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Chapter 7: Problem 14

Consider the Gractz problem for heat transfer in a pipe $$ \begin{gathered} r u^{\prime}+u^{\prime}+\lambda^{2} r\left(1-r^{2}\right) f(r) u=0 \\ \quad u(1)=0, \quad u(0)<\infty \end{gathered} $$ where \(\lambda \gg 1\) and \(f(r)>0\). (a) Determine an expansion valid away from \(r=0\) and \(r=1 ;\) (b) determine expansions valid near \(r=0\) and \(r=1 ;\) (c) match these three expansions, hence determine \(\lambda\), and form a uniformly valid composite expansion (Sellers, Tribus, and Klein, 1956).

Short Answer

Expert verified
Solve the problem by finding three expansions for different regions, match them to determine \( \lambda \), then form a composite solution incorporating all regions.

Step by step solution

01

Understand the Problem

We have a differential equation involving a function \( u(r) \), which depends on \( r \), with a large parameter \( \lambda \). We need to find solutions (expansions) for \( u(r) \) in three regions for the radial coordinate \( r \). These are away from \( r = 0 \), near \( r = 0 \), and near \( r = 1 \).
02

Expansion Away from Singularities

Consider the case when \( r \) is neither near 0 nor 1. The dominant balance occurs with the terms \( r u' + u' + \lambda^2 r(1-r^2)f(r)u = 0 \). Since \( \lambda \) is large, scale \( u(r) = u_0(r) + \frac{u_1(r)}{\lambda} + \ldots \). The leading order gives \( \lambda^2 r(1-r^2)f(r)u_0 = 0 \), so \( u_0 = 0 \). Next order will introduce corrections as needed later.
03

Expansion Near r=0

For \( r \approx 0 \), the terms involving the small \( r \) and the large \( \lambda \) must be resolved. Assume \( u(r) = v(s), \; s = \lambda r \). Substitute into the differential equation considering dominant balance and expand to solve asymptotically. The result will be given similarly as a series in \( \frac{1}{\lambda} \). Provide \( v(s) = s \cdot \text{functions of } (s, \lambda) \).
04

Expansion Near r=1

Near \( r = 1 \), let \( r = 1 - \epsilon \), where \( \epsilon \) is small, and rescale the radial part accordingly with a stretched variable \( \epsilon = \frac{x}{\lambda} \). This will dominate the behavior as \( r \to 1 \). Solve resulting equation by expansion \( u(x) = A \cdot g(x, \lambda) \), determining functions of \( x \).
05

Match and Determine \( \lambda \)

To ensure these solutions are consistent and match between the regions, apply matched asymptotic expansions: match outer solution with inner near \( r = 0 \) and inner near \( r = 1 \). Since leading order outside is \( u_0 = 0 \), corrections from both inner zones give the constraint to determine \( \lambda \).
06

Composite Solution

Formulate a uniformly valid composite expansion by combining expressions from Steps 2, 3, and 4, ensuring consistency across overlap regions. Include terms like outer plus correction from both inner expansions minus double counted inner leading orders. Define the composite function so it approaches correct limits at critical \( r \) and uniformly combines solutions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Asymptotic Analysis
Asymptotic analysis is a powerful mathematical method used to approximate complex equations, especially in contexts where parameters are very small or very large. In this case, we are considering the Gractz problem for heat transfer, where the parameter \( \lambda \) is very large (\( \lambda \gg 1 \)). By using asymptotic analysis, we aim to break the problem down into simpler parts that are easier to solve.
Advantages of Asymptotic Analysis:
  • It offers a way to handle differential equations that are otherwise difficult or impossible to solve exactly.
  • By focusing on the behavior of the solution as \( \lambda \to \infty \), we identify dominant balances in various parts of the domain.
  • The solutions obtained can be highly accurate and reveal insights about the underlying mechanics or physics of the problem.
In practice, we find separate asymptotic expansions for different regions (in this exercise, away from \( r=0 \), near \( r=0 \), and near \( r=1 \)), and later match and combine them to form a complete solution.
Gractz Problem
The Gractz problem involves a differential equation related to heat transfer within a pipe. Specifically, it describes how heat distributes along the radial direction \( r \) under certain boundary conditions and a significant parameter \( \lambda \) affecting the equation.
Key Elements of the Gractz Problem:
  • The differential equation incorporates a function \( f(r) \) that is positive in the given context, indicating an involved heat conduction pattern.
  • Boundary conditions \( u(1) = 0 \) and \( u(0) < \infty \) impose constraints at specific locations (pipe wall and center, respectively).
  • The problem models behavior where the parameter \( \lambda \) is particularly large, further influencing the nature of the solution.
To solve this problem, it's essential to tackle different radial sections individually because equations behave differently near \( r=0 \), \( r=1 \), and in between.
Matched Asymptotic Expansions
Matched asymptotic expansions are a critical technique to bridge gaps between different expansions obtained for different regions of the problem domain. Once asymptotic solutions have been found for various segments (i.e., away from the singularities, near \( r=0 \), and near \( r=1 \)), they need to be "matched" or connected smoothly.
Purpose and Process:
  • This approach ensures that solutions overlap correctly in intermediate regions so that the complete solution behaves consistently across the entire domain.
  • The method involves identifying regions where different expansions overlap and adjusting parameters like \( \lambda \) to harmonize the behavior of solutions between regions.
  • For the Gractz problem, the inner solutions near \( r=0 \) and \( r=1 \) are aligned with the outer solution where the leading order from each region often aligns precisely.
Ultimately, this technique gives confidence that the asymptotic expansions are compatible and valid across the entire radial domain of interest.
Composite Expansion
A composite expansion is an elegant way to unify and construct a solution that is uniformly valid over the entire domain of the problem. After deriving asymptotic expansions for sections of the pipe, and matching them through asymptotic methods, these results are combined into a composite solution.
Characteristics of Composite Expansion:
  • It synthesizes insights from localized (inner and outer) expansions, ensuring it approximates the original differential equation well everywhere within the domain \( [0,1] \).
  • The composite solution avoids redundancies, ensuring that any over-counted leading order terms in the matched expansions are subtracted appropriately.
  • The approach is designed to naturally capture behavior in intermediate regions, smoothly transitioning from one asymptotic expansion to another.
The result is a practical approach to solving complex differential equations that balances precision with simplicity, providing an answer accessible even for large \( \lambda \) values, hence meeting the boundary conditions effectively.

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Most popular questions from this chapter

Consider the problem $$ \vec{u}-i\left[\omega_{1}(e t)+\omega_{2}(e t)\right] \hat{a}-\omega_{1}(e t) \omega_{2}(\epsilon t) u=f(\epsilon t) $$ where overdots denote differentiation with respect to \(t\) and \(e\) is a small parameter. Determine uniform asymptotic approximations to the solution of this equation if (a) \(f(\epsilon t)=0\), (b) \(f(\epsilon t)\) is a bounded nonperiodic function of \(t\), (c) \(f(\epsilon t)=k \exp (i \phi)\) with \(\phi=\lambda(\epsilon t) \neq \omega_{i}\), and (d) \(f(\epsilon t)\) has the same form as in (c) but \(\lambda=\omega_{1}\) at some \(t=t_{0^{-}}\)

Determine uniform asymptotic expansions for small e for the solutions of $$ y^{\prime \prime}+\frac{x(1+x)-\epsilon x}{(1+x)^{2}} y=0, \quad-1

Given that \(f(x)>0\) and \(\lambda \gg 1\), determine first approximations to the following eigenvalue problems (a) \(y^{\prime \prime}+\lambda^{2}(1-x) f(x) y=0\) \(y(0)=0\) and \(y(\infty)<\infty\) (b) \(x y^{\prime}+y^{\prime}+\lambda^{2} x f(x) y=0\) \(y(1)=0\) and \(y(0)<\infty\) (c) \(y^{*}+\lambda^{2}(1-x)^{n} f(x) y=0, \quad n\) is a positive integer \(y(0)=0\) and \(y(\infty)<\infty\) (d) \(x y^{\prime \prime}+y^{\prime}+\lambda^{2} x^{n} f(x) y=0, \quad n\) is a positive integer \(y(1)=0\) and \(y(0)<\infty\) (e) \(y^{*}+\lambda^{2}(x-1)^{n}(2-x)^{m} f(x) y=0, \quad m\) and \(n\) are positive integers \(y<\infty \quad\) for all \(x\).

Bessel functions \(J_{n}(n x), Y_{n}(n x), H_{n}^{(1,2)}(n x)\) are solutions of the differential equation $$ x^{2} y^{\prime \prime}+x y^{\prime}+n^{2}\left(x^{2}-1\right) y=0 $$ Determine a uniform asymptotic expansion for the general solution of this equation for large \(n\).
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