Collaboration diagram for Evolve in time 3D heat equation with a linear term:
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The source is here.
- Points of interest
- application of a function of a function to exponentiate the potential
- use of a functor to compute the solution at an arbitrary future time
- convolution with the Green's function
- Background
This adds to the complexity of the other heat equation example by including a linear term. Specifically, we solve
![\[ \frac{\partial u(x,t)}{\partial t} = c \nabla^2 u(x,t) + V_p(x,t) u(x,t) \]](form_135.png)
If 
time evolution operator is
![\[ G_0(x,t) = \frac{1}{\sqrt{4 \pi c t}} \exp \frac{-x^2}{4 c t} \]](form_137.png)
For non-zero 
the time evolution is performed using the Trotter splitting
![\[ G(x,t) = G_0(x,t/2) * \exp(V_p t) * G_0(x,t/2) + O(t^3) \]](form_139.png)
In order to form an exact solution for testing, we choose 
but the solution method is not limited to this choice.
