E1 DETEST

Equations:

$$\begin{align}\dot{x}_{0}&=1.0\\\dot{x}_{1}&=x_{2}\\\dot{x}_{2}&=- x_{1} \cdot \left(1.0 - \frac{0.25}{\left(x_{0} + 1.0\right)^{2}}\right) - \frac{1.0 \cdot x_{2}}{x_{0} + 1.0}\end{align}$$

Description:

Derived from Bessel's equation of order 1/2 with shifted origin (a linear second-order equation with variable coefficients).
Link to DETEST description

Keywords:

variable coefficients

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Plots:

C5 WPD

IVP 64

Class: PILBNU

Initial conditions:

$$\begin{align}x_{0}(0)&=0.0\\x_{1}(0)&=0.671396707141803\\x_{2}(0)&=0.0954005144747446\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&=t\\x_{1}(t)&=\frac{0.4310988680456461 \cdot \sin {t} + 0.671396707141803 \cdot \cos {t}}{\sqrt{t + 1.0}}\\x_{2}(t)&=\frac{\left(0.4310988680456461 - \frac{0.3356983535709015}{t + 1}\right) \cdot \cos {t} - \left(0.671396707141803 + \frac{0.215549434022823}{t + 1}\right) \cdot \sin {t}}{\sqrt{t + 1.0}}\end{align}$$

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=0.00900
c5: t_us=0.00820 , e_us=1.038058528024521e-14
c6: t_bd=10.000000 , e_bd=2.191753722957657e-13

VNODE-LP

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

c4: t_c=0.01680
c5: t_us=0.01370 , e_us=1.554312234475219e-15
c6: t_bd=10.000000 , e_bd=5.101821742847790e-15

VNODE-LP

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

c4: t_c=0.02510
c5: t_us=0.02080 , e_us=9.436895709313831e-16
c6: t_bd=10.000000 , e_bd=3.821595817576906e-15

VNODE-LP

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

c4: t_c=0.06090
c5: t_us=0.05520 , e_us=1.203481758693670e-13
c6: t_bd=10.000000 , e_bd=2.657023906449396e-13

Generate graphics:

VNODE-LP

WPD for Problem 64

IVP 65

Class: PILBU

Initial conditions:

$$\begin{align}x_{0}(0)&=0.0\\x_{1}(0)&\subset~\left[ 0.67, 0.68\right]\\x_{2}(0)&=0.0954005144747446\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&=t\\x_{1}(t)&\subset~\frac{\sin {t} \cdot \left[ 0.430400514474745, 0.435400514474745\right] + \cos {t} \cdot \left[ 0.67, 0.68\right]}{\sqrt{t + 1.0}}\\x_{2}(t)&\subset~\frac{\left(\left[ 0.430400514474745, 0.435400514474745\right] + \frac{\left[ -0.34, -0.335\right]}{t + 1}\right) \cdot \cos {t} - \left(\left[ 0.67, 0.68\right] + \frac{\left[ 0.215200257237372, 0.217700257237372\right]}{t + 1}\right) \cdot \sin {t}}{\sqrt{t + 1.0}}\end{align}$$