update examples

examples/problems/first_order_ode.py

@@ -0,0 +1,46 @@
+from pina.problem import SpatialProblem
+from pina import Condition, Span
+from pina.operators import grad
+import torch
+
+# ===================================================== #
+#                                                       #
+#  This script implements a simple first order ode.     #
+#  The FirstOrderODE class is defined inheriting from   #
+#  SpatialProblem. We  denote:                          #
+#           y --> field variable                        #
+#           x --> spatial variable                      #
+#                                                       #
+# ===================================================== #
+
+class FirstOrderODE(SpatialProblem):
+
+    # variable domain range
+    x_rng = [0, 5]
+    # field variable
+    output_variables = ['y']
+    # create domain
+    spatial_domain = Span({'x': x_rng})
+
+    # define the ode
+    def ode(input_, output_):
+        y = output_
+        x = input_
+        return grad(y, x) + y - x
+
+    # define initial conditions
+    def fixed(input_, output_):
+        exp_value = 1.
+        return output_ - exp_value
+
+    # define real solution
+    def solution(self, input_):
+        x = input_
+        return x - 1.0 + 2*torch.exp(-x)
+
+    # define problem conditions
+    conditions = {
+        'bc': Condition(location=Span({'x': x_rng[0]}), function=fixed),
+        'dd': Condition(location=Span({'x': x_rng}), function=ode),
+    }
+    truth_solution = solution