"The theory of the heat equation was first developed by Joseph Fourier in 1822."
A PDE that models the diffusion of heat through a solid material.
Definition and Interpretation of Heat Equation: This topic describes the fundamental properties of the heat equation and its physical interpretation.
Boundary Conditions: This topic covers the different types of boundary conditions that may arise in solving the heat equation, including Dirichlet, Neumann, and mixed boundary conditions.
Fourier Series: This topic discusses the use of Fourier series in solving the heat equation, including orthogonality of Fourier series and boundary value problems, and the calculation of Fourier coefficients.
Separation of Variables: This topic explains the technique of separation of variables in solving the heat equation, including the general form of solutions and the application of boundary conditions.
Green's Functions: This topic discusses the use of Green's functions in solving the heat equation, including the definition of Green's function, its properties, and its application to boundary value problems.
Numerical Methods: This topic explains the use of numerical methods in solving the heat equation, including finite difference methods, finite element methods, spectral methods, and meshless methods.
Nonlinear Heat Equation: This topic covers the nonlinear versions of the heat equation, including the reaction-diffusion equation, and the application of analytical and numerical techniques to solve them.
Heat Equation with Varying Coefficients: This topic describes how to solve the heat equation when coefficients such as diffusivity or thermal conductivity vary in space or time.
Applications: This topic discusses the various applications of the heat equation, including heat conduction in materials, heat transfer in fluids, and thermal diffusion in biological systems.
Heat Equation in Multiple Dimensions: This topic covers the extension of the heat equation to multiple dimensions, including the derivation of the three-dimensional heat equation, and the application of various techniques to solve it.
Perturbation Methods: This topic explains the use of perturbation methods to solve the heat equation with small parameter coefficients.
Asymptotic Analysis: This topic covers the application of asymptotic methods such as WKB approximation and boundary layer theory to obtain approximate solutions to the heat equation.
Variational Methods: This topic explains the use of variational methods to solve the heat equation, including the principle of minimum potential energy, the Rayleigh-Ritz method, and the Galerkin method.
Inverse Heat Conduction Problem: This topic covers the inverse problem of determining the unknown boundary or initial conditions of the heat equation from measured data.
Analytical Solutions: This topic explains the derivation of analytical solutions to the heat equation, including the method of images, Green's function approach, and similarity solutions.
Classical Heat Equation: Describes the flow of heat in a homogeneous medium over time.
Nonlinear Heat Equation: Includes additional terms that lead to nonlinearity, such as the presence of an external field or chemical reactions.
Fractional Heat Equation: Describes the flow of heat in a medium where the heat flux density is not proportional to the temperature gradient.
Generalized Heat Equation: Incorporates additional effects that are not captured by the classical Heat Equation, such as convection, radiation, or additional boundary conditions.
Porous Media Heat Equation: Describes the flow of heat in porous media, which typically exhibit complex flow regimes and heat transfer mechanisms.
Stochastic Heat Equation: Represents the evolution of temperature in a medium subjected to random fluctuations or uncertain parameters.
Phase-Field Heat Equation: Uses a phase-field function to describe the evolution of temperature and phase transitions in multi-component systems.
Multi-Scale Heat Equation: Incorporates multiple scales of observation, such as microscale fluctuations or macroscopic heat reservoirs.
Hyperbolic Heat Equation: Represents the flow of heat in a hyperbolic medium, which is characterized by a high degree of wave-like behavior.
Integral Heat Equation: Expresses the temperature field as an integral equation rather than a differential equation.
"...for the purpose of modeling how a quantity such as heat diffuses through a given region."
"As the prototypical parabolic partial differential equation..."
"...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."
"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."
"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."
"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."
"...beginning in the 1950s with work of Jim Douglas, D.W. Peaceman, and Henry Rachford Jr."
"Solutions of the heat equation are sometimes known as caloric functions."
"...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..."
"...culminating in the proof of the Poincaré conjecture by Grigori Perelman in 2003."
"...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."
"The heat equation is among the most widely studied topics in pure mathematics."