15.2. Fourier Series in Solving PDEs

We focus on solving three classical PDEs using Fourier series: A. Heat Equation B. Wave Equation C. Laplace Equation

This chunk introduces the main types of Partial Differential Equations (PDEs) that will be solved using Fourier series. These include the Heat Equation, which models heat distribution; the Wave Equation, which models wave motion; and the Laplace Equation, often used in steady-state problems.

By identifying these equations, we set the stage for applying Fourier series methods, which can simplify and provide solutions to complex problems in physics and engineering.

Think of the Heat Equation in the context of cooling a cup of coffee. Initially, the coffee is hot, but over time it cools down to room temperature. The Heat Equation can help us predict how quickly the temperature changes over time, similar to measuring the temperature changes in your cup as it slowly cools.

Form: ∂𝑢/∂𝑡 = 𝛼² ∂²𝑢/∂𝑥², 0 < 𝑥 < 𝐿, 𝑡 > 0 Boundary Conditions: 𝑢(0,𝑡) = 0, 𝑢(𝐿,𝑡) = 0 Initial Condition: 𝑢(𝑥,0) = 𝑓(𝑥)

This chunk presents the mathematical formulation of the Heat Equation, which describes how heat evolves over time in a given space. The equation states that the rate of change of temperature with respect to time (∂u/∂t) relates to the second spatial derivative of temperature (∂²u/∂x²), scaled by a constant (α²), which is related to the material properties.

The boundary conditions specify that the ends of the domain (at x=0 and x=L) are kept at zero temperature, while the initial condition gives the temperature distribution at time t=0. These constraints set up a well-defined problem that can be solved using Fourier series.

Imagine a metal rod that is heated only in the middle; the ends are held in ice. The temperature at both ends will always remain at zero due to the ice, and how the temperature in the rod changes over time can be described by the Heat Equation. It allows us to predict how heat will flow and distribute along the rod.

Solution Approach:

1. Separation of Variables: Assume 𝑢(𝑥,𝑡) = 𝑋(𝑥)𝑇(𝑡)
2. Substituting in the PDE and separating variables gives: 1/𝛼²𝑇(𝑡) d𝑇/d𝑡 = 1/𝑋(𝑥) d²𝑋/d𝑥² = -𝜆
3. Solve two ODEs:...
4. General solution: 𝑢(𝑥,𝑡) = ∑𝐵 sin(𝑛𝜋𝑥/𝐿)e^{-𝛼²(𝑛𝜋/𝐿)²𝑡}

In this chunk, we describe the method of separation of variables, which is a powerful technique for solving PDEs. We assume that the solution can be decomposed into two functions: one dependent on x and the other on t. This allows us to create two ordinary differential equations (ODEs) from the original PDE. After solving these ODEs, we combine the results to formulate the general solution which expresses the temperature distribution over time.

The final solution includes a series that uses sine functions to respect the boundary conditions and reflects how the initial temperature distribution evolves over time.

Imagine you're playing two musical notes on a piano simultaneously, one for the x position of a string and another for the time. By breaking down the complex sounds (the PDE) into individual notes (the ODEs), you can easily play them one by one, eventually creating the full melody (the general solution).

Form: ∂²𝑢/∂𝑡² = 𝑐² ∂²𝑢/∂𝑥² Boundary Conditions: 𝑢(0,𝑡) = 𝑢(𝐿,𝑡) = 0 Initial Conditions: 𝑢(𝑥,0) = 𝑓(𝑥), ∂𝑢/∂𝑡 = 𝑔(𝑥)

The Wave Equation describes how waves propagate in a medium, whether it be sound waves, water waves, or vibrations in a string. It shows that the acceleration of a wave at a point in time and space is proportional to the spatial curvature of the wave profile.

The boundary conditions set that the ends of a string (or medium) are fixed, meaning that the wave must also be zero at those points. The initial conditions, given by two different functions, define the shape of the wave at the start and its initial velocity.

Consider a guitar string. When you pluck it, the string vibrates, creating sound waves. The Wave Equation can help describe how that vibration travels along the string and how it starts at the moment you pluck it, just like registering where your finger is on the string and how quickly it moves at the beginning.

Form (in 2D): ∂²𝑢/∂𝑥² + ∂²𝑢/∂𝑦² = 0 Used in steady-state heat conduction problems. The method of separation of variables leads to Fourier sine or cosine series in one direction, depending on boundary conditions.

The Laplace Equation is crucial in steady-state problems, including heat conduction over time when the system has stabilized and temperatures are no longer changing. The equation itself reflects that the sum of the second derivatives in two dimensions will equal zero, signifying a balance. By solving it, we often utilize separation of variables, leading to Fourier series solutions, either through sine or cosine functions, based on the problem's symmetry and boundary conditions.

Think of a lake that, after a while, reaches a uniform temperature throughout its water without any new heat being added. The Laplace Equation describes how that even temperature state is achieved, and just like measuring the temperature at various points in the lake, we can use Fourier series to find out what those uniform temperatures would be across the surface and depth.

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