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In mathematics and physics, the heat equation is a certain partial differential equation.

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In mathematics and physics, the heat equation is a certain partial differential equation.

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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.

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The heat equation can be considered on Riemannian manifolds, leading to many geometric applications.

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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.

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In probability theory, the heat equation is connected with the study of random walks and Brownian motion via the Fokker–Planck equation.

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The Black–Scholes equation of financial mathematics is a small variant of the heat equation, and the Schrodinger equation of quantum mechanics can be regarded as a heat equation in imaginary time.

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In image analysis, the heat equation is sometimes used to resolve pixelation and to identify edges.

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The coefficient in the equation takes into account the thermal conductivity, specific heat, and density of the material.

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Heat equation implies that peaks of will be gradually eroded down, while depressions will be filled in.

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Heat flow, the heat equation follows from the physical laws of conduction of heat and conservation of energy .

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Heat equation is a consequence of Fourier's law of conduction .

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Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object.

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Heat equation is the prototypical example of a parabolic partial differential equation.

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Heat equation governs heat diffusion, as well as other diffusive processes, such as particle diffusion or the propagation of action potential in nerve cells.

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Heat equation is, technically, in violation of special relativity, because its solutions involve instantaneous propagation of a disturbance.

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Heat equation flow is a form of energy flow, and as such it is meaningful to speak of the time rate of flow of heat into a region of space.

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Fundamental solution, called a heat kernel, is a solution of the heat equation corresponding to the initial condition of an initial point source of heat at a known position.

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Since the heat equation is linear, solutions of other combinations of boundary conditions, inhomogeneous term, and initial conditions can be found by taking an appropriate linear combination of the above Green's function solutions.

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Heat equation arises in the modeling of a number of phenomena and is often used in financial mathematics in the modeling of options.

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The Black–Scholes option pricing model's differential equation can be transformed into the heat equation allowing relatively easy solutions from a familiar body of mathematics.

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The equation describing pressure diffusion in a porous medium is identical in form with the heat equation.

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The heat equation is widely used in image analysis and in machine-learning as the driving theory behind scale-space or graph Laplacian methods.

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The heat equation can be efficiently solved numerically using the implicit Crank–Nicolson method of .

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