C1 DETEST

Equations:

$$\begin{align}\dot{x}_{0}&=- x_{0}\\\dot{x}_{1}&=x_{0} - x_{1}\\\dot{x}_{2}&=x_{1} - x_{2}\\\dot{x}_{3}&=x_{2} - x_{3}\\\dot{x}_{4}&=x_{3} - x_{4}\\\dot{x}_{5}&=x_{4} - x_{5}\\\dot{x}_{6}&=x_{5} - x_{6}\\\dot{x}_{7}&=x_{6} - x_{7}\\\dot{x}_{8}&=x_{7} - x_{8}\\\dot{x}_{9}&=x_{8}\end{align}$$

Description:

A radioactive decay chain with all constants equal to 1. Please note that the supplied exact solution is correct only for certain initial conditions (e.g., those from the IVPs).
Link to DETEST description

Keywords:

decoupled

Download file

Plots:

C5 WPD

IVP 58

Class: PILBNU

Initial conditions:

$$\begin{align}x_{0}(0)&=1.0\\x_{1}(0)&=0.0\\x_{2}(0)&=0.0\\x_{3}(0)&=0.0\\x_{4}(0)&=0.0\\x_{5}(0)&=0.0\\x_{6}(0)&=0.0\\x_{7}(0)&=0.0\\x_{8}(0)&=0.0\\x_{9}(0)&=0.0\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&=1.0 \cdot e^{- t}\\x_{1}(t)&=1.0 \cdot t \cdot e^{- t}\\x_{2}(t)&=0.5 \cdot t^{2} \cdot e^{- t}\\x_{3}(t)&=0.166666666666667 \cdot t^{3} \cdot e^{- t}\\x_{4}(t)&=0.0416666666666667 \cdot t^{4} \cdot e^{- t}\\x_{5}(t)&=0.00833333333333333 \cdot t^{5} \cdot e^{- t}\\x_{6}(t)&=0.00138888888888889 \cdot t^{6} \cdot e^{- t}\\x_{7}(t)&=0.000198412698412698 \cdot t^{7} \cdot e^{- t}\\x_{8}(t)&=2.48015873015873 \cdot 10^{-5} \cdot t^{8} \cdot e^{- t}\\x_{9}(t)&=- 2.48015873015873 \cdot 10^{-5} \cdot \left(t^{8} + 8.0 \cdot t^{7} + 56.0 \cdot t^{6} + 336.0 \cdot t^{5} + 1680.0 \cdot t^{4} + 6720.0 \cdot t^{3} + 20160.0 \cdot t^{2} + 40320.0 \cdot t + 40320.0\right) \cdot e^{- t} + 1.0\end{align}$$

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=0.01960
c5: t_us=0.01770 , e_us=1.283786216506430e-15
c6: t_bd=10.000000 , e_bd=5.269118474870993e-13

VNODE-LP

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

c4: t_c=0.02920
c5: t_us=0.02510 , e_us=8.865406814966090e-17
c6: t_bd=10.000000 , e_bd=2.498001805406602e-14

VNODE-LP

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

c4: t_c=0.03620
c5: t_us=0.03140 , e_us=1.767334244446098e-18
c6: t_bd=10.000000 , e_bd=1.187938636348917e-14

VNODE-LP

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

c4: t_c=0.57180
c5: t_us=0.56620 , e_us=7.184503812457822e-16
c6: t_bd=10.000000 , e_bd=6.971534460831208e-12

Generate graphics:

VNODE-LP

WPD for Problem 58

IVP 59

Class: PILBU

Initial conditions:

$$\begin{align}x_{0}(0)&\subset~\left[ 0.5, 1.5\right]\\x_{1}(0)&=0.0\\x_{2}(0)&=0.0\\x_{3}(0)&=0.0\\x_{4}(0)&=0.0\\x_{5}(0)&=0.0\\x_{6}(0)&=0.0\\x_{7}(0)&=0.0\\x_{8}(0)&=0.0\\x_{9}(0)&=0.0\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&\subset~e^{- t} \cdot \left[ 0.5, 1.5\right]\\x_{1}(t)&\subset~t \cdot e^{- t} \cdot \left[ 0.5, 1.5\right]\\x_{2}(t)&\subset~t^{2} \cdot e^{- t} \cdot \left[ 0.25, 0.75\right]\\x_{3}(t)&\subset~t^{3} \cdot e^{- t} \cdot \left[ 0.0833333333333333, 0.25\right]\\x_{4}(t)&\subset~t^{4} \cdot e^{- t} \cdot \left[ 0.0208333333333333, 0.0625\right]\\x_{5}(t)&\subset~t^{5} \cdot e^{- t} \cdot \left[ 0.00416666666666667, 0.0125\right]\\x_{6}(t)&\subset~t^{6} \cdot e^{- t} \cdot \left[ 0.000694444444444444, 0.00208333333333333\right]\\x_{7}(t)&\subset~t^{7} \cdot e^{- t} \cdot \left[ 9.92063492063492 \cdot 10^{-5}, 0.000297619047619048\right]\\x_{8}(t)&\subset~t^{8} \cdot e^{- t} \cdot \left[ 1.24007936507937 \cdot 10^{-5}, 3.7202380952381 \cdot 10^{-5}\right]\\x_{9}(t)&\subset~\left(t^{8} + 8.0 \cdot t^{7} + 56.0 \cdot t^{6} + 336.0 \cdot t^{5} + 1680.0 \cdot t^{4} + 6720.0 \cdot t^{3} + 20160.0 \cdot t^{2} + 40320.0 \cdot t + 40320.0\right) \cdot e^{- t} \cdot \left[ -3.7202380952381 \cdot 10^{-5}, -1.24007936507937 \cdot 10^{-5}\right] + \left[ 0.5, 1.5\right]\end{align}$$