# What is Heat equation: Definition and 282 Discussions

In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.
As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations. The heat equation can also be considered on Riemannian manifolds, leading to many geometric applications. Following work of Subbaramiah Minakshisundaram and Åke Pleijel, the heat equation is closely related with spectral geometry. A seminal nonlinear variant of the heat equation was introduced to differential geometry by James Eells and Joseph Sampson in 1964, inspiring the introduction of the Ricci flow by Richard Hamilton in 1982 and culminating in the proof of the Poincaré conjecture by Grigori Perelman in 2003. Certain solutions of the heat equation known as heat kernels provide subtle information about the region on which they are defined, as exemplified through their application to the Atiyah–Singer index theorem.The heat equation, along with variants thereof, is also important in many fields of science and applied mathematics. In probability theory, the heat equation is connected with the study of random walks and Brownian motion via the Fokker–Planck equation. The Black–Scholes equation of financial mathematics is a small variant of the heat equation, and the Schrödinger equation of quantum mechanics can be regarded as a heat equation in imaginary time. In image analysis, the heat equation is sometimes used to resolve pixelation and to identify edges. Following Robert Richtmyer and John von Neumann's introduction of "artificial viscosity" methods, solutions of heat equations have been useful in the mathematical formulation of hydrodynamical shocks. Solutions of the heat equation have also been given much attention in the numerical analysis literature, beginning in the 1950s with work of Jim Douglas, D.W. Peaceman, and Henry Rachford Jr.

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1. ### Model CO2 diffusing across the wall of a cylindrical alveolar blood vessel

TL;DR Summary: Solve heat equation in a disc using fourier transforms Carbon dioxide dissolves in the blood plasma but is not absorbed by red blood cells. As the blood returns to an alveolus, assume that it is well-mixed so that the concentration of dissolved CO2 is uniform across a...
2. ### I Heat Equation: Solve with Non-Homogeneous Boundary Conditions

Imagine you have a plane wall with constant thermal conductivity, that is the intermediate between two thermal reservoirs: The reservoir on the left is being kept at temp ##T_s##, and it is a fluid that has very high convective coefficient ##h##. As a result, the boundary condition at the...
3. ### I PDE - Heat Equation - Cylindrical Coordinates.

Would method of separation of variables lead to a solution to the following PDE? $$\frac{1}{r} \frac{ \partial}{\partial r} \left( kr \frac{ \partial T}{ \partial r}\right) = \rho c_p \frac{\partial T }{ \partial t }$$ This would be for the transient conduction of a hollow cylinder, of wall...
4. ### A Solve the heat equation having Dirichlet boundary conditions

I am going through these notes...they are pretty easy to follow. I would like more insight on the initial condition. In this problem, (attachment below), i guess the choice of initial condition is convenient as its easier to plug in the values of ##n=2## and ##b=3## (highlighted on the...
5. ### I Kappa factor in heat equation and heat Flux

I am doing a project, actually it is a simulation. And we aim to determine the spatial and heat flux evolution of the system. The system consists of two concentric cylinders separated by an insulating material. I change the value of kappa of the insulator but the heat flux remains always the...
6. ### A Can the Solution of a Heat Equation Only Get Smoother as Time Increases?

how to show or prove that the shape of the solution of a heat equation can only go smoother and smoother but not the opposite as time increases?
7. ### Steady state heat equation in a rectangle with a punkt heat source

I have checked several textbooks about the heat equation in a rectangle and I have found none that deals with my exact problem. I have though to use separation of variables first (to no avail), then Green's function (to no avail), then simplifying the problem for example by defining a new...
8. ### I Understanding relationship between heat equation & Green's function

Given a 1D heat equation on the entire real line, with initial condition ##u(x, 0) = f(x)##. The general solution to this is: $$u(x, t) = \int \phi(x-y, t)f(y)dy$$ where ##\phi(x, t)## is the heat kernel. The integral looks a lot similar to using Green's function to solve differential...
9. ### I General solution of heat equation?

We know $$K(x,t) = \frac{1}{\sqrt{4\pi t}}\exp(-\frac{x^2}{4t})$$ is a solution to the heat equation: $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$$ I would like to ask how to prove: $$u(x,t) = \int_{-\infty}^{\infty} K(x-y,t)f(y)dy$$ is also the solution to...
10. ### Solving the heat equation numerically using Python

I want to solve the heat equation numerically. The equation is: $$k \dfrac{\partial^2 T}{\partial x^2} = \dfrac{\partial T}{\partial t}.$$ This is a parabolic PDE. Following this pdf (specifically, equation 7 given on page 3), I wrote the following Python function to implement the explicit...
11. ### Thin disc with high thermal conductivity

I've tried to explicitly solve the Fourier's equation in cylindrical coordinates but I'm getting some messy integrals which cannot be solved analytically. Additionally my instructor said that there's a neat trick for this problem and it's possible to obtain the answer in a rather elementary...
12. ### Verify this function is a solution of the heat equation

I spent hours looking at this and cannot figure out where the error is. I'm wondering if there is an error before the boxed expression. @Orodruin and @PeroK may I ask for your assistance?Consider a solution ##u:[0,\infty)\times \mathbb{R}^n\rightarrow \mathbb{R}## of the heat equation, ie...

50. ### I PDE, heat equation lambda =,<,> 0 question

So I have been studying solving separation of variable, heat equation and came across 2 set of lambda equation. and lambda = 0 have the same equation. Is it different?