18.3. General Steps in the Eigenfunction Expansion Method

Separate variables by assuming:

$u(x,t) = X(x)T(t)$

Substitute into the PDE and separate variables to get two ODEs:

• One in $x$: Eigenvalue problem $L X = \lambda X$
• One in $t$: First-order ODE in $T$

In this initial step of the Eigenfunction Expansion Method, we start by assuming that the solution can be written as a product of two functions: one that depends only on the spatial variable (x) and the other that depends only on the time variable (t). This assumption allows us to separate the variables in the partial differential equation (PDE) into two ordinary differential equations (ODEs). The first ODE represents an eigenvalue problem for the spatial component, while the second gives us a first-order equation for the time component. The goal is to solve these two simpler problems, which helps in constructing the overall solution to the original PDE.

Think of this step like a chef preparing a dish where you decide to make the sauce and the pasta separately. By handling each component (the spatial and time parts) independently, you ensure that when they come together later in the dish, they combine harmoniously to create the final meal.

Solve the spatial ODE under given boundary conditions to get:

• $\phi_n(x)$: Eigenfunctions
• $\lambda_n$: Eigenvalues

In this step, we focus on solving the spatial ordinary differential equation obtained from the separation of variables. This involves applying appropriate boundary conditions applicable to the problem at hand. Solving this ODE allows us to determine the eigenfunctions, which are specific solutions representing distinct modes of the system, and the eigenvalues, which are associated constants indicating the scaling of these modes. The eigenfunctions are essential because they form the basis for constructing the final solution to the PDE.

Consider an orchestra tuning its instruments. Each instrument (like the eigenfunctions) has its own specific note it can play (eigenvalues) that resonates at certain frequencies. Finding the right pitches allows the orchestra to harmonize as a whole, just like how the eigenfunctions and eigenvalues align to solve the PDE.

Given initial condition:

$u(x,0) = f(x)$

Expand $f(x)$ using the eigenfunctions:

$f(x) = \sum_{n=1}^{\infty} c_n \phi_n(x)$

Where the coefficients are:

$c_n = \frac{\int_a^b f(x) \phi_n(x) w(x) dx}{\int_a^b \phi_n^2(x) w(x) dx}$

Here, we deal with the initial condition provided for our PDE setup. The initial state of the system is expressed using the eigenfunctions that were found in the previous step. This is done by expanding a given function, $f(x)$, as an infinite series of eigenfunctions, with each eigenfunction weighted by a coefficient $c_n$. These coefficients are calculated by taking an inner product between the function $f(x)$ and the eigenfunctions, normalized by the integral of the square of the eigenfunctions. This step is crucial as it establishes the connection between the initial condition and the solution space created by the eigenfunctions.

Imagine you're trying to recreate a painting by a famous artist. You take parts of different known paintings (the eigenfunctions) to represent your complete image (the initial condition). The coefficients represent how much of each painting you need. By calculating these weights accurately, you can ensure your recreated artwork closely resembles the original style.

From separation of variables:

$\frac{dT}{dt} + \lambda_n T = 0 \Rightarrow T_n(t) = c_n e^{-\lambda_n t}$

In this step, we focus on the time component of the PDE obtained during separation of variables. The resulting first-order ordinary differential equation is solved to obtain the time-dependent solution. It yields an exponential decay form for $T(t)$, where the decay rate is determined by the eigenvalue $\lambda_n$. The $c_n$ here are constants that can be determined based on the specific initial and boundary conditions of the problem. This results in a dynamic description of how the system evolves over time.

Think of this step like watching a candle burn. The time-dependent part (the candle's height over time) can be modeled using an exponential function. As the candle burns down (time evolves), its height diminishes at a rate that depends on how quickly it burns (the eigenvalue). Just as each candle burns differently, depending on its size and the material it’s made of, each mode of the overall system behaves according to its specific eigenvalue.

Combine the solutions:

$u(x,t) = \sum_{n=1}^{\infty} c_n e^{-\lambda_n t} \phi_n(x)$

This is the eigenfunction expansion solution of the PDE.

The final step involves combining all the pieces obtained from the previous steps to express the overall solution of the original PDE. The solution is represented as an infinite series where each term incorporates the corresponding spatial eigenfunction, weighted by the time-dependent decay term derived from the eigenvalues and the coefficients calculated for the initial condition. This series represents the complete solution of the system over time and space.

Imagine assembling a complex LEGO structure. Each separated piece (eigenfunction) contributes to the whole structure (solution) when combined with the correct joint (coefficient) to hold each piece together. The final assembled structure represents all the interactions together as time progresses; just as each LEGO piece shows its unique role in the overall construction, each term in the series contributes to the complete solution.

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