Heat equation Definition and 282 Threads

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. user123abc

    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. E

    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. E

    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. chwala

    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. G

    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. F

    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. fluidistic

    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. L

    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. L

    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. Wrichik Basu

    Python 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. Rlwe

    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. D

    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...
  13. D

    Solving the heat equation in 1D

    (3) To solve the initial value problem $$\begin{cases} \partial_t\phi-\partial^2_x\phi=0 & \text{in}\quad (0,\infty)\times R \\ \phi(0,\cdot)=\psi & \text{on}\quad \{t=0\}\times R \end{cases}$$ we use the fundamental solution in 1D $$\Phi_1(t,x)=\frac{1}{\sqrt{4\pi...
  14. Shackleford

    A Solution of nonhomogeneous heat equation problem

    This is from Evans page 50. I'm sure it's something simple, but I don't follow the change from $$ \frac{\partial}{\partial t} \quad \text{to} \quad -\frac{\partial}{\partial s}$$ and from $$ \Delta_x \quad \text{to} \quad \Delta_y$$. \begin{gather*} \begin{split} u_t(x,t) - \Delta u(x,t) & =...
  15. K

    Help solving this Heat Equation please

    I want to solve the heat equation below: I don't understand where the expression for ##2/R\cdot\int_0^R q\cdot sin(k_nr)\cdot r \, dr## came from. The r dependent function is calculated as ##sin(k_nr)/r## not ##sin(k_nr)\cdot r##. I don't even know if ##sin(k_nr)/r## are orthogonal for...
  16. Neothilic

    I Fourier Integral of the Schrodinger Equation

  17. S

    Heat Equation with Periodic Boundary Conditions

    I'm solving the heat equation on a ring of radius ##R##. The ring is parameterised by ##s##, the arc-length from the 3 o'clock position. Using separation of variables I have found the general solution to be: $$U(s,t) = S(s)T(t) = (A\cos(\lambda s)+B\sin(\lambda s))*\exp(-\lambda^2 kt)$$...
  18. B

    Heat distribution in a piece of glass receiving protons

    First of all, I didn't know whether to pick this subforum or the engineering/compsci one, I understand this might need to be moved to a more appropriate subforum. The general approach is fairly obvious, use implicit method to construct the tridiagonal matrix for Thomas method and solve. However...
  19. patricio ramos

    Question about initial and boundary conditions with the heat equation

    I am seeing the heat conduction differential equation, and I was wondering about a boundary condition when the equation is of transient (unsteady) nature. When analyzing boundary conditions at the surface of say, a sphere, the temperature does not depend on time. For example, if you have...
  20. H

    A Crank-Nicholson solution to the cylindrical heat equation

    Hi, I am solving the radially symmetric heat equation with an internal heat source(this is meant to model the heating of a cylindrical battery). It's meant to model heat in a cylinder with conduction to the environment, so my outer boundary condition is Newton's law of cooling. The $T$ in the...
  21. Leonardo Machado

    A Boundary conditions for the Heat Equation

    Hello guys. I am studying the heat equation in polar coordinates $$ u_t=k(u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}) $$ via separation of variables. $$u(r,\theta,t)=T(t)R(r)\Theta(\theta)$$ which gives the ODEs $$T''+k \lambda^2 T=0$$ $$r^2R''+rR+(\lambda^2 r^2-\mu^2)R=0$$...
  22. F

    A 3D Heat equation with elementary Dirichlet BC

    Hi, I am looking for the solution of the following heat conduction problem (see figure below): the geometry is the semi-infinite domain such that (x,y)∈R2 and z∈[0,∞[ ; the thermal diffusivity is constant; the domain is initially at a temperature of 0; At t>0, a small square of the surface...
  23. H

    A Irreversible processes. Heat equation, diffusion equation

    Both the heat equation and the diffusion equation describe processes which are irreversible, because the equations have an odd time derivative. But how can these equations describe the real world when we know that all processes in nature are reversible, information is always conserved? But these...
  24. B

    I Solution of the 1D heat equation

    $$\frac{\partial T}{\partial t}=\alpha\frac{\partial^2 T}{\partial^2 t}$$ with an initial condition and boundary conditions $$T(x,0)=T_0$$ $$T(L,t)=T_0$$ $$-k\left.\frac{\partial T}{\partial x}\right|_{x=0}=2A\cos^2\left(\frac{\omega t}{2}\right)=A(\cos\omega t+1)$$ where $A=V_0^2/(8RhL)$...
  25. F

    I Heat Equation Problem: Solving c + 3d = 0

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  26. evinda

    MHB Find formula for solution heat equation

    Hello! (Wave) Let $\phi \in C^1(\mathbb{R})$ and periodic. We consider the problem $u_t=u_{xx}, x \in \mathbb{R}, \ 0<t<\infty$, with initial data $\phi$. I want to compute a formula for a solution $u$ and I want to prove strictly that this formula solves the initial value problem. I also...
  27. fluidistic

    I How come electrons and phonons lead to the heat equation?

    Phonons on their own lead to the common heat equation. One sees that for example in insulators or non doped semiconductors. However in metals (or conductors), the electrons are the ones that are mostly responsible for the heat transfer, which extremely surprisingly to me, is also of the form of...
  28. H

    MATLAB Crank-Nicholson solution of 1D heat equation

    I wish to numerically compute solutions of the 1D heat equation using the Crank-Nicholson scheme: The equation is: \partial_{t}u=\partial^{2}_{x}u I use the discretisation: u_{i+1,j}-u_{i,j}=s(u_{i+1,j+1}-2u_{i+1,j}+u_{i+1,j-1})+s(u_{i+1,j+1}-2u_{i+1,j}+u_{i+1,j-1}) Where s=\delta...
  29. M

    I What is the solution to the heat equation with a constant added?

    I have seen how to solve the heat equation: $$ \frac{ \partial^2 u(x,t) }{\partial x^2} = k^2 \frac{ \partial u(x,t) }{\partial t} $$ With boundary conditions. I use separation variables to find the result, but i don't know how to solve the equation plus a...
  30. S

    A Heat equation on infinite domain

    Hi everyone, I'm reading this paper about the solution of the heat equation inside an infinite domain: https://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006/lecture-notes/fourtran.pdf 1) Please let me know if the following discussion is correct. The...
  31. P

    Heat equation and stainless-steel wire

    Homework Statement A stainless-steel wire is 0.1 mm in diameter and 1 m long. If the outside of the wire is held fixed at 20◦C, estimate the steady-state current passing through the wire when the stainless steel at the centre of the wire begins to melt.Homework EquationsThe Attempt at a...
  32. P

    How Is Heat Transport in a Rod Influenced by Its Radius and Material Properties?

    Homework Statement show ## \rho c_m \frac{\partial T}{\partial t} = \kappa \frac{\partial^2T}{\partial x^2} -\frac{2}{a}R(T)## where ##R(T)=A(T-T_0) ## a) Obtain an expression for T as a function of x for the case of an infinitely long rod whose hot end has temperature ##T_m## b) Show that the...
  33. A

    Modeling analytical solution of 1D heat equation

    I am trying to write code for analytical solution of 1D heat conduction equation in semi-infinite rod. The analytical solution is given by Carslaw and Jaeger 1959 (p305) as $$ h(x,t) = \Delta H .erfc( \frac{x}{2 \sqrt[]{vt} } ) $$ where x is distance, v is diffusivity (material property) and t...
  34. A

    A Understanding dummy variable in solution of 1D heat equation

    The solution of 1D diffusion equation on a half line (semi infinite) can be found with the help of Fourier Cosine Transform. Equation 3 is the https://ibb.co/ctF8Fw figure is the solution of 1D diffusion equation (eq:1). I want to write a code for this equation in MATLAB/Python but I don't...
  35. J

    A What is the method for calculating the dampening of thermal oscillations?

    Hello, I am attempting to solve the 1 d heat equation using separation of variables. 1d heat equation: ##\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}## I used the standard separation of variables to get a solution. Without including boundary conditions right now...
  36. P

    Periodic BC's of heat equation

    Homework Statement I have the heat equation $$u_t=u_{xx}$$ $$u(0,t)=0$$ $$u(1,t) = \cos(\omega t)$$ $$u(x,0)=f(x)$$ Find the stable state solution. The Attempt at a Solution I used a transformation to complex to solve this problem, and then I can just take the real part to the complex...
  37. G

    General solution for the heat equation of a 1-D circle

    Homework Statement Modify the initial conditions (for the diffusion equation of a circle) to have the initial conditions ## g(\theta)= \sum_{n=-\infty}^{\infty}d_{n}e^{2\pi in\theta} ## Using the method of Green's functions, and ## S(\theta,t)= \frac{1}{\sqrt{4\pi...
  38. Telemachus

    I Heat equation with variable coefficients

    ##\displaystyle \frac{\partial u(x,t)}{\partial t}-p(x) \frac{\partial^2 u(x)}{\partial x^2}-\frac{\partial p(x)}{\partial x}\frac{\partial u(x)}{\partial x}=f(x,t)##. With periodic boundary conditions: ##u(x,t)=u(x+2\pi,t)##. The right hand side is also a periodic function of space, and it can...
  39. A

    Solving the heat equation using FFCT (Finite Fourier Cosine Trans)

    Homework Statement Solve the following heat Eq. using FFCT: A metal bar of length L is at constant temperature of Uo, at t=0 the end x=L is suddenly given the constant temperature U1, and the end x=0 is insulated. Assuming that the surface of the bar is insulated, find the temperature at any...
  40. F

    Trouble with Galilean transform problem heat equation

    Homework Statement 1. The common form of the heat-diffusion equation governing the temperature distribution $$\rho C_p \frac{\partial T}{\partial t}=k\nabla^2T$$ Is this equation valid in any inertial frame of reference? (i.e. does it have the property of Galilean invariance?) If not, can it...
  41. Vitani11

    Help with understanding BVP for the Heat equation (PDE)?

    Homework Statement Find the steady state (equilibrium) solution for the following boundary value problem: ∂u/∂t = (1/2)∂2u/∂x2 Boundary condition: u(0,t) = 0 and u(1,t) = -1 Initial condition: u(x,0) = 0 Homework Equations u(x,t) = Φ(x)G(t) The Attempt at a Solution I have found the solution...
  42. A

    Which PDE should I use to simulate different kinds of groundwater flow?

    I have learned that diffusion/heat equation can be used to model groundwater flow in confined conditions. Recently I read a paper where they used linear Boussinesq equation (equation 1 in linked paper) to model groundwater flow in unconfined aquifer. Then in another paper, the auther said, he is...
  43. maistral

    A RK method for heat equation with dependent variables

    Hello. How do I solve this equation without killing the k(y) term: I managed to derive an analytical solution for this one. I intend to run the numerical solution via Runge-Kutta but I can't stop myself from killing the k(y) term. I'm starting to think I'm doing something wrong... It goes...
  44. T

    Black Scholes heat equation form) Crank Nicolson

    Dear all: I am studying the finite difference method for black scholes equation(heat equation form), and my Prof. asked me to use the Crank Nicolson and Compact Scheme to do it. But I cannot reach second order for the space, i.e., x . The error only reach 1st order and sometimes not stable, it...
  45. dumbdumNotSmart

    Heat equation integral - Fourier Series coefficient is zero

    Homework Statement WE have a thermally insulated metallic bar (from enviroment/surroundings) . It has a temperature of 0 ºC. At t=0 two thermal sources are applied at either end, the first being -10 ºC and the second being 10 ºC. Find the equation for the temperature along the bar T(x,t), in...
  46. F

    Algorithmic differentiation in unsteady heat equation

    Homework Statement Okey her we go I was given a base code called heat_equation_primer. The goal is to implement a optimizer into the program. The two methods that are going to be used is the Quasi-Newton and Steepest descent with search line. So I need gradients. So I tried to differentiate as...
  47. P

    Verify orthogonality integral by direct integration

    This is a heat equation related math problem. 1. Homework Statement The complete question is: Verify the orthogonality integral by direct integration. It will be necessary to use the equation that defines the λ_n: κ*λ_n*cos(λ_n*a) + h*sin(λ_n*a)=0. Homework Equations κ*λ_n*cos(λ_n*a) +...
  48. N

    Change of variables in Heat Equation (and Fourier Series)

    Q: Suppose ##u(x,t)## satisfies the heat equation for ##0<x<a## with the usual initial condition ##u(x,0)=f(x)##, and the temperature given to be a non-zero constant C on the surfaces ##x=0## and ##x=a##. We have BCs ##u(0,t) = u(a,t) = C.## Our standard method for finding u doesn't work here...
  49. F

    Why is the wave equation different from the heat equation

    I have been thinking about this. For a wave equation, the acceleration of a point on a drumhead is proportional to the height of its neighbors $$U_{tt}=\alpha^2\nabla^2U$$ The heat equation, change in concentration or temperature is equal to the average of its neighbors...
  50. K

    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?
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