I’ll help you solve a Partial Differential Equation (PDE) example using $Python$.
Let’s solve a 2D heat equation, which is a classic PDE problem.
1 | import numpy as np |
Let me explain this code and the problem we’re solving:
The Problem: We’re solving the 2D heat equation:
- Equation: $∂u/∂t = α(∂²u/∂x² + ∂²u/∂y²)$
- Initial condition: $u(x,y,0) = sin(πx)sin(πy)$
- Boundary conditions: $u = 0$ at all boundaries
- $α$ is the thermal diffusivity (set to $1.0$ in our case)
Solution Method:
- We use the Finite Difference Method (FDM)
- Space is discretized into a grid with steps $dx$ and $dy$
- Time is discretized with step $dt$
- The solution uses an explicit scheme for time-stepping
Code Structure:
- Creates a mesh grid for $x$ and $y$ coordinates
- Sets initial conditions using sine functions
- Implements time-stepping loop with boundary conditions
- Uses $numpy$ for efficient array operations
Visualization:
- Left plot: 3D surface plot showing temperature distribution
- Right plot: 2D contour plot showing temperature from top view
- Colors indicate temperature (darker blue = cooler, yellow = warmer)
Stability parameter: 0.020 (should be < 0.5)
- Results Interpretation:
- The initial sine wave pattern gradually diffuses
- Heat spreads from warmer regions to cooler regions
- The solution maintains zero temperature at boundaries
- The pattern becomes smoother over time due to diffusion
The visualization shows both a 3D surface plot and a 2D contour plot to help understand how heat distributes across the 2D surface.
The color gradient represents temperature variations, with warmer colors indicating higher temperatures.