Problem 13 - Advanced Engineering Mathematics

Question

Use an appropriate Fourier integral transform to solve the given boundary- value problem. Make assumptions about boundedness where necessary. Find the steady-state temperature \(u(x, y)\) in a plate defined by \(x \geq 0, y \geq 0\) if the boundary \(x=0\) is insulated and, at \(y=0\) $$ u(x, 0)=\left\\{\begin{array}{lr} 50, & 01 \end{array}\right. $$

Step-by-Step Solution

Answer

Use a Fourier sine transform to solve Laplace's equation and meet the boundary conditions.

Step 1: Understand the Problem

We are tasked with finding the steady-state temperature distribution in a half-plane, defined for \(x \geq 0\) and \(y \geq 0\). The boundary conditions are an insulated boundary at \(x=0\) and a Dirichlet condition at \(y=0\). The goal is to use a Fourier integral transform to solve this problem.

Step 2: Set Up the Problem

The problem setup involves solving Laplace's equation \( abla^2 u = 0 \) in the half-plane, subject to the given boundary conditions. This implies solving \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \) for \(x \geq 0\), \(y \geq 0\) with the given boundary conditions.

Step 3: Use Fourier Transform

Apply the Fourier sine transform with respect to \(y\) because \(y\) ranges from 0 to \(\infty\). The Fourier transform of the partial differential equation (PDE) leads to an ordinary differential equation (ODE) in terms of the transformed variable \(U(k, x)\). This simplifies solving the boundary-value problem.

Step 4: Solve Fourier Transformed Equation

With Fourier-transformed ODE \( \frac{d^2 U}{dx^2} - k^2 U = 0 \), solve this second-order differential equation. The general solution is \( U(k, x) = A(k) e^{-kx} + B(k) e^{kx} \). To ensure boundedness as \(x \to \infty\), we discard the \( B(k) e^{kx} \) term.

Step 5: Apply Boundary Conditions

Using the boundary condition at \(y=0\), transform \(u(x, 0)=f(x)\) into the Fourier domain, taking the transform of \( f(x) \) given as \( f(x) = 50 \) for \(0 < x < 1\) and \(0\) for \(x > 1\). This results in \(A(k) = \int_0^1 50 e^{-ikx} dx \).

Step 6: Perform Inverse Fourier Transform

The steady-state temperature \(u(x, y)\) is obtained by performing an inverse Fourier sine transform on \(U(k, x)\). This integral reconstructs \(u(x, y)\) from its frequency components and solves the original boundary-value problem.

Step 7: Write Final Solution

The solution \(u(x, y)\) after inverse transformation is an integral over all frequency components, representing the temperature distribution in the plate in terms of \(x\) and \(y\). Specific integration steps and simplifications yield the final function.

Key Concepts

steady-state temperature distribution

The concept of a steady-state temperature distribution is central to understanding how heat behaves in a given material over time, particularly when conditions have reached equilibrium. In our problem, we are investigating the steady-state temperature in a semi-infinite plate. This means the temperature at every point in the plate remains constant over time, provided there are no changes in external conditions.

To achieve a steady state, the heat input through any part of the plate must equal the heat output. This equilibrium results in a stable temperature distribution that we want to evaluate. We are interested in finding out how the temperature varies across the plate, given an initial heat input at the boundary.

When focusing on a steady state, any time-dependent changes or fluctuations are ignored, letting us focus purely on spatial variables like the plate's physical boundaries and the heat applied.

Laplace's equation

Laplace's equation is essential in mathematical physics and engineering, particularly when solving problems related to steady-state phenomena such as our heat distribution problem. This equation is a second-order partial differential equation (PDE) represented as: \[ abla^2 u = 0 \]

For two-dimensional spaces, it expands to: \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \]

This form of the equation describes a field where there is no net change, such as a steady-state temperature situation without internal sources or sinks. Solving Laplace's equation provides us with the temperature distribution across the plate, considering the conditions set by the boundaries.

The importance of solving this equation lies in its ability to define how temperature varies in the space, allowing for design and analysis of thermal systems where equilibrium is desired.

boundary-value problem

Boundary-value problems involve solving differential equations with specified values or expressions known as boundary conditions. These conditions dictate the behavior of the solution at the boundaries of the domain.

In the given exercise, the boundary-value problem consists of Laplace's equation alongside boundary specifications: the boundary at \(x=0\) is insulated and hence does not allow heat flow, and the condition at \(y=0\) specifies distinct temperatures based on the location along the x-axis.

Solving this type of problem typically involves transforming the problem, simplifying it, and applying the given boundary conditions to determine the solution's behavior. Successful resolution tells us the inside values of the domain based on the behavior specified at the boundary, essentially giving us a complete view of the temperature distribution from the edges inward.

Fourier sine transform

The Fourier sine transform is a powerful tool for solving partial differential equations, especially when working with infinite or semi-infinite domains like our half-plane plate problem. This mathematical technique transforms a function into its frequency components, easing the process of solving differential equations.

In this particular problem, we use the Fourier sine transform with respect to \(y\), as \(y\) extends from 0 to \(\infty\). The transformation simplifies the original problem into a more manageable ordinary differential equation (ODE) by focusing on the sinusoidal components of the solution.

After transforming the problem using the Fourier sine transform, the solution involves handling a simpler equation in the frequency domain. We then perform an inverse Fourier transform to return to the spatial domain, retrieving the solution that describes the steady-state temperature distribution \(u(x, y)\). This method is especially handy in physics and engineering, where analytical solutions are necessary for complex boundary conditions.