Add Gray-Scott model implementations and update references

- Implemented the Gray-Scott model in `gray_scott.py` with Laplacian calculations and simulation controls.
- Created `gray_scott_art.py` for an interactive version of the Gray-Scott model using image masks.
- Updated references in documentation to point to the new file structure for Gierer-Meinhardt and Gray-Scott models.
- Enhanced the user interface with sliders for parameter adjustments and mouse interactions for perturbations.

Author: Daniel Precioso, PhD

amlab/pdes/__init__.py

@@ -0,0 +1,21 @@
+"""Shared PDE models and interactive demos for the Applied Math Lab package."""
+
+from .gierer_meinhardt_1d import (
+    find_unstable_spatial_modes as find_unstable_spatial_modes_1d,
+)
+from .gierer_meinhardt_1d import gierer_meinhardt_pde as gierer_meinhardt_pde_1d
+from .gierer_meinhardt_1d import is_turing_instability
+from .gierer_meinhardt_2d import (
+    find_unstable_spatial_modes as find_unstable_spatial_modes_2d,
+)
+from .gierer_meinhardt_2d import gierer_meinhardt_pde as gierer_meinhardt_pde_2d
+from .gray_scott import gray_scott_pde
+
+__all__ = [
+    "find_unstable_spatial_modes_1d",
+    "find_unstable_spatial_modes_2d",
+    "gierer_meinhardt_pde_1d",
+    "gierer_meinhardt_pde_2d",
+    "gray_scott_pde",
+    "is_turing_instability",
+]

amlab/pdes_1d/gierer_meinhardt_1d.py amlab/pdes/gierer_meinhardt_1d.pyamlab/pdes_1d/gierer_meinhardt_1d.py renamed to amlab/pdes/gierer_meinhardt_1d.py

@@ -57,7 +57,7 @@ Include a brief paragraph that answers these questions:
 - Reuse your 2D Laplacian in both Gierer-Meinhardt and Gray-Scott.
 
 ::: {.callout-tip}
-Reference code is available in [amlab/pdes_1d/gierer_meinhardt_1d.py](https://github.com/daniprec/BAM-Applied-Math-Lab/blob/main/amlab/pdes_1d/gierer_meinhardt_1d.py), [amlab/pdes_2d/gierer_meinhardt_2d.py](https://github.com/daniprec/BAM-Applied-Math-Lab/blob/main/amlab/pdes_2d/gierer_meinhardt_2d.py), and [amlab/pdes_2d/gray_scott.py](https://github.com/daniprec/BAM-Applied-Math-Lab/blob/main/amlab/pdes_2d/gray_scott.py).
+Reference code is available in [amlab/pdes/gierer_meinhardt_1d.py](https://github.com/daniprec/BAM-Applied-Math-Lab/blob/main/amlab/pdes/gierer_meinhardt_1d.py), [amlab/pdes/gierer_meinhardt_2d.py](https://github.com/daniprec/BAM-Applied-Math-Lab/blob/main/amlab/pdes/gierer_meinhardt_2d.py), and [amlab/pdes/gray_scott.py](https://github.com/daniprec/BAM-Applied-Math-Lab/blob/main/amlab/pdes/gray_scott.py).
 :::
 
 \vspace{1cm}

amlab/pdes_2d/gierer_meinhardt_2d.py amlab/pdes/gierer_meinhardt_2d.pyamlab/pdes_2d/gierer_meinhardt_2d.py renamed to amlab/pdes/gierer_meinhardt_2d.py

@@ -29,7 +29,7 @@ where $u(x,t)$ is the activator, $v(x,t)$ is the inhibitor, and $d$ is the diffu
 import matplotlib.pyplot as plt
 import numpy as np
 
-from amlab.pdes_1d.gierer_meinhardt_1d import gierer_meinhardt_pde
+from amlab.pdes.gierer_meinhardt_1d import gierer_meinhardt_pde
 
 np.random.seed(3)
 
@@ -71,6 +71,34 @@ $$
 \Delta u(x) \approx \frac{u(x+dx) - 2u(x) + u(x-dx)}{dx^2}.
 $$ {#eq-pde-gm-1d-fd}
 
+:::{.callout-tip}
+## Where does this formula come from?
+
+The second derivative is the limit of the difference quotient:
+
+$$
+\frac{d^2 u}{dx^2} = \lim_{dx \to 0} \frac{u(x+dx) - 2u(x) + u(x-dx)}{dx^2}.
+$$
+
+This is a direct consequence of the Taylor expansion of $u$ around $x$:
+
+$$
+\begin{aligned}
+u(x+dx) &= u(x) + u'(x) dx + \frac{1}{2} u''(x) dx^2 + \mathcal{O}(dx^3), \\
+u(x-dx) &= u(x) - u'(x) dx + \frac{1}{2} u''(x) dx^2 + \mathcal{O}(dx^3).
+\end{aligned}
+$$
+
+Adding these two expansions gives
+
+$$
+u(x+dx) + u(x-dx) = 2u(x) + u''(x) dx^2 + \mathcal{O}(dx^3),
+$$
+
+which can be rearranged to give the finite difference formula, ignoring the higher-order terms $\mathcal{O}(dx^3)$.
+:::
+
+
 Write a helper that computes this second derivative for a 1D field.
 
 ```python
@@ -289,5 +317,5 @@ You now have a working 1D solver. The next question is theoretical: for which pa
 [Turing Instability](turing-instability.qmd){.btn .btn-primary}
 
 ::: {.callout-tip}
-Reference code is available in [amlab/pdes_1d/gierer_meinhardt_1d.py](https://github.com/daniprec/BAM-Applied-Math-Lab/blob/main/amlab/pdes_1d/gierer_meinhardt_1d.py).
+Reference code is available in [amlab/pdes/gierer_meinhardt_1d.py](https://github.com/daniprec/BAM-Applied-Math-Lab/blob/main/amlab/pdes/gierer_meinhardt_1d.py).
 :::

amlab/pdes_2d/gm2d_template.py amlab/pdes/gm2d_template.pyamlab/pdes_2d/gm2d_template.py renamed to amlab/pdes/gm2d_template.py

@@ -29,7 +29,7 @@ but now $u$ and $v$ depend on $(x, y, t)$.
 import matplotlib.pyplot as plt
 import numpy as np
 
-from amlab.pdes_2d.gierer_meinhardt_2d import gierer_meinhardt_pde
+from amlab.pdes.gierer_meinhardt_2d import gierer_meinhardt_pde
 
 np.random.seed(3)
 
@@ -270,5 +270,5 @@ The Gierer-Meinhardt model is a good training ground, but the final target of th
 [Gray-Scott](gray-scott.qmd){.btn .btn-primary}
 
 ::: {.callout-tip}
-Reference code is available in [amlab/pdes_2d/gierer_meinhardt_2d.py](https://github.com/daniprec/BAM-Applied-Math-Lab/blob/main/amlab/pdes_2d/gierer_meinhardt_2d.py).
+Reference code is available in [amlab/pdes/gierer_meinhardt_2d.py](https://github.com/daniprec/BAM-Applied-Math-Lab/blob/main/amlab/pdes/gierer_meinhardt_2d.py).
 :::

amlab/pdes_2d/gray_scott.py amlab/pdes/gray_scott.pyamlab/pdes_2d/gray_scott.py renamed to amlab/pdes/gray_scott.py

@@ -186,5 +186,5 @@ At this point you have all the ingredients of the session: a 1D solver, the Turi
 [Assignment](assignment.qmd){.btn .btn-secondary}
 
 ::: {.callout-tip}
-Reference code is available in [amlab/pdes_2d/gray_scott.py](https://github.com/daniprec/BAM-Applied-Math-Lab/blob/main/amlab/pdes_2d/gray_scott.py).
+Reference code is available in [amlab/pdes/gray_scott.py](https://github.com/daniprec/BAM-Applied-Math-Lab/blob/main/amlab/pdes/gray_scott.py).
 :::