Wrong log scale in Y axis and how to add fit equation to each plot with ax_b

I have a repetitive label in legend. How can I fix that? and how can I assign each label to its specific curve? I also want to modify the curves themselves by changing the colors. line width and style

import numpy as np
import matplotlib.pyplot as plt
import math
from scipy.special import erfc

pdi = 0
TD=[0.05, 0.1, 1]
def Dirichlet_Dirichlet(td_arr = [ 0.05, 0.1, 1], 
           xd_arr = np.arange(0,1,0.001)):
   for td in td_arr:   
       xd = xd_arr       
       sumBound = np.arange(1,100,1)
       sumTerm1 = np.zeros(len(xd))
       sumTerm2 = np.zeros(len(xd))
       for n in sumBound:
           sumTerm1 += (((-1)**n) / n)*np.sin(n*np.pi*xd)*np.exp(-(n**2)*(np.pi**2)* td)
           sumTerm2 += (((-1)**n) / n)*np.sin(n*np.pi*(xd - 1))*np.exp(-(n**2)*(np.pi**2)* td)               
       
       res = 1 - xd - (2*pdi/np.pi)*sumTerm1 - (2*(1-pdi)/np.pi)*sumTerm2           
       for td in TD:
           plt.plot(xd, res, label='$t_D$={}'.format(round(td,3)))
              
   plt.xlabel("'$\\xi_D$'")
   plt.ylabel("Dimensionless Pressure, $P_D$")
   plt.legend()
   plt.xlim(0, 1)
   plt.ylim(0, 1)
   plt.title('Plot Dimensionless Pressure vs. Distance: Dirichlet_Dirichlet')
   plt.tight_layout()   
   plt.title('Analytical Solution')       
   plt.show()
    
Dirichlet_Dirichlet()