Partial Differential Equations in 1D

Reaction-Diffusion Models

In this session we move from ODEs to reaction-diffusion PDEs in one spatial dimension. We will discretize space with finite differences, advance in time with Euler’s method, and visualize pattern formation.

Case Studies

Gierer-Meinhardt 1D Turing Instability Assignment

Goals

Implement the 1D Laplacian with finite differences.
Simulate a reaction-diffusion system with explicit time stepping.
Visualize patterns and understand when they emerge.
Connect parameter choices to stability and pattern formation.
Build animations and add simple user interaction.

What do we need?

`numpy`

We use NumPy arrays to represent spatial fields and np.roll() to compute the discrete Laplacian.

`matplotlib.pyplot`

Initialize the figure and axes and plot static elements (labels, titles, limits).

`matplotlib.animation`

Update line objects frame-by-frame to create animations.

`matplotlib.backend_bases`

Handle mouse clicks to let the user interact with the simulation.

`scipy`

Great for ODEs, but we will not use it for PDEs in this session. We need to enforce boundary conditions at every step.

---
title: "Partial Differential Equations in 1D"
subtitle: "Reaction-Diffusion Models"
format: html
---

In this session we move from ODEs to **reaction-diffusion PDEs** in one spatial dimension. We will discretize space with finite differences, advance in time with Euler's method, and visualize pattern formation.

## Case Studies

[Gierer-Meinhardt 1D](gierer-meinhardt.qmd){.btn .btn-primary} [Turing Instability](turing-instability.qmd){.btn .btn-primary} [Assignment](assignment.qmd){.btn .btn-secondary}

## Goals

- Implement the 1D Laplacian with finite differences.
- Simulate a reaction-diffusion system with explicit time stepping.
- Visualize patterns and understand when they emerge.
- Connect parameter choices to stability and pattern formation.
- Build animations and add simple user interaction.

## What do we need?

### `numpy`

We use NumPy arrays to represent spatial fields and `np.roll()` to compute the discrete Laplacian.

### `matplotlib.pyplot`

Initialize the figure and axes and plot static elements (labels, titles, limits).

### `matplotlib.animation`

Update line objects frame-by-frame to create animations.

### `matplotlib.backend_bases`

Handle mouse clicks to let the user interact with the simulation.

### `scipy`

Great for ODEs, but we will **not** use it for PDEs in this session. We need to enforce boundary conditions at every step.