ex4

ex4/code/task_a.py

@@ -0,0 +1,57 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+# PDE:
+#  -u''(x) + a*u(x) = f(x)         x in (0,1)
+#              u(0) = 0
+#             u'(1) = \alpha*(g_b - u(1))
+
+# parameters
+a = 1
+f = 3
+alpha = 1
+gb = 1
+n = 10       # elements
+
+# mesh
+m = n+1      # nodes
+h = 1.0/n
+x = np.linspace(0,1,m)
+
+# local stiffness matrix
+K_loc = np.zeros((2,2))
+A = (1.0/h) * np.array([[ 1,-1], [-1, 1]])
+B = (a*h/6) * np.array([[ 2, 1], [ 1, 2]])
+K_loc = A+B
+
+# Assembling
+K = np.zeros((m,m))
+F = np.zeros(m)
+for i in range(n):
+    K[i:i+2,i:i+2] += K_loc
+    F[i:i+2] += np.full(2, f*h/2)
+
+# Boundary conditions
+# Dirichlet: u(0) = 0
+K[0,:] = 0
+K[0,0] = 1
+F[0]   = 0   
+# Neumann: u'(1) = \alpha*(g_b - u(1))
+A[-1,-1] += alpha
+F[-1] += alpha*gb
+
+u = np.linalg.solve(K, F)
+plt.plot(x, u, "-o", label="u_h")
+plt.title(f"n = {n} | h = {h} | a = {a} | f(x) = {f} | alpha = {alpha} | gb = {gb}")
+plt.xlabel("x")
+plt.ylabel("u(x)")
+plt.legend()
+plt.grid(True)
+
+print("K = ", K)
+print("f = ", F)
+print("u = ", u)
+
+plt.tight_layout()
+plt.savefig("../task_a.png", dpi=300)
+plt.show()