Damped vibrations

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Damped vibrations

Equations:

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

Description:

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Plots:

C5 WPD

Initial value problems:

45 (PILANU)
115 (PILAU)

IVP 45

Class: PILANU

Initial conditions:

$$\begin{align}x_{0}(0)&=0.0\\x_{1}(0)&=-1.0\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&=- 0.707106781186547 \cdot e^{- t} \cdot \sin {1.4142135623731 \cdot t}\\x_{1}(t)&=\left(0.707106781186547 \cdot \sin {1.4142135623731 \cdot t} - 1.0 \cdot \cos {1.4142135623731 \cdot t}\right) \cdot e^{- t}\end{align}$$

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=0.00450
c5: t_us=0.00330 , e_us=1.149080830487037e-14
c6: t_bd=10.000000 , e_bd=9.300347221952859e-15

VNODE-LP

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

c4: t_c=0.00850
c5: t_us=0.00650 , e_us=8.604228440844963e-16
c6: t_bd=10.000000 , e_bd=2.436337806846489e-16

VNODE-LP

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

c4: t_c=0.01240
c5: t_us=0.00850 , e_us=6.383782391594650e-16
c6: t_bd=10.000000 , e_bd=4.004771774618332e-18

VNODE-LP

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

c4: t_c=0.03700
c5: t_us=0.03280 , e_us=1.197375532058231e-13
c6: t_bd=10.000000 , e_bd=5.840933883479241e-14

Generate graphics:

VNODE-LP

WPD for Problem 45

IVP 115

Class: PILAU

Initial conditions:

$$\begin{align}x_{0}(0)&\subset~\left[ 0.0, 0.1\right]\\x_{1}(0)&\subset~\left[ -1.05, -0.9\right]\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&\subset~\left(\sin {1.4142135623731 \cdot t} \cdot \left[ -0.742462120245875, -0.565685424949238\right] + \cos {1.4142135623731 \cdot t} \cdot \left[ 0.0, 0.1\right]\right) \cdot e^{- t}\\x_{1}(t)&\subset~\left(\sin {1.4142135623731 \cdot t} \cdot \left[ 0.424264068711928, 0.742462120245875\right] + \cos {1.4142135623731 \cdot t} \cdot \left[ -1.05, -0.9\right]\right) \cdot e^{- t}\end{align}$$