Question

Consider the problem $$ \begin{array}{r} e y^{\prime \prime}-y^{\prime}+y=0 \\ y(0)=\alpha, \quad y(1)=\beta, \quad y^{\prime}(1)=y \end{array} $$ (a) Show that boundary layers exist at both ends which are characterized by the stretching transformations $$ \eta=x / \epsilon \text { and } \zeta=(1-x) / \epsilon $$ (b) Determine a second-order uniformly valid solution using the MMAE. (c) Determine a second-order expansion using the MCE by letting $$ y=F(x ; \epsilon)+G(\eta ; \epsilon)+H(\zeta ; \epsilon) $$ where \(G \rightarrow 0\) as \(\eta \rightarrow \infty\) and \(H \rightarrow 0\) as \(\zeta \rightarrow \infty\)

Short answer

Boundary layers occur at both ends due to transformations \( \eta \) and \( \zeta \). Use MAE for a uniformly valid solution and MCE for expansion accounting for boundary conditions.

Step-by-step solution

Characterize Boundary Layers

For part (a), we need to demonstrate the existence of boundary layers. First, examine the original differential equation:\[ e y'' - y' + y = 0 \]Boundary layers typically occur when there is a significant change in the behavior of a dynamic system in a small region. The stretching transformations \( \eta = x / \epsilon \) and \( \zeta = (1-x) / \epsilon \) suggest that as \( \epsilon \to 0 \), rapid changes near \( x = 0 \) (at \( \eta \)) and \( x = 1 \) (at \( \zeta \)) are expected, characterizing the boundary layers.

Apply Matched Asymptotic Expansions

For (b), to find a second-order uniformly valid solution using the Matched Asymptotic Expansions (MAE), we assume the outer solution, inner solution at the left boundary, and inner solution at the right boundary:1. **Outer Solution**: Solve the equation by letting \( \epsilon \rightarrow 0 \), giving the reduced equation \( -y' + y = 0 \). This simplifies to \( y = C e^x \), where \( C \) is a constant.2. **Inner Solution at \(x = 0\)**: Substitute \( \eta = x / \epsilon \) into the original differential equation and solve the resultant equation, considering \( \eta \rightarrow \infty \).3. **Inner Solution at \(x = 1\)**: Substitute \( \zeta = (1-x) / \epsilon \) and solve analogously, evaluating at \( \zeta \rightarrow \infty \).4. Use boundary conditions to find any constants in the solutions and match the inner and outer solutions for a consistent solution across the entire domain.

Use Composite Expansion

For (c), a second-order expansion using the Method of Composite Expansions (MCE) involves:1. Expressing the solution as \( y = F(x; \epsilon) + G(\eta; \epsilon) + H(\zeta; \epsilon) \).2. Each part handles different scales: \( F \) is the regular perturbation for the outer solution, \( G \) manages the inner expansion at the boundary near \( x = 0 \), and \( H \) for the boundary condition near \( x = 1 \).3. Collect terms from expanding each component to derive system behaviors as \( \epsilon \to 0 \), ensuring boundary conditions are satisfied and internal consistency between these expansions.

Key concepts

Perturbation Methods

Perturbation methods are invaluable tools in applied mathematics for handling problems that cannot be solved by exact methods. The primary idea behind perturbation methods is to start with a simple problem that approximates the more complex, original problem and gradually introduce corrections. This is typically done when a small parameter, such as \( \epsilon \) in our original exercise, controls the difference in behavior between a simple baseline problem and the full problem.

In our specific exercise, a perturbation method allows us to separate the simpler, dominant behavior from the difficult, detailed behavior. For instance, when \( \epsilon \to 0 \), rapid changes occur at the boundaries. To address this, perturbation methods help identify boundary layers and enable us to examine behaviors at different regions within the domain by transforming the problem strategically using small parameters.

Matched Asymptotic Expansions (MMAE)

Matched Asymptotic Expansions (MMAE) are a fascinating technique used within perturbation theory to find solutions in problems where different regions of the domain exhibit distinct behaviors. The method involves developing separate approximations—inner solutions for regions like boundary layers and an outer solution for the rest of the domain—and then matching these solutions in an overlap region.

In our differential equation, we used MMAE to identify and describe solutions within boundary layers at both ends of the domain. We consider an outer solution that works well away from boundaries and inner solutions that detail behaviors at the critical points \( x = 0 \) and \( x = 1 \). The outer solution becomes \( y = C e^x \), where \( C \) is the constant determined by boundary conditions. The inner solutions are derived by substituting transformations such as \( \eta = x / \epsilon \) and \( \zeta = (1-x) / \epsilon \). Eventually, all solutions are matched to ensure continuity and accuracy across the domain.

Composite Expansions

Composite Expansions provide a harmonious blend of outer and inner solutions, consolidating them into a single, unified expression. This process is crucial because it delivers a uniformly valid approximation across the entire problem domain, addressing both boundary layers and standard regions uniformly.

In the exercise, we express the solution as a sum, \( y = F(x; \epsilon) + G(\eta; \epsilon) + H(\zeta; \epsilon) \). Each term in this expansion handles different parts of the problem:

• \( F(x; \epsilon) \) is the outer solution.
• \( G(\eta; \epsilon) \) describes the left boundary layer.
• \( H(\zeta; \epsilon) \) focuses on the right boundary layer.

This construct smoothly transitions between different regions and assures that boundary conditions are met as \( \epsilon \) trends to zero.

Differential Equations

Differential equations are mathematical expressions that relate a function to its derivatives, reflecting how quantities change. They are critical in modeling real-world phenomena, rooted in physics, engineering, and economics, where processes can be naturally described by rates of change and accumulations.

In our original exercise, we explore a second-order linear differential equation that represents boundary layer phenomena. The differential equation \( e y'' - y' + y = 0 \) indicates a dynamic system where perturbation methods are applicable. The use of transformations and expansions is motivated by the specific conditions of this equation, such as the presence of small, but significant, correction terms signaled by \( e \). Solving such equations involves analyzing behavior under various conditions, which reveals the role of boundary layers and the asymptotic expansion methods to handle transitions and sharp changes efficiently.