Determine the symmetries of the heat pde

\[ \frac{\partial T}{\partial t}-\frac{\partial^{2} T}{\partial x^{2}}=0 \]

defined on \(\mathcal{D}=[0, \infty] \times R^{1}\).

2.5.1 Change the notation to \(x \rightarrow x_{1}, t \rightarrow x_{2}, T \rightarrow y_{1}\) and determine the prolonged operator \(X^{(2)}\) and the condition for symmetry transformations.

2.5.2 Compute the total derivatives determining the infinitesimals appearing in the condition for symmetry transformations.

2.5.3 Establish the equations for the infinitesimals.

2.5.5 Solve the equations for the infinitesimals.

2.5.6 Choose a set of values for the parameters and compute the global transformation.