%pylab inline
import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(0.,1)
v = 0*x
u = np.exp(-100*(x-0.5)**2)
plt.plot(x,u)

tfinal = .2
dx = x[1]-x[0]
dt = 0.5*dx
nsteps = int(round(tfinal/dt))
for i in range(nsteps):
    # Diffusion
    v[1:-1] = v[1:-1] + dt/dx**2 * (u[2:] - 2*u[1:-1] + u[:-2])
    # Reaction
    u = u + dt*v

plt.plot(x,u)