From 40f52ba78fb265a22743163831aaa2fd2fa37145 Mon Sep 17 00:00:00 2001
From: mrshnra <mrshnra@gmail.com>
Date: Sun, 22 Nov 2015 07:54:27 -0500
Subject: [PATCH] added the spolution for the wave equation via leap frog

---
 PDE.ipynb | 45 +++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 45 insertions(+)

diff --git a/PDE.ipynb b/PDE.ipynb
index b1665e7..2bb9ba8 100644
--- a/PDE.ipynb
+++ b/PDE.ipynb
@@ -335,3 +335,48 @@
  "nbformat": 4,
  "nbformat_minor": 0
 }
+
+import numpy as np
+import matplotlib.pyplot as plt
+%matplotlib inline
+
+Nx, Nt = 200, 5000
+
+v = 1
+dt = 0.04
+dx = 0.1
+delta =   v**2 * dt**2 / dx**2
+
+u = np.zeros((Nt,Nx))
+
+#u[0,:] = np.sin(2*np.pi*np.arange(Nx)/Nx)
+u[0,:] = np.exp(-(np.arange(Nx)-100)**2/1000)
+#u[0,100]=1
+#u[0,101]=1
+
+#u[1,100]=1
+#u[1,101]=1
+
+#u[0,80:110] =1 
+
+#u[1,:] = np.exp(-(np.arange(Nx)-100)**2/500)
+for j in range(Nx):
+    u[1,j]=2*u[0,j-1]  - (u[0,(j+1)%Nx] - u[0,j-1]) * delta
+    #u[1,j]=u[0,j-1]
+
+for n in range(2,Nt):
+    for j in range(Nx):
+        u[n,j] = - u[n-2,j] + 2*u[n-1,j] + (u[n-1,(j+1)%Nx] + u[n-1,j-1] - 2*u[n-1,j]) * delta
+        
+plt.imshow(u)
+plt.show()
+plt.plot(u[0,:])
+plt.show()
+plt.plot(u[200,:])
+plt.show()
+plt.plot(u[800,:])
+plt.show()
+plt.plot(u[1000,:])
+plt.show()
+plt.plot(u[1200,:])
+plt.show()