MATLAB Script for Solving Heat Transfer Equation Using Explicit Difference Method

The explicit difference method is a classical numerical approach for solving heat transfer equations (heat conduction equations). Its core concept involves discretizing time and space to transform partial differential equations into difference equations, then obtaining numerical solutions through iterative computation. This method is straightforward to implement but requires attention to stability conditions to prevent solution divergence.

### Method Overview The general form of heat transfer equation is ∂u/∂t = α(∂²u/∂x²), where u represents temperature distribution and α is thermal diffusivity. The explicit difference method employs forward difference for time derivative approximation and central difference for spatial second derivative, effectively discretizing the equation. In MATLAB implementation, this translates to using simple matrix operations for efficient computation.

### Stability Condition Stability of the explicit difference method is governed by the Courant-Friedrichs-Lewy (CFL) condition: Δt ≤ (Δx)² / (2α). Excessive time step Δt may cause numerical instability, leading to oscillations or divergence. When coding in MATLAB, proper selection of Δt and Δx is crucial, typically implemented through conditional checks at the beginning of the script.

### Implementation Approach Spatial and temporal discretization: Divide the solution domain into grid points, define spatial step Δx and time step Δx, initialize temperature field using zeros() or ones() arrays. Boundary condition handling: Implement Dirichlet (fixed temperature) or Neumann (adiabatic) boundaries through array indexing operations, e.g., u(1,:)=constant for left boundary. Iterative computation: Update temperature values time-step by time-step using explicit difference formula, typically involving nested loops for time and space dimensions. Stability verification: Include automatic CFL condition checking using if statements with warning messages for unstable parameters. Result visualization: Utilize MATLAB plotting functions like plot() for 1D results or surf()/pcolor() for 2D temperature evolution displays, with proper labeling and color mapping.

### Important Considerations While computationally simple, explicit methods may require small Δt for long-duration simulations, increasing computational load. For higher efficiency or unconditional stability, consider implicit methods like Crank-Nicolson scheme, though requiring matrix inversion operations. During debugging, start with simple test cases (e.g., constant boundary conditions) to validate result rationality before progressing to complex scenarios. Code optimization tips: Preallocate arrays using zeros() to improve performance, vectorize operations where possible to reduce loop usage.

This approach enables efficient solving of heat transfer equations with physically meaningful numerical solutions. MATLAB's visualization capabilities further facilitate intuitive analysis of temperature evolution patterns, supporting research and optimization processes. Typical script structure includes parameter initialization, stability check, main computation loop, and visualization section.