Question

All sides of the plate of Sect. \(19.5 .2\) are held at \(T=0\). Find the temperature distribution for all time if the initial temperature distribution is given by (a) \(T(x, y, 0)=\left\\{\begin{array}{ll}T_{0} & \text { if } \frac{1}{4} a \leq x \leq \frac{3}{4} a \text { and } \frac{1}{4} b \leq y \leq \frac{3}{4} b, \\ 0 & \text { otherwise. }\end{array}\right.\) (b) \(T(x, y, 0)=\frac{T_{0}}{a b} x y, \quad\) where \(0 \leq x

Short answer

The solutions to the problem provided form a series of sine waves for part (a), a series of exponential functions for part (b) and a solution to a 1-D heat conduction problem for part c. Each of the temperature distributions are reliant on the initial conditions specified in each part of the problem.

Step-by-step solution

Part (a) Solution

We have a plate with boundaries at \(x=a\), \(x=0\), \(y=b\), and \(y=0\). So, we know the solution will be a Fourier series of sine functions (since boundaries are at zero).The first step is to write down the heat equation:\[ \frac{\partial T}{\partial t} = \alpha \left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) \]Next, using the initial condition \(T(x, y, 0)\) as given in the problem statement and the fact that \(T(x, y, t)\) must go to zero as \(x\) or \(y\) goes to \(a\) or \(b\), respectively, we can find the coefficients by integrating the series obtained over \(x\) and \(y\) respectively from \(0\) to \(a\) and \(0\) to \(b\).Details of the procedure will depend on the exact form of the initial condition given.

Part (b) Solution

For \(T(x, y, 0)=\frac{T_{0}}{a b} x y\), this is a separable problem and can be solved using the method of separation of variables. We assume that \(T(x, y, t)=X(x)Y(y)T(t)\). We now insert the initial condition into the heat equation and make it a function only relying on time. Next, by normalizing the solution, the temperature distribution can be obtained through an infinite sum of terms. The final solution will be an expression dependent on \(x\), \(y\), \(t\), \(a\), \(b\) and a set of constants \(C_n\), \(D_n\) which are determined by the initial condition.

Part (c) Solution

For \(T(x, y, 0)=\frac{T_{0}}{a} x\), it is essentially the 1-D heat conduction problem with x varying from 0 to a. Since the function is only dependent on \(x\), we need to solve the 1-D heat equation using the method of separation of variables. First, solve the spatial part, then using the initial condition solve the time part. This should provide the desired temperature distribution.

Key concepts

Temperature Distribution

Temperature distribution refers to how heat spreads within an object over time. In the context of the heat equation, it predicts how the initial temperatures given in different scenarios evolve as time progresses. When considering a 2D plate with boundaries at zero temperature, the heat distribution changes according to initial conditions and boundary constraints. The goal is to find a function \(T(x, y, t)\) that represents the temperature at any point \((x, y)\) at any time \(t\).
Understanding temperature distribution requires:

• The initial temperature distribution at \(t = 0\). For example, a square area within a plate might be initially at a higher temperature than the adjacent areas.
• Boundary conditions, like holding the edges at zero temperature, which affect how the heat will dissipate.
• The heat equation, a partial differential equation representing how heat diffuses across space and time.

Managing these factors allows us to compute how temperature levels at various points will change over time, providing insights into the thermal behavior of materials.

Fourier Series

A Fourier series allows us to break down any periodic function into a sum of sines and cosines. In solving heat equations, Fourier series are fundamental. They help express complex temperature distributions in terms of simple trigonometric terms that can be easily manipulated.
Here's how it works:

• You express the initial temperature distribution as a series of sinusoidal functions.
• For a plate with zero boundary conditions, we use sine functions since they satisfy these conditions by default.
• Each term in the Fourier series represents a 'mode' of heat distribution — these modes are combined to model the full temperature profile.
• Solving for the coefficients in these sinusoidal terms involves integrating the initial conditions over the domain of interest.

This approach simplifies the task of solving partial differential equations, transforming them into manageable algebraic problems involving summation of series, paving the way for solving complex thermal processes efficiently.

Separation of Variables

Separation of variables is a mathematical method used to simplify partial differential equations, like the heat equation, by splitting them into simpler, ordinary differential equations. This technique assumes that the solution can be written as a product of functions, each depending only on a single variable:
\( T(x, y, t) = X(x)Y(y)T(t) \).
Here's the procedure:

• The heat equation is transformed, with each derivative corresponding to a separate equation related to one of the variables.
• Each of these simpler equations can be solved independently. For example, solve a spatial component first — find \(X(x)\) and \(Y(y)\).
• Then solve the time-dependent part for \(T(t)\).
• Integrating the solutions together, you reproduce the entire temperature distribution over time and space.
• Finally, constants of integration and initial conditions are applied to ensure the solution fits the physical scenario.

By using separation of variables, complex multi-variable equations become easier to tackle individually, allowing for a systematic solution of the heat distribution problem.