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Application of Variations of Parameters and Duhamel's Principle: A Simple Example

In the previous post, we explored the theoretical differences and similarities between Variations of Parameters and Duhamel's Principle. Now, let’s bring these ideas to life with a simple example, demonstrating how these methods work in practice. We'll aim for clarity and connection, using an example that connects to everyday phenomena like heat distribution.

A Problem: Modeling Heat Transfer in a Rod

Imagine a metal rod that is heated at one end while the rest is kept cool. Over time, heat spreads along the rod. Mathematically, this situation can be modeled with the heat equation, a partial differential equation (PDE):

∂u/∂t = α ∂²u/∂x²

where:

  • u(x, t) is the temperature at position x and time t,
  • α is a constant representing how well the material conducts heat,
  • ∂u/∂t is the rate of change of temperature with time,
  • ∂²u/∂x² is the second derivative of temperature with position, representing how temperature changes spatially.

Suppose the rod initially has no heat (u(x, 0) = 0) and we apply a heat source at x = 0, modeled as g(t) = sin(t). How does the temperature u(x, t) evolve?

Approach 1: Variations of Parameters

For PDEs like the heat equation, Variations of Parameters can seem complex, but the core idea remains: use the homogeneous solution and adjust it to fit the specific heat source. Here’s a simplified overview:

  1. Solve the Homogeneous Equation:
    The solution to the heat equation without any heat source (g(t) = 0) is:

    u_h(x, t) = ∫₀^∞ A(k) e^(-αk²t) cos(kx) dk

    where A(k) is determined by boundary conditions.

  2. Adjust for the Non-Homogeneous Term:
    Assume that A(k) is not a constant but depends on t: A(k, t). This adjustment ensures the solution reflects the applied heat source g(t).

  3. Find A(k, t):
    Differentiating and substituting back into the heat equation lets us solve for A(k, t) in terms of g(t).

Approach 2: Duhamel’s Principle

Duhamel’s Principle offers a more intuitive method by thinking of the heat source g(t) as being applied incrementally over time.

  1. Impulse Response:
    Solve the heat equation for a very short pulse of heat applied at t = 0, which gives the response:

    u_impulse(x, t) = 1 / √(4παt) * e^(-x² / (4αt))

  2. Superposition of Responses:
    Use the idea of superposition to combine the responses over time, weighted by g(t). The solution is given by:

    u(x, t) = ∫₀^t u_impulse(x, t - τ) g(τ) dτ

  3. Apply g(t) = sin(t):
    Substituting the heat source g(t) = sin(t) into the integral gives the complete solution:

    u(x, t) = ∫₀^t (1 / √(4πα(t - τ))) * e^(-x² / (4α(t - τ))) sin(τ) dτ

Key Takeaways

  • Variations of Parameters adjusts the constants in the homogeneous solution to account for the heat source.
  • Duhamel’s Principle constructs the solution as a sum of the system’s response to each part of the heat source.

In this example, Duhamel’s Principle provides a more direct approach for solving the heat equation with a time-dependent heat source. Its reliance on convolution integrals makes it particularly suited to problems involving dynamic inputs, like our g(t) = sin(t).

A Visualization: Heat Distribution

If we plot u(x, t), we’d see the heat spreading along the rod over time, with the oscillations of g(t) = sin(t) influencing the distribution. This connection between mathematics and physical behavior demonstrates the power of these methods.

Whether you’re solving PDEs in engineering or physics, mastering these techniques equips you to tackle dynamic systems and uncover the beautiful patterns within.


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