Partial differential equation describing the evolution of temperature in a region
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.[1]
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.
^Berline, Nicole; Getzler, Ezra; Vergne, Michèle. Heat kernels and Dirac operators. Grundlehren der Mathematischen Wissenschaften, 298. Springer-Verlag, Berlin, 1992. viii+369 pp. ISBN 3-540-53340-0
In mathematics and physics, the heatequation is a certain partial differential equation. Solutions of the heatequation are sometimes known as caloric...
book was Fourier's proposal of his heatequation for conductive diffusion of heat. This partial differential equation is now a common part of mathematical...
Specific heat of melting (Enthalpy of fusion) Specific heat of vaporization (Enthalpy of vaporization) Frenkel line Heat capacity ratio HeatequationHeat transfer...
nonlinear. For example, Fisher's equation is a nonlinear PDE that includes the same diffusion term as the heatequation but incorporates a linear growth...
In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian...
diffusion equation is a special case of the convection–diffusion equation when bulk velocity is zero. It is equivalent to the heatequation under some...
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heatequation on a specified domain with appropriate...
Heat transfer can be modeled in various ways. The heatequation is an important partial differential equation that describes the distribution of heat...
The porous medium equation, also called the nonlinear heatequation, is a nonlinear partial differential equation taking the form: ∂ u ∂ t = Δ ( u m )...
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when...
difference method used for numerically solving the heatequation and similar parabolic partial differential equations. It is a first-order method in time, explicit...
usual heatequation for non-relativistic heat conduction must be modified, as it leads to faster-than-light signal propagation. Relativistic heat conduction...
mechanics Heat capacity ratio Statistical mechanics Thermodynamic equations Thermodynamic databases for pure substances HeatequationHeat transfer coefficient...
and spatial modeling. One of the most studied SPDEs is the stochastic heatequation, which may formally be written as ∂ t u = Δ u + ξ , {\displaystyle \partial...
methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial...
u=(C_{\text{v}})(\delta T)} Inserting the specific heatequation into the thermal efficiency equation (Equation 2) yields. η = 1 − ( C v ( T 4 − T 1 ) C v (...
Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry) List of equations in classical...
Fourier series were first used by Joseph Fourier to find solutions to the heatequation. This application is possible because the derivatives of trigonometric...
way. Energy portal History of heatHeatequationHeat diffusion Heat transfer coefficient Relativistic heat conduction Heat death of the Universe Effect...
differential equations with boundary and initial conditions, such as the heatequation, wave equation, Laplace equation, Helmholtz equation and biharmonic...
The Fourier number arises naturally in nondimensionalization of the heatequation. The general definition of the Fourier number, Fo, is: Fo=timetime scale...