12.5. Example Problem

Problem: Solve the heat equation with: • 𝑢(0,𝑡) = 𝑢(𝐿,𝑡) = 0 • 𝑢(𝑥,0) = 𝑓(𝑥) = 𝑥(𝐿−𝑥)

In this example problem, we are tasked with solving the heat equation under specific boundary and initial conditions. The first part of the problem, indicated by 𝑢(0,𝑡) = 𝑢(𝐿,𝑡) = 0, tells us that the temperatures at both ends of the rod are kept at zero temperature for all times. This is known as Dirichlet boundary conditions. The second part, 𝑢(𝑥,0) = 𝑓(𝑥) = 𝑥(𝐿−𝑥), provides the initial temperature distribution along the rod at time 𝑡=0, showing that it's shaped like a parabola peaking at the middle point of the rod.

Imagine a metal rod being cooled down at both ends by water while it is heated from its center. At the start, the temperature is highest at the center and lowest at both ends. Over time, this setup will lead to heat spreading along the rod until it stabilizes, simulating the described conditions.

Solution Outline:

1. Compute Fourier sine coefficients of 𝑓(𝑥).

The first step in solving this problem is to compute the Fourier sine coefficients of the initial condition function, 𝑓(𝑥) = 𝑥(𝐿−𝑥). These coefficients, usually denoted as 𝐵ₙ, will be determined using integrals that project the initial condition onto the eigenfunctions of the system, which in this case are sine functions due to the boundary conditions at both ends being zero.

Think of a musician tuning a guitar string. Each note corresponds to a frequency that can be expressed in terms of its fundamental and harmonic frequencies. Similarly, when calculating Fourier coefficients, we are essentially breaking down the initial temperature distribution into a series of 'notes' (sine functions) that together reproduce the shape of 𝑓(𝑥).

2. Plug into general solution.

After calculating the Fourier sine coefficients, the next step is to insert these coefficients back into the general solution format for the heat equation, which combines the spatial and temporal components derived from the previously solved ordinary differential equations. This forms a series solution that accounts for the evolving temperature distribution over time.

Imagine a recipe where you have various ingredients (Fourier coefficients) that you mix together to create a dish (the solution). Each ingredient contributes a unique flavor to the overall taste. Similarly, each coefficient affects how the temperature distribution evolves over time, resulting in a complete picture of the thermal conduction process.

3. Write final form of 𝑢(𝑥,𝑡).

In the final step, we compile all previous calculations and representations into the explicit form of 𝑢(𝑥,𝑡), which represents the temperature at position 𝑥 along the rod at time 𝑡. This can be expressed as a series involving the Fourier coefficients and the exponential decay terms determined from the time-dependent part of the solution.

Consider the final moments of baking a cake. You’ve mixed all the batter and added the remaining ingredients, and now it’s time to see the finished product rise in the oven. The end result of the cake is analogous to 𝑢(𝑥,𝑡), showcasing how all the parts (initial conditions, coefficients, and eigenfunctions) come together to create the outcome of the heat distribution.

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