Assignment

PDEs in 1D: Reaction-Diffusion

Implement a full 1D reaction-diffusion workflow for the Gierer-Meinhardt model. Your submission should include code and figures that demonstrate pattern formation.

Required

Implement the PDE solver using an explicit Euler time step.
Enforce Neumann boundary conditions at each step.
Use \(L=40\), \(dx=0.5\), \(dt=0.001\), \(a=0.40\), \(b=1.00\), \(d=20\), \(\gamma=1\).
Create an animation of \(v(x)\) that updates in time.

Turing Instability

Implement is_turing_instability(a, b, d) and plot the Turing space diagram.
Mark the point \((a, d)\) you are using on the diagram.

Extra Mile (Optional)

Compute unstable spatial modes and display the leading mode on the animation.
Make the diagram interactive: clicking on \((a, d)\) updates the simulation.
Try other boundary conditions and compare.

Tips for Success

Start simple: Verify your Laplacian and boundary conditions first.
Test incrementally: Plot intermediate results to detect instability early.
Be careful with dt: If the simulation explodes, reduce \(dt\) or increase \(dx\).
Document choices: Report the parameters you used and the patterns you observed.

Good luck and enjoy your coding!

---
title: "Assignment"
subtitle: "PDEs in 1D: Reaction-Diffusion"
format: html
---

Implement a full 1D reaction-diffusion workflow for the **Gierer-Meinhardt model**. Your submission should include code and figures that demonstrate pattern formation.

## Required

1. Implement the PDE solver using an explicit Euler time step.
2. Enforce **Neumann** boundary conditions at each step.
3. Use $L=40$, $dx=0.5$, $dt=0.001$, $a=0.40$, $b=1.00$, $d=20$, $\gamma=1$.
4. Create an animation of $v(x)$ that updates in time.

## Turing Instability

5. Implement `is_turing_instability(a, b, d)` and plot the Turing space diagram.
6. Mark the point $(a, d)$ you are using on the diagram.

## Extra Mile (Optional)

- Compute unstable spatial modes and display the leading mode on the animation.
- Make the diagram interactive: clicking on $(a, d)$ updates the simulation.
- Try other boundary conditions and compare.

## Tips for Success

- **Start simple:** Verify your Laplacian and boundary conditions first.
- **Test incrementally:** Plot intermediate results to detect instability early.
- **Be careful with dt:** If the simulation explodes, reduce $dt$ or increase $dx$.
- **Document choices:** Report the parameters you used and the patterns you observed.

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**Good luck and enjoy your coding!**