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Chapter 13: Problem 12

Set up the boundary-value problem for the steady-state temperature \(u(x, y)\). A semi-infinite plate coincides with the region defined by \(0 \leq x \leq \pi, y \geq 0\). The left end is held at temperature \(e^{-y}\), and the right end is held at temperature \(100^{\circ}\) for \(01\). The bottom of the plate is held at temperature \(f(x)\).

Short Answer

Expert verified
Boundary-value problem is formulated using Laplace's equation and specified boundary conditions.

Step by step solution

01

Define the boundaries of the region

The semi-infinite plate lies within the region where the coordinates satisfy the conditions: \(0 \leq x \leq \pi\) and \(y \geq 0\). These define the extents of the plate for the problem.
02

Establish the governing equation

The steady-state temperature distribution \(u(x, y)\) in such a plate is governed by the Laplace equation for two dimensions: \[\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\]. This equation will be satisfied in the interior of the specified region.
03

Identify boundary conditions on edges of the plate

For boundary conditions: 1. At \(x = 0\), the left edge is held at \(u(0, y) = e^{-y}\).2. At \(x = \pi\), the right edge's temperature is \(u(\pi, y) = 100\) for \(0 < y \leq 1\) and zero for \(y > 1\).3. At \(y = 0\), the bottom edge's temperature is \(u(x, 0) = f(x)\).
04

Summary of the boundary-value problem setup

The problem setup consists of solving \[\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\] for \(0 \leq x \leq \pi\) and \(y \geq 0\), with boundary conditions:- \(u(0, y) = e^{-y}\) on \(x = 0\).- \(u(\pi, y) = 100\) for \(0 < y \leq 1\) and \(u(\pi, y) = 0\) for \(y > 1\).- \(u(x, 0) = f(x)\) on \(y = 0\).

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Key Concepts

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

Laplace Equation
The Laplace equation is a fundamental partial differential equation that plays a crucial role in many areas of science and engineering. It is represented as \(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\). This equation is used to describe steady-state scenarios, where a field (like temperature) does not change with time.
  • The equation is applicable in regions where there is no generation or absorption of energy.
  • It assumes that any point's value is the average of its immediate surroundings.
For a given region, like our semi-infinite plate, solving the Laplace equation helps us understand how temperature distributes across the plate without changing over time.
Steady-State Temperature
In our context, steady-state temperature refers to a condition where the temperature distribution does not change over time. This is a result of the system reaching thermal equilibrium.
  • This means that the amount of heat entering any area equals the amount leaving it.
  • In a steady-state condition, the temperature field is time-invariant. Hence, it simplifies our calculations as we are only concerned with the spatial distribution.
Understanding steady-state temperature is important, as it allows for analyzing various heat flow problems in materials like the semi-infinite plate described in this problem.
Boundary Conditions
Boundary conditions are essential when solving partial differential equations like the Laplace equation. They define the behavior of a solution at the boundary of the region.
  • In our problem, they specify the temperatures at the edges of the plate, guiding the problem-solving process.
  • The boundaries given are: \(u(0, y) = e^{-y}\) on the left edge, \(u(\pi, y) = 100\) for \(0 < y \leq 1\) and \(u(\pi, y) = 0\) for \(y > 1\) on the right edge, and \(u(x, 0) = f(x)\) at the bottom of the plate.
By adhering to these conditions, we can determine the complete temperature distribution across the plate.
Semi-Infinite Plate
A semi-infinite plate is a theoretical concept where one dimension of the plate extends to infinity, while other dimensions are finite. In this exercise, it is defined by the region \(0 \leq x \leq \pi\) and \(y \geq 0\).
  • The nature of a semi-infinite plate allows for simplifying assumptions, such as extending infinitely in one direction.
  • This means that solutions involving a semi-infinite plate often focus on the area of interest, as boundaries are set over a finite part of the region.
In problems like ours, using the concept of a semi-infinite plate helps in examining the effects of extended boundary conditions on temperature distribution without the complexities of a truly infinite domain.

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Most popular questions from this chapter

The vertical displacement \(u(x, t)\) of an infinitely long string is determined from the initial-value problem $$ \begin{aligned} &a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}, \quad-\infty0 \\ &u(x, 0)=f(x),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=g(x) \end{aligned} $$ This problem can be solved without separating variables. (a) Show that the wave equation can be put into the form \(\partial^{2} u / \partial \eta \partial \xi=0\) by means of the substitutions \(\xi=x+a t\) and \(\eta=x-a t .\) (b) Integrate the partial differential equation in part (a), first with respect to \(\eta\) and then with respect to \(\xi\), to show that \(u(x, t)=F(x+a t)+G(x-a t)\), where \(F\) and \(G\) are arbitrary twice differentiable functions, is a solution of the wave equation. Use this solution and the given initial conditions to show that $$ \begin{aligned} &F(x)=\frac{1}{2} f(x)+\frac{1}{2 a} \int_{x_{0}}^{x} g(s) d s+c \\ &\text { and } \quad G(x)=\frac{1}{2} f(x)-\frac{1}{2 a} \int_{x_{0}}^{x} g(s) d s-c, \end{aligned} $$ where \(x_{0}\) is arbitrary and \(c\) is a constant of integration. (c) Use the results in part (b) to show that \(u(x, t)=\frac{1}{2}[f(x+a t)+f(x-a t)]+\frac{1}{2 a} \int_{x-a t}^{x+a t} g(s) d s .\) (14) Note that when the initial velocity \(g(x)=0\) we obtain $$ u(x, t)=\frac{1}{2}[f(x+a t)+f(x-a t)],-\infty

Classify the given partial differential equation as hyperbolic, parabolic, or elliptic.\(\frac{\partial^{2} u}{\partial x^{2}}-\frac{\partial^{2} u}{\partial x \partial y}-3 \frac{\partial^{2} u}{\partial y^{2}}=0\)

Use separation of variables to find, if possible, product solutions for the given partial differential equation. $$ k \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t}, k>0 $$

A rod of length \(L\) coincides with the interval \([0, L]\) on the \(x\) -axis. Set up the boundary-value problem for the temperature \(u(x, t)\) The ends are insulated, and there is heat transfer from the lateral surface of the rod into the surrounding medium held at temperature \(50^{\circ} .\) The initial temperature is \(100^{\circ}\) throughout.

A string is stretched and eecured on the \(x\)-axis at \(x=0\) and \(x=\pi\) for \(t>0\). If the transverse vibrations take place in a medium that imparts a resistance proportional to the instantaneous velocity, then the wave equation takes on the form $$ \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}+2 \beta \frac{\partial u}{\partial t}, \quad 0<\beta<1, \quad t>0 $$ Find the displacement \(u(x, t)\) if the string starts from rest from the initial displacement \(f(x)\).
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