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. R

    Heat Equation With Seperable Variables

    Homework Statement du/dt = d2u/dx2 + u Bc: u'(0) = u'(1) = 0 Ic: u(x,0) = 1 Homework Equations Using sturm-liouville to solve for eigenvalues. The Attempt at a Solution After first separating variables in the equation we get G'/G - 1 = F'' = λ after using Sturm-Liouville we...
  2. N

    Steady State 2-D Heat Equation with Mixed Boundary Conditions

    Homework Statement I am trying to solve the Laplacian Equation with mixed boundary conditions on a rectangular square that is 1m x 1m. Homework Equations \nabla2T=0 .....T=500K ....________ ....|@@@@| T=500K...|@@@@|...T=500K ....|@@@@| ....|______.| ....Convection ....dT...
  3. J

    Heat Equation 2D: Conservation of Energy

    Homework Statement heat equation of for 2 dimensional body(stationary)...heat is supplied to a body per unit volume and per unit time and by using conservation of energy principle the following equation is derived.. \intQt dA= \intqt d\ell the intergral on the right is a line integral on a...
  4. J

    Solution of Heat Equation Through Fourier Series.

    OK, so I was trying to solve the Heat Equation with Inhomogeneous boundary conditions for a rod through Fourier Series when I got stuck at the solution for the coefficient c_n, the part where I'm stuck is highlighted in red. The following is just a step-by-step solution of how I got to c_n...
  5. T

    How can the Heat Equation be solved for a periodic heating scenario?

    thanks allot they worked out fine, just another quick question if could help. A semi-infinite bar 0<x<infinity is subject to periodic heating at x=0; the temperature at x=0 is T_0cos\omegat and is zero at x=infinity. By solving the heat equation show that T(x,t)=...
  6. J

    Can the Non-Homogenous Heat Equation be Solved Using Eigenfunctions?

    Consider the following non-homogenous heat equation on 0 \leq x \leq \pi u_t = k u_{xx} - 1 with u(x,0) = 0, u(0,t) = 0, u(\pi, t) = 0 Find a solution of the form \displaystyle \sum_1^{\infty} b_n(t) \phi_n (x) where \phi_n(x) are the eigenfunctions of an appropriate homogenous...
  7. B

    ODE's for 2 space Heat equation

    Homework Statement The Heat equation in two space is \alpha ^2 \left[\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2} \right]=\frac{\partial u}{\partial t} Assuming separation solution of the form u(x,y,t)=F(x)G(y)H(t) find ordinary differential equations satisfied by...
  8. G

    1D Heat equation, numerical solution with ONLY one heat source

    Hi, I have the following problem. I am tried to numerically solve the 1D heat equation for a metal bar of length L. Using the forward time, centered space equation a(t+1) = a(t)+(alpha*deltaA/(deltaX)^2)*(a(x+1,t)-2*a(x,t)+a(x-1,t)) The problem is that I only have ONE heat source at...
  9. G

    Heat equation - theta function?

    du/dt=d2u/dx2 Show that u(x,t)=(t^a) * theta(xi) where xi=x/sqrt(t) and a is a constant, then theta(xi) satisfys the ODE a*theta - 0.5 * xi * dtheta/dxi = d2theta/dxi2 Not sure how to start this. Any help most appreciated (sorry if question isn't easy to ready)
  10. G

    Heat equation - separation of variables

    Homework Statement du/dt=d2u/dx2, u(0,t)=0, u(pi,t)=0 u(x,0) = sin^2(x) 0<x<pi Find the solution Also find the solution to the initial condition: du/dt u(x,0) = sin^2(x) 0<x<pi The Attempt at a Solution From separation of variables I obtain u(x,t) = B.e^(-L^2t).sin(Lx)...
  11. C

    Hom. heat equation in cylindrical coordinates using Fourier & Laplace transforms

    I'm trying to solve the homogeneous heat equation of a semi-infinite cylinder in cylindrical coordinates for a semi-infinite cable (no theta dependence): \frac{\partial U}{\partial t}=D\left(\frac{\partial^{2} U}{\partial r^{2}}+\frac{1}{r}\frac{\partial U}{\partial r}+\frac{\partial^{2}...
  12. S

    Non-Homogeneous Heat Equation (Insulated Bar Question)

    Homework Statement Find U(x,t) dU/dt = d2U/dx2 + sin x Boundary Conditions: dU/dx (0,t) = 0 and U(1,t) = 0 Initial Condition: U(x,0) = cos 7*π*x 2. The attempt at a solution I start off with: d2(Un)/dx2 = λnUn (as an initial value problem) [d(Un)/dx](0) = 0...
  13. E

    Heat Equation - Maximum Principle proof

    Hi, i'm just going through my lecture notes reading a proof of the maximum principle for the heat equation. It goes roughly like this: Maximum Principle: Heat equation: u_{t}=ku_{xx} u(x,0)=\Psi(x) x\in[0,L], t\in[0,T]. Given a C2 solution of the HE then the maximum u(x,t) is attained at...
  14. S

    Understanding the Physical Representation of Special Cases in the Heat Equation

    I got a solution to the heat equation using Fourier transforms with the special case g(x) = GH(x-a) u(x,t) = G/2[1+erf(x-a/(2\sqrt{t}))]. But I just wanted to know what this special case represents physically. I should probably ask what does any special case to the heat equation represent...
  15. P

    What would be my boundary conditions? Heat Equation

    1. I have a rod of length 4,cross section 1 and thermal conductivity 1.Nothing is mentioned about the end at the origin x=0, but at the opposite end x=4, the rod is radiating heat energy at twice the difference between the temperature of that end and the air temperature of 23 celcius. Find the...
  16. P

    Heat Equation for No Heat Loss at x=a

    The question :Two rods L1 and L2 of different materials( hence different thermal conductivities) and different cross-sectional areas,are joined at x=a. The temperature is continuous, And NO HEAT ENERGY IS LOST AT a, so all heat energy that flows from L1 flows into L2. ? What equation...
  17. C

    Archived Heat equation and energy transport.

    Homework Statement I have a rod of density \rho and length l. It's located at 0\leq x\leq l . The density of internal energy per mass is E = c(T-T_0) + E_0 where T is the tempertature in Kelvin, E_0 is a constant and c is the specific heat capacity. We assume that the temperature is not...
  18. L

    Two dimensional heat equation with source

    I have the following PDE PDE: ut=A2(uxx+uyy)+f(x,y,t) such that A is a constant BC: ux(1,y,t)=0 ux(0,y,t)=0 uy(x,1,t)=0 u(x,0,t)=0 IC: u(x,y,0)=0 I've set up my ODEs using separation of variables to get X''/X=k1 Y''/Y=k2 T'/(A2)T=k1+k2 where k1 and k2 are constants. How do I account for my...
  19. Y

    Heat equation (cylindrical)

    I have tried to solve the cylindrical case of the heat equation and reached the second order differential equation for the function R(r): R'' + (1/r)*R' + (alfa/k)*R = 0 (alfa, k are constants) I couldn't find material on the web for non-constant coefficients, does anyone know how to...
  20. B

    How can the heat equation be derived for a long circular cylinder?

    Consider heat flow in a long circular cylinder where the temperature depends only on t and on the distance r to the axis of the cylinder. Here r=\sqrt{x^{2}+y^{2}} is the cylindrical coordinate. From the three dimensional heat equation derive the equation u_{t}=k(u_{rr}+\frac{u_{r}}{r}). My...
  21. N

    Modified heat equation in n dimensions

    helloo! i'm trying to find the Green's function for the modified heat equation $u_t = t * grad(u) $ in n dimensions using scaling arguments. I know how to do this for the regular heat equation, by switching to polar coordinates and noticing that the equation and initial conditions are...
  22. M

    Solving the PDE 1-d Heat Equation for a Flipped Rod

    regarding 1-d Head Equations on rods. I am aware of how to long a rod with length x=0 to x=L. and initial conditions of u(0,t)=0 degrees and u(L,t)=100 degrees. But how does the problem change if before t=0 the rod at x=0 was at 100 degrees and x=L was at 0 degrees. So at time=0 the rod was...
  23. E

    Heat Equation Initial Conditions

    Greetings all, I have a question in regards to my initial conditions. The problem as given is: ut=uxx with u' = 0 at x=0 and u=0 at x=L I was also given u={1 0<x<L/2, 0 L/2<x<L I understand the set up of the problem and the solving of it for the most part, however I'm having...
  24. D

    Equilibrium heat equation in 2D cylindrical coordinates

    Homework Statement Plate in the shape of the circular halo (inner radius a, outer radius b>a), the inner edge is being kept at a constant temperature T_0, and the outer at the temperature given by the function f(\phi)=T_0\cos(2\phi). Find the equilibrium distribution of the heat everywhere...
  25. E

    What the heck is the heat equation?

    My professor asked us to use the heat equation to compare temperature change with the specific heat of three metals. I have no idea of what the heat equation is. I read a little bit about it on Wikipedia, but the equation there makes no sense to me. I'd really appreciate if anyone could help me...
  26. T

    Heat equation: partial differential equations

    solve the heat equation ut = kuxx -infinity < x < infinity and 0 < t < infinity with u(x,0)= x2 and uxxx(x,0)= 0 first i showed that uxxx(x,t) solves the equation (easy part) the next step is to conclude that u(x,t) must be of the form A(t)x2 + B(t)x + C(t). i...
  27. T

    Help solving the heat equation

    help solving the heat equation! solve the heat equation ut = kuxx -infinity < x < infinity and 0 < t < infinity with u(x,0)= x2 and uxxx(x,0)= 0 first i showed that uxxx(x,t) solves the equation (easy part) the next step is to conclude that u(x,t) must be of...
  28. S

    Heat equation advice for a small ice rink.

    Hi everyone, I am a Cybernetics student and as part of a project I need to determine the feasability of building a small ice rink (approximately 2m x 4m) The plan is to build a frame and exterior (the sides and bottom of a box) out of wood, and have a layer of internal insulation such as...
  29. T

    Change of variables in heat equation

    dTB/dt = -k(TB-TM). TM is held constant. (TB-TM) = Q, so: dTB/Q = -kdt. How would I change this equation so that instead of integrating wrt to TB, I can instead integrate wrt Q?
  30. S

    Deriving the Heat Equation with Exponential Term: Can We Solve for exp(theta)?

    Homework Statement I'm having trouble deriving the following equation \frac {\partial^2 {\theta}}{\partial {x'^2}} = -y^2*exp(\theta) and y = x/x' my main problem is the exponent Homework Equations The Attempt at a Solution Normally i would use the equation (x')'' + k^2*x' = 0 x' = c1...
  31. G

    Heat equation with radiation effect

    Homework Statement Let's say you have a 3m long copper pipe, 3mm in thickness with a diameter of 170mm. You fix one end at 1K and insulate it to prevent conduction or convection between the air and the pipe itself. There is still radiation. Assume that the inside of the pipe has no effect...
  32. K

    Heat Equation with insulated endpoints.

    Homework Statement Assume that a bar is insulated at the endpoints. If it loses heat through its lateral surface at a rate per unit length proportional to the difference u(x,t) - T, where T is the temperature of the medium surrounding the bar, the equation of heat propagation is now u_{t} =...
  33. R

    Solving the Heat Equation for Initial Conditions

    Problem: u (sub t) = (1/2)u (sub xx) find the solution u(x,t) of the heat equation for the following initial conditions: u(x,0) = x u(x,0) = x^2 u(x,0) = sinx u(x,0) = 0 for x < 0 and 1 for x>=0 i'm really flying blind here. I've taken differential equations years ago but nothing...
  34. D

    Heat equation with periodic BC

    I got the following PDE. {\frac {\partial }{\partial t}}u \left( y,t \right) = \nu \cdot {\frac {\partial ^{2}}{\partial {y}^{2}}}u \left( y,t \right) With boundary conditions: y=0: u(0,t)=0 y=h: u(h,t)=U_0 \cdot cos(\omega \cdot t) Now I need to show by using substitution that...
  35. P

    Heat Equation + 2 Robin Boundary Conditions

    Homework Statement Find the temperature distribution in the long thin bar −a ≤ x ≤ a with a given initial temperature u(x,0) = f(x). The side walls of the bar are insulated, while heat radiates from the ends into the surrounding medium whose temperature is u = 0. The radiation is taken...
  36. M

    Consider the heat equation in a radially symmetric sphere of radius

    Consider the heat equation in a radially symmetric sphere of radius unity: u_t = u_{rr}+{2 \over r}u_r \ for \ (r,t) \in (0,1) x (0,\infty) with boundary conditions \lim_{r \rightarrow 0}u(r,t) < \infty ; \ u(1,t)=0\ for \ t >0 Now, using separation of variables u=R(r)T(t) leads to the...
  37. M

    Heat Equation B.Cs: Solve & Determine Coefficients

    Hello there, hope you are having a good one. My problem is to solve the heat equtaion in cylindrical coordinates. This has been done by others for me, so a closed form solution is available, please see attached (please note the problem is 1 - D due to initial conditions depending only on...
  38. H

    One-Dimensional Heat Equation Problem

    Hi, I need help to solve this problem, about 1-D heat equation \partialu / \partialt = k (\partial2u / \partialx2)-2u (0< x <1) u(x,0)=e-x u(0,t)=e-2t u(1,t)=0 I need to solve it with separation variable
  39. A

    Heat Equation, Neumann, Fourier

    Hi, it's been a while since I touched mathematics and I'm a little rusty... I'm looking at a problem right now that I find difficult to understand, conceptually. Any insight would be greatly appreciated. (A direct solution would help immensely as well, not only because that's what I need to...
  40. M

    Is the Fourier Heat Equation's Speed of Propagation Infinite?

    One critic of the Fourier heat equation \frac{\partial T}{\partial t}=k\nabla^2 T that I recently came across is that it gives rise to infinite speed of heat propagation. I understand that the speed cannot be infinite because it contradict special relativity that no speed should be...
  41. A

    Forward difference method for heat equation

    I don't be able to convert the following code(HEAT EQUATION BACKWARD-DIFFERENCE ALGORITHM in the Burden-faires numerical analysis book).I need heat EQUATION FORWARD-DIFFERENCE ALGORITHM C like following code.I don't be able to convert FORWARD-DIFFERENCE the following code .Please help me. if...
  42. A

    Forward difference method for heat equation

    I don't be able to convert the following code(HEAT EQUATION BACKWARD-DIFFERENCE ALGORITHM in the Burden-faires numerical analysis book).I need HEAT EQUATION FORWARD-DIFFERENCE ALGORITHM C like following code.I don't be able to convert FORWARD-DIFFERENCE the following code .Please help me. /*...
  43. D

    Heat equation applied to a rod

    Homework Statement http://img444.imageshack.us/img444/7641/20240456gw8.png Homework Equations http://img14.imageshack.us/img14/5879/63445047rj2.png Note that the rightside of the rod is insulated. The Attempt at a Solution I get this model: \frac{ \partial{u} }{ \partial{t} } = \kappa...
  44. D

    Solving Heat Equation Homework

    Homework Statement http://img3.imageshack.us/img3/5020/84513876dm0.png The Attempt at a Solution I found that f(t) =exp \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right) Is this correct?
  45. clope023

    Finding u(x) for the 1-D Heat Equation

    Homework Statement If there is heat radiation in a rod of length L, then the 1-D heat equation might take the form: u_t = ku_xx + F(x,t) exercise deals with the steady state condition => temperature u and F are independent of time t and that u_t = 0. u_t = partial derivative with...
  46. J

    Solving the 1D Heat Equation with Given Parameters

    Problem:IF there is heat radiation within the rod of length L , then the 1 dimensional heat equation might take the form u_t = ku_xx + F(x,t) Find u(x) if F = x , k = 1 , , u(0)=0 , u(L) = 0 the problem is that i am not sure what this is asking me , how can i find u(x) if i have...
  47. F

    Heat equation PDE for spherical case

    Hello, I believe this is my first post. I would like to solve the heat equation PDE with some special (but not complicated) initial conditions, my scenario is as follows: A perfectly spherical mass of water, where the outer surface is at some particular temperature at t=0 (but not held at...
  48. B

    Heat Equation- Constant Dirichlet and Neumann BCs

    Hi all, I need to solve the heat equation (Ut=C*Uzz) with the following boundary conditions: U(max z,t)=0 and Uz(0,t)=-B. where B is a constant. My initial condition is U(z,0)=Uo where Uo is a constant. I know how to solve the equation for simple 0 boundary conditions. The Neumann BC...
  49. M

    Solve initial-value problem for heat equation and find relaxation time

    Homework Statement Solve the initial-value problem for the heat equation ut = K\nabla2u in the column 0< x < L1, 0< y < L2 with the boundary conditions u(0,y;t)=0, ux(L1,y,t)=0, u(x,0;t)=0, uy(x,L2;t)=0 and the initial condition u(x,y;0)=1. Find the relaxation time. Can anyone please explain...
  50. Μ

    Fundamental Solution for Nonhomogeneous Heat Equation?

    Homework Statement So I'm trying to solve Evans - PDE 2.5 # 12... "Write down an explicit formula for a solution of u_t - \Delta u + cu=f with (x,t) \in R^n \times (0,\infty) u(x,0)=g(x)" Homework Equations The Attempt at a Solution I figure if I can a fundamental solution...
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