18.2.1. General Procedure

1. Assume a solution of the form: u(x,t)=X(x)T(t)

In the method of separation of variables, the first step is to assume that the solution to the partial differential equation (PDE) can be expressed as a product of two functions. The function u(x,t) represents the solution that depends on two variables: x and t. By setting u(x,t) as the product of X(x), which depends only on x, and T(t), which depends only on t, we simplify our task of solving the PDE. This approach allows us to tackle the problem in a more manageable way, focusing on one variable at a time.

Imagine you're trying to solve a puzzle that’s too complicated to just look at once. Instead, you break it down into smaller sections. By focusing on one piece at a time (like focusing on X(x) and T(t)), it becomes easier to manage and eventually solve the entire puzzle.

2. Substitute into the PDE.

After assuming the solution has the form u(x,t) = X(x)T(t), the next step is to substitute this form into the original PDE. This means replacing u in the PDE with X(x)T(t) and deriving the new equation resulting from this substitution. This new equation will typically involve partial derivatives of T(t) with respect to time and X(x) with respect to space. The goal here is to create an equation where each side can be separated into functions of a single variable.

Think of substituting values into a recipe. If a recipe calls for 'x' cups of flour, and you've decided to break down the recipe into parts, you would replace 'x' in the recipe with the specific amount you need in that particular step. This makes the process clearer and allows you to focus on individual ingredients, just like we are focusing on the variables in our PDE.

3. Separate the equation such that each side depends on only one variable: 1 dT 1 d2X = =−λ T(t) dt X(x) dx2 • where λ is the separation constant.

In this step, we rearrange the substituted PDE to isolate terms that only involve T(t) on one side and those that only involve X(x) on the other side. This is critical because it leads to two ordinary differential equations (ODEs). The introduction of the separation constant λ allows us to state that the relationship between the two variables is proportional, enabling us to solve them independently. This separation is key to simplifying the problem and making it more tractable.

Consider a seesaw balanced in the middle. The left side only contains the weight of the left child (representing X(x)), and the right side only contains the weight of the right child (representing T(t)). By adjusting the weights on either side, you can analyze how each child affects the seesaw independently, just like how we separate variables to understand their individual contributions.

4. Solve the resulting ODEs: • One in x • One in t

Once we have successfully separated the PDE into two distinct equations—one involving only the spatial variable x and the other involving the temporal variable t—we can now solve these ordinary differential equations (ODEs) independently. This involves applying standard techniques for solving ODEs, which may include methods like integrating factors, characteristic equations, or even known solutions for specific types of equations. This step is crucial, as it provides us with functions that describe the behavior of our system over time and space.

Think of a music piece that is divided into sections for different instruments. Each instrument plays its part independently, allowing musicians to focus on their specific sections before bringing the entire composition together. Similarly, we solve each ODE separately to understand different aspects of the original problem.

5. Apply boundary conditions to determine allowed values of λ and corresponding eigenfunctions.

After finding the solutions to our ODEs, we now need to apply the boundary conditions relevant to our original problem to find specific values for the separation constant λ and the eigenfunctions associated with these values. These boundary conditions are essential because they ensure that the solution meets the physical realities of the problem—like temperatures being set at certain values at specific points or the behavior of a system at its edges. This step often yields a discrete set of allowed values for λ, which corresponds to unique solutions that fit the original PDE.

Imagine a gardener who has specific plots of land defined by fences (the boundary conditions) that dictate where flowers can grow. Each flower type may only thrive in certain segments of the garden. Similarly, applying boundary conditions helps us discover where our mathematical solutions can 'grow' and fit into the larger context of our problem.

6. Construct the general solution as a series using the principle of superposition.

The final step in the separation of variables procedure is to construct the general solution to the original PDE using the solutions derived from the ODEs and the identified eigenfunctions. This is done through a process known as superposition, where we add together these individual solutions, each scaled by corresponding coefficients based on the initial conditions. This combined solution represents the overall behavior of the system over the defined domain and satisfies all imposed boundary and initial conditions.

Think of creating a smoothie by blending different fruits together. Each fruit adds its unique flavor and contributes to the overall taste of the smoothie. By combining them, you create a delicious mix that is greater than the sum of its parts, much like how we use the principle of superposition to combine various solutions into one comprehensive answer.

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