I'm trying to model the Black-Scholes Equation (transformed in into a heat equation) using method of lines in Python.

The transformed formula is basically 

\begin{equation*}
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + (k-1)\frac{\partial u}{\partial x} - ku
\end{equation*}

where $k$ is a constant and with initial condition 

\begin{equation*}
u(x,0) = \max(e^a - 1, 0)
\end{equation*}

and boundary conditions

\begin{equation*}
u(a,t) = \alpha \hspace{35pt} u(b,t) = \beta
\end{equation*}

 
In Python, the resulting graph is very strange, and I don't believe it is correct. I can't seem to find where I went wrong. Any insight on the Python code would be really helpful. 


    import numpy as np
    from scipy.integrate import odeint
    import matplotlib.pyplot as plt
    from scipy.fftpack import diff as psdiff
    
    N = 40
    r = 0.065;
    sigma = 0.8;
    k = float(0.5*sigma**2);
    a = np.log(float(2)/5)
    b = np.log(float(7)/5)
    t0 = 0;
    tf = 5;
    
    x = np.linspace(a, b, N);
    xmesh = np.max(np.exp(a)-1,0)*np.ones(x.shape)
    
    def odefunc(u, t):
        dudt = np.zeros(xmesh.shape)
        
        #boundary conditions
        dudt[0] = u[0]
        dudt[-1] = (7 - 5*np.exp(-k*t))/5 
        
        #step size
        h = b/(N-1)
    
        for i in range(1, N-1):
            dudt[i] = ((u[i + 1] - 2*u[i] + u[i - 1]) / h**2) + (k-1)*((u[i] - u[i-1]) / h) - (k*u[i])
        return dudt
    
    tspan = np.linspace(t0, tf, 20);
    sol = odeint(odefunc, xmesh, tspan)
    
    for i in range(0, len(tspan), 5):
        plt.plot(xmesh, sol[i], label='t={0:1.2f}'.format(tspan[i]))
    
    # put legend outside the figure
    plt.legend(loc='center left', bbox_to_anchor=(1, 0.5))
    plt.xlabel('X position')
    plt.ylabel('Temperature')
    
    # adjust figure edges so the legend is in the figure
    plt.subplots_adjust(top=0.89, right=0.77)
    plt.savefig('pde.png')
    
    # Make a 3d figure
    from mpl_toolkits.mplot3d import Axes3D
    fig = plt.figure()
    ax = fig.add_subplot(111, projection='3d')
    
    SX, ST = np.meshgrid(xmesh, tspan)
    ax.plot_surface(SX, ST, sol, cmap='jet')
    ax.set_xlabel('x')
    ax.set_ylabel('t')
    ax.set_zlabel('u')
    ax.view_init(elev=30, azim=100) # adjust view so it is easy to see
    plt.savefig('pde-transient-heat-3d.png')

[![3d Plot][2]][2]


  [1]: http://www.math.uwaterloo.ca/~hwolkowi//henry/reports/talks.d/t09talks.d/09waterloomatlab.d/mfileshigham.d/bs.m
  [2]: https://i.sstatic.net/iNlJM.png