Problem 27

IVP 83

Class: PILBU

Initial conditions:

$$\begin{align}x_{0}(0)&\subset~\left[ 1.5, 2.0\right]\\x_{1}(0)&\subset~\left[ -2.0, -1.5\right]\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&\subset~e^{- 2.0 \cdot t} \cdot \left[ -8.0, -6.0\right] + e^{- t} \cdot \left[ 7.5, 10.0\right]\\x_{1}(t)&\subset~e^{- 2.0 \cdot t} \cdot \left[ -12.0, -9.0\right] + e^{- t} \cdot \left[ 7.5, 10.0\right]\end{align}$$

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=0.00570
c5: t_us=0.00400 , e_us=5.136927532188673e-01
c6: t_bd=10.000000 , e_bd=1.134936409647350e-04

VNODE-LP

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

c4: t_c=0.00980
c5: t_us=0.00660 , e_us=5.136927532187840e-01
c6: t_bd=10.000000 , e_bd=1.134936409459186e-04

VNODE-LP

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

c4: t_c=0.01370
c5: t_us=0.00910 , e_us=5.136927532187787e-01
c6: t_bd=10.000000 , e_bd=1.134936409454506e-04

VNODE-LP

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

c4: t_c=0.06590
c5: t_us=0.06020 , e_us=5.136927532244828e-01
c6: t_bd=10.000000 , e_bd=1.134936413732187e-04

Generate graphics:

VNODE-LP

WPD for Problem 83

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

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