Problem 12 - Advanced Engineering Mathematics

Question

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 \(f(x)\).

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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 \(u(x, 0) = f(x)\).

Step 4: 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 \(y = 0\).

Key Concepts

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.