One-Dimensional Heat Equation

Concept Introduction

• Scenario: A thin, insulated rod with endpoints touching ice (0°C), while the rod's initial temperature is 20°C.
• Objective: Determine the progression of temperature distribution over time (t).

Definitions and Assumptions

• u(x, t): Temperature at location x and time t.
  • Initial: u(x, 0) = 20°C (uniform temperature).
  • Ends are kept at 0°C due to ice.
• Rod's Length: 0 to L.
• Boundary Conditions: Temperature at x=0 and x=L is 0°C for all time.

Expected Temperature Behavior

• Initially, temperature is uniform at 20°C along the rod.
• Over time, temperature dips at the endpoints due to the ice.
• Intermediate times show decreasing temperatures, most significant near the endpoints.

Governing Equation

• Newton's Cooling Law: Change in temperature equals ambient temperature minus the current temperature.
• Derivation uses intuitive argument rather than detailed derivation from Newton's law.

Intuitive Argument

• Second derivative analysis (concavity concept):
  • Concave Up: Second derivative (∂²u/∂x² > 0) => temperature (u) increases.
  • Concave Down: Second derivative (∂²u/∂x² < 0) => temperature (u) decreases.
• Relation: Rate of change of temperature (∂u/∂t) is proportional to concavity (∂²u/∂x²).
• Equation Formation:
  • ∂u/∂t = k * ∂²u/∂x² (k is a proportionality constant dependent on material properties).
• Often written as:
  • uₜ = k * uₓₓ (using indices).
• Material Dependent Constant (k):
  • High k: Fast temperature change (e.g., metals).
  • Low k: Slow temperature change.

Conclusion

• Heat Equation: uₜ = k * uₓₓ.
• Next Steps: Solve the heat equation using boundary conditions.