A cube is originally at \(100^{\circ}\). From \(t=0\) on, the faces are held at \(0^{\circ}\). Find the timedependent temperature distribution. Hint : This problem leads to a triple Fourier series; see the double Fourier series in Problem 9 and generalize it to three dimensions.

Heat conduction is the process by which heat energy is transferred within a material without any movement of the material itself. It occurs due to the temperature gradient within the material.
In the context of the given problem, a cube initially at 100°C cools down as heat flows from its interior to its surfaces, which are maintained at 0°C.
The rate at which heat is conducted depends on various factors, including the thermal conductivity of the material, the temperature difference, and the distance over which the temperature difference occurs. Understanding heat conduction is critical to solving the heat equation.

A Fourier series is a way to represent a function as the sum of simple sine waves. In solving the heat equation, the temperature distribution is expressed as a Fourier series because it helps break down complex functions into simpler, periodic components.
For the three-dimensional heat equation, we use a triple Fourier series to account for variations in temperature in all three spatial dimensions. The general form of the Fourier series for temperature in the cube is:
\[ T(x, y, z, t) = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \sum_{p=1}^{\infty} A_{nmp} \sin\left(\frac{n \pi x}{a}\right) \sin\left(\frac{m \pi y}{b}\right) \sin\left(\frac{p \pi z}{c}\right) e^{-(\lambda_{nmp}^2 \alpha t)} \].
This allows for an accurate description of how temperature changes over time within the cube.

Partial differential equations (PDEs) involve multiple independent variables and their partial derivatives. The heat equation is a PDE that describes the distribution of temperature in a given region over time.
The three-dimensional heat equation is given by:
\[ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) \].
Here, \(T\) is the temperature, \(t\) is time, and \(\alpha\) is the thermal diffusivity. Solving this PDE helps find the temperature distribution in the cube over time.

Boundary conditions specify the behavior of a solution at the boundaries of a domain. They are essential for solving PDEs as they define how the variables behave at the edges.
In the given problem, the boundary conditions are that the faces of the cube are held at 0°C from \(t = 0\) onwards. Mathematically, this is expressed as:
\[ T(0, y, z, t) = T(a, y, z, t) = 0 \]
\[ T(x, 0, z, t) = T(x, b, z, t) = 0 \]
\[ T(x, y, 0, t) = T(x, y, c, t) = 0 \].
These conditions ensure that the heat flows out to the surfaces and that the solution is consistent with the physical setup.

Thermal diffusivity, denoted by \(\alpha\), is a measure of how quickly heat spreads through a material. It is given by:
\[ \alpha = \frac{k}{\rho c_p} \]
where \(k\) is the thermal conductivity, \(\rho\) is the density, and \(c_p\) is the specific heat capacity of the material.
In the heat equation, thermal diffusivity appears in the term that governs the rate of temperature change:
\[ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) \].
High thermal diffusivity means heat spreads out quickly, while low thermal diffusivity means heat spreads out slowly.