Mathematical modeling of heat flow

From: 3blue1brown

Mathematical modeling of heat flow involves understanding how temperature distribution changes over time, often studied in a simplified one-dimensional case like a rod [00:00:03]. This process provides a foundational example for an partial differential equation (PDE) [00:00:14].

The Heat Equation

The heat equation in the one-dimensional case describes that the rate at which temperature at a given point changes over time depends on the second derivative of that temperature at that point with respect to space [00:00:18]. Essentially, where there is curvature in space, there is a change in time [00:00:28].

Constraints in Solving the Heat Equation

Solving the heat equation is more complex than just satisfying the PDE itself, as it represents only one of three essential constraints for accurately describing heat flow [00:00:33]:

1. The PDE: The core equation relating temperature change over time to spatial curvature [00:00:38].
2. Boundary Conditions: Specific conditions that the temperature function must satisfy at the physical boundaries of the system [00:00:45].
3. Initial Condition: The initial temperature distribution at time t=0, which is not chosen but given [00:00:52].

These added constraints are where the primary challenge in solving the problem lies [00:00:57].

Joseph Fourier’s Contribution

In 1822, Joseph Fourier made a key contribution by developing a method to control the vast number of functions that solve the PDE and satisfy boundary conditions, allowing for the selection of a particular solution that fits a given initial condition [00:01:12]. His solution is based on three fundamental observations [00:01:27]:

1. Certain sine waves offer simple solutions to the heat equation [00:01:33].
2. If multiple solutions are known, their sum is also a solution [00:01:37].
3. Any function can be expressed as a sum of sine waves, potentially infinitely many [00:01:43]. This last idea is the essence of Fourier series [00:02:03].

Sine waves are particularly useful because their relatively clean second derivatives make it easier to solve the heat equation for each wave component [00:02:23]. Since a sum of solutions to the equation yields another solution, this provides a recipe for solving the heat equation for any complex initial temperature distribution [00:02:35].

Sine Waves and Exponential Decay in Solutions

Consider a simple initial temperature function u(x,0) = sin(x) for a rod [00:02:50].

• The second derivative of sin(x) with respect to x is -sin(x) [00:03:31].
• This means the amount the wave curves is equal and opposite to its height at each point [00:03:37].
• At time t=0, each point changes its temperature at a rate proportional to its own temperature, with a consistent proportionality constant [00:03:45].
• This implies that sine waves will be uniformly scaled down over time, maintaining their sine curve shape but looking like some constant times sin(x) for all times t [00:04:16].

When a value changes at a rate proportional to itself, it points to an exponential relationship [00:04:29]. The derivative of e^(kt) is k times e^(kt) [00:04:46]. For a sine wave, the right-hand side of the heat equation becomes -α times the temperature function itself, where α is a constant [00:05:27]. Therefore, the solution is proposed to be scaled down by a factor of e^(-αt) [00:05:36].

A function of x and t of the form sin(x) * e^(-αt) satisfies the partial differential equation [00:05:47]:

• The second partial derivative with respect to x is -sin(x) * e^(-αt) [00:06:10].
• The partial derivative with respect to t is -α * sin(x) * e^(-αt) [00:06:21].
• This confirms the function satisfies the heat equation [00:06:31].

Boundary Conditions for Heat Flow

The simple sine wave solution doesn’t fully describe actual heat flow because it doesn’t account for how the boundaries of the rod behave [00:06:52]. For example, if no heat flows into or out of the rod, the slope of the temperature curve at either end must be zero [00:08:50]. This is described precisely as the partial derivative of the temperature function with respect to x at 0T and LT (where L is the length of the rod) being zero for all times T > 0 [00:09:28]. This is a crucial “boundary condition” [00:09:41].

To incorporate boundary conditions:

• A cosine function cos(x) can be used instead of sin(x), as its second derivative is also -cos(x), and it naturally starts flat at x=0 [00:10:07]. So, cos(x) * e^(-αt) also satisfies the PDE [00:10:23].
• To satisfy the boundary condition at the right side (x=L), the frequency of the wave must be adjusted [00:10:35].
• Introducing a constant ω multiplied by x (e.g., cos(ωx)), the second derivative becomes -ω² * cos(ωx) [00:11:12].
• This means the exponential decay term in the solution must also include ω², becoming e^(-αω²t) [00:11:20]. This makes intuitive sense: sharper curves (higher frequency) decay more quickly towards equilibrium [00:11:30].
• For the rightmost point to be flat (slope zero) at x=L, ω must be chosen such that sin(ωL) = 0. The frequencies that satisfy this are integer multiples of π/L (i.e., ω = nπ/L, where n is a whole number) [00:11:46]. These are analogous to harmonics [00:12:09].

This leads to an infinite family of functions that satisfy both the PDE and the boundary conditions, each looking like a cosine curve in space and exhibiting exponential decay in time [00:12:31]. This framework then allows for constructing a more general solution in the next step [00:13:03].

General Themes in Differential Equations

The approach used in modeling heat distribution and change over time highlights several common themes in the field of differential equations [00:13:14]:

• Modeling boundaries with special rules distinct from the main differential equation for the interior is a regular theme, especially in PDEs [00:13:20].
• The strategy of breaking down general situations into simpler, idealized cases is ubiquitous [00:13:32].
• These simpler cases frequently involve mixtures of sine curves and exponentials, which is not unique to the heat equation [00:13:42].