from gekko import GEKKO

t_data = [0, 0.1, 0.2, 0.4, 0.8, 1]
x_data = [2.0,  1.6,  1.2, 0.7,  0.3,  0.15]

m = GEKKO(remote=False)
m.time = t_data
x = m.CV(value=x_data); x.FSTATUS = 1  # fit to measurement
k = m.FV(); k.STATUS = 1               # adjustable parameter
m.Equation(x.dt()== -k * x)            # differential equation

m.options.IMODE = 5   # dynamic estimation
m.options.NODES = 5   # collocation nodes
m.solve(disp=False)   # display solver output
k = k.value[0]

import numpy as np
import matplotlib.pyplot as plt  # plot solution
plt.plot(m.time,x.value,'bo',\
         label='Predicted (k='+str(np.round(k,2))+')')
plt.plot(m.time,x_data,'rx',label='Measured')
# plot exact solution
t = np.linspace(0,1); xe = 2*np.exp(-k*t)
plt.plot(t,xe,'k:',label='Exact Solution')
plt.legend()
plt.xlabel('Time'), plt.ylabel('Value')
plt.show()