Heat equation

In mathematics and physics, especially in thermodynamics, the heat equation is a special kind of math problem. It helps us understand how things spread out over time. It was first created by a mathematician named Joseph Fourier in 1822.

Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. The height and redness indicate the temperature at each point. The initial state has a uniformly hot hoof-shaped region (red) surrounded by uniformly cold region (yellow). As time passes the heat diffuses into the cold region.

The heat equation shows us how heat, or energy, moves from places where there is a lot to places where there is less. This idea is important in many areas of science and math. It helps scientists and engineers solve many kinds of problems, from designing buildings to understanding how the Earth’s climate works.

Because it is so useful, the heat equation is studied in both basic math and many practical fields. It gives us a way to predict how things like temperature, sound, and some chemical processes will behave over time.

Definition

The heat equation is a way to show how heat moves through space and time. It was first made by Joseph Fourier in 1822 to help us understand how heat spreads out.

In simple terms, the heat equation shows how the temperature at any point changes over time. If temperatures are different next to each other, heat will move from the warmer area to the cooler one. This continues until everything is the same temperature. This idea is important in many areas of science and engineering.

∂ ∂ t v ( t , x ) = ∂ ∂ t u ( t α , x ) = α − 1 ∂ u ∂ t ( t α , x ) = ∇ 2 u ( t α , x ) = ∇ 2 v ( t , x ) {\displaystyle {\frac {\partial }{\partial t}}v(t,x)={\frac {\partial }{\partial t}}u\left({\frac {t}{\alpha }},x\right)=\alpha ^{-1}{\frac {\partial u}{\partial t}}\left({\frac {t}{\alpha }},x\right)=\nabla ^{2}u\left({\frac {t}{\alpha }},x\right)=\nabla ^{2}v(t,x)}

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Interpretation

Solution of a 1D heat partial differential equation. The temperature ( u {\displaystyle u} ) is initially distributed over a one-dimensional, one-unit-long interval (x = [0,1]) with insulated endpoints. The distribution approaches equilibrium over time.

The heat equation helps us learn how heat moves through things. It tells us that heat always moves from warm places to cooler ones. This movement changes based on how much warmer or cooler the places are and how well the material lets heat pass through.

When heat moves into something, that thing gets warmer. The heat equation shows how fast a spot gets warm or cool depends on how much the nearby areas are different in temperature. This helps explain why temperatures become the same in objects and spaces after some time.

Specific examples

The heat equation shows how heat spreads in materials. It was first made by Joseph Fourier in 1822 to explain how temperatures change in objects over time.

When heat moves through a thin, even rod, the heat equation comes from simple physics ideas. In easy cases, the equation shows how the temperature at each spot in the rod changes as heat moves from warm areas to cooler ones. This spreading makes temperatures more even over time.

The heat equation also works in three dimensions. It helps us learn how heat moves in objects with length, width, and height. By studying this equation, scientists can guess how temperatures will change in many different places.

Solving the heat equation using Fourier series

Joseph Fourier, a mathematician, found a way to solve the heat equation in 1822. This method helps us learn how heat moves through things like rods. The heat equation tells us how heat spreads out over time.

The equation uses two main things: x, which shows where you are on the rod, and t, which shows time. We start with a known amount of heat at the beginning and set conditions at the ends of the rod.

Fourier’s method, called separation of variables, makes the problem easier to solve. By doing this, we can find answers that fit the problem’s conditions. This method works for many kinds of equations and helps us understand how heat moves in different shapes and sizes.

u t = α u x x {\displaystyle \displaystyle u_{t}=\alpha u_{xx}}

u ( x , 0 ) = f ( x ) ∀ x ∈ [ 0 , L ] {\displaystyle u(x,0)=f(x)\quad \forall x\in [0,L]}

u ( 0 , t ) = 0 = u ( L , t ) ∀ t > 0 {\displaystyle u(0,t)=0=u(L,t)\quad \forall t>0} .

u ( x , t ) = X ( x ) T ( t ) . {\displaystyle u(x,t)=X(x)T(t).}

T ′ ( t ) = − λ α T ( t ) {\displaystyle T'(t)=-\lambda \alpha T(t)}

X ″ ( x ) = − λ X ( x ) . {\displaystyle X''(x)=-\lambda X(x).}

Fundamental solutions

Applications

The heat equation is an important idea in math and science. It helps us understand how heat moves through different materials. This idea started with the work of Joseph Fourier in 1822 and is used in many areas of math and science.

People use the heat equation in many ways. It connects to ideas about chance and how particles move in a liquid. It also helps in understanding how prices change in money markets and how to make images clearer. Scientists use it to study how heat moves in different shapes and materials, which helps make things like rubber and plastics better.