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. J

    Heat Equation: Boundary Value Problem

    http://img821.imageshack.us/img821/7901/heatp.png Uploaded with ImageShack.us I'm having difficulty with the boundary conditions on this problem. I don't need a solution or a step by step. I've just never solved a boundary condition like this. Its the u(pi,t) = cos(t) that is giving me...
  2. M

    MHB Solve Heat Equation with Initial Conditions

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  3. M

    Looking for method to use in final step in heat equation problem

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  4. fluidistic

    Heat equation, Fourier cosine transform

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  5. M

    MHB Solving a Heat Equation with $\sin \pi x$

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  6. N

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  7. R

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  8. fluidistic

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  9. C

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  10. M

    Heat Equation for Compressible Fluids: Valid or Not?

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  11. H

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  12. A

    Heat equation in the first quadrant.

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  13. L

    Calculating Latent Heat of Vaporisation: 3 kW Kettle, 2.0 kg Water @ 100oC

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  14. C

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  15. J

    Help with similarity solutions to Heat Equation

    I'm trying to solve the Heat equation by assuming a similarity solution of the form U=f(z) where z = x / √t also subject to U=H(x) at t=0 *H(x) is heaviside function. The question want the answers to be given in terms of the error function and also checked by using the fundamental solution of...
  16. P

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  17. R

    Using Crank-Nicolson Method to solve Heat Equation

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  18. M

    Solving the heat equation on the real line using fourier transforms

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  19. T

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  20. T

    Heat Equation - Trouble Finding a General Solution

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  21. B

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  22. M

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  23. K

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  24. W

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  25. D

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  26. Telemachus

    Heat equation with boundary conditions

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  27. U

    How can I graph a heat equation with multiple variables in Maple?

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  28. R

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  29. S

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  30. C

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  31. M

    Can the heat equation apply to gases?

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  32. D

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  33. B

    Heat Equation with movable point source

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  34. T

    What is the Relationship Between Heat Energy and Temperature?

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  35. U

    Analytical solution for heat equation with simple boundary conditions

    I am trying to solve the following heat equation ODE: d^2T/dr^2+1/r*dT/dr=0 (steady state) or dT/dt=d^2T/dr^2+1/r*dT/dr (transient state) The problem is simple: a ring with r1<r<r2, T(r1)=T1, T(r2)=T2. I have searched the analytical solution for this kind of ODEs in polar coordinate...
  36. Y

    Find an equilibrium solution of heat equation Please help

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  37. T

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  38. J

    Is this an acceptable method to solve a convectihe heat equation?

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  39. B

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  40. S

    Fourier Transform to solve heat equation in infinite domain

    I'm having trouble following a step in my notes: first off the heat equation is given by: \frac{\partial u}{\partial t}=k^{2}\frac{\partial^{2}u}{\partial x^{2}} then take the Fourier transform of this w.r.t.x, where in this notation the Ftransform of u(x,t) is denoted by U(alpha,t)...
  41. O

    Heat equation and taylor's approximation

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  42. T

    Heat equation with nonhomogeneous boundary conditions

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  43. M

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  44. D

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  45. W

    Is there an ODE with only time dependence and constant position?

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  46. J

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  47. W

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  48. S

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  49. M

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  50. N

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