7.1.3. General Steps for the Method

1. Assume a separable solution: 𝑢(𝑥,𝑡) = 𝑋(𝑥)𝑇(𝑡) (or appropriate variables).

In this first step, we start by making an assumption about the form of the solution for the partial differential equation (PDE). Instead of looking for a complex function directly, we assume that the solution can be expressed as a product of two separate functions: one that depends only on the spatial variable (𝑥) and another that depends only on the temporal variable (𝑡). This simplifies our problem because we focus on each variable independently.

Think of trying to solve a cooking recipe. Instead of preparing the entire dish at once, you separate the tasks: first, you chop all the vegetables (focusing on one aspect), and then you cook them (focusing on another). This methodical approach helps ensure that each part is done correctly before combining them.

2. Substitute into the given PDE.

After stating the form of the solution, we substitute this product form into the original partial differential equation. This substitution replaces the complex relationship defined by the PDE with a simpler expression involving the product of functions, which is the essence of the separation of variables approach.

Returning to our cooking analogy, this step is like integrating the chopped vegetables into your pot as you prepare the dish. By combining ingredients (the functions), you can see how their characteristics change, which allows for a clearer understanding of how they interact within the overall dish (the PDE).

3. Separate variables, and equate both sides to a constant (say, −𝜆).

In this step, we manipulate the substituted equation to separate the variables completely. We aim to isolate terms that only depend on 𝑥 on one side and terms that depend only on 𝑡 on the other side. By doing this, we can introduce a separation constant (often denoted as -𝜆). This is crucial because it allows us to formulate two ordinary differential equations (ODEs), one for each independent variable.

Imagine sorting fruits into two baskets: one basket for apples (spatial variables) and another for bananas (temporal variables). By doing so, you ensure that each type of fruit, just like each variable in the equation, maintains its characteristics, making it easier to analyze and work with each category separately.

4. Solve each resulting ordinary differential equation.

Once we have the separated forms of the equations, we solve these ordinary differential equations individually. Each ODE can typically be solved using standard techniques, leading us to terms that describe the behavior of our original functions independently for both space and time.

This is like completing the recipes for your chopped vegetables and cooked components separately. You follow each set of instructions without interruptions, ensuring that each dish component is perfected before combining them back together in the final meal.

5. Apply boundary and initial conditions to determine constants and eigenvalues.

After solving the ODEs, we use specific conditions called boundary and initial conditions to fine-tune our solutions. These conditions help us determine any constants or eigenvalues that arise during the process. They are critical for ensuring that the solutions are not just mathematical abstractions but align with the physical context of the problem we are solving.

This step is akin to tasting and adjusting the seasoning of your dish—adding salt or spices until it meets the desired flavor profile. In mathematical terms, boundary and initial conditions ensure that our solution meets the physical requirements of the problem at hand.

6. Construct the final solution by summing (or integrating) the product solutions if needed: ∞ 𝑢(𝑥,𝑡) = ∑𝑋 (𝑥)𝑇 (𝑡) 𝑛 𝑛 𝑛=1

In the final step, if necessary, we combine all the individual product solutions from our ODEs to form the complete solution to the original PDE. This is often represented as a series summation, showcasing how various modes or components contribute to the overall solution. It might also involve integrating over specific intervals depending on the problem's nature.

This step is like bringing together all the cooked components of the meal and serving them together on a plate. You take individual parts that have been perfected in isolation and combine them into a coherent and flavorful dish that represents the whole.

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