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Partial differential equation describing the evolution of temperature in a region
specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph
Heat_equation
Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry) List of equations in classical
List_of_equations
Fundamental solution to the heat equation, given boundary values
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate
Heat_kernel
Process by which heat is transferred within an object
Convection diffusion equation R-value (insulation) Heat pipe Fick's law of diffusion Relativistic heat conduction Churchill–Bernstein equation Fourier number
Thermal_conduction
Type of differential equation
Acoustic wave equation Burgers' equation Continuity equation Heat equation Helmholtz equation Klein–Gordon equation Jacobi equation Lagrange equation Lorenz
Partial_differential_equation
Entropy production in Newtonian fluids
In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian
General equation of heat transfer
General_equation_of_heat_transfer
Heat required to raise the temperature of a given unit of mass of a substance
fusion (latent heat of melting) Enthalpy of vaporization (latent heat of vaporization) Frenkel line Heat capacity ratio Heat equation Heat transfer coefficient
Specific_heat_capacity
Second-order partial differential equation
dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation. In general, Laplace's equation describes situations of
Laplace's_equation
Equation that describes density changes of a material that is diffusing in a medium
diffusion equation is a special case of the convection–diffusion equation when bulk velocity is zero. It is equivalent to the heat equation under some
Diffusion_equation
Class of second-order linear partial differential equations
mathematics. Examples include the heat equation, time-dependent Schrödinger equation and the Black–Scholes equation. To define the simplest kind of parabolic
Parabolic partial differential equation
Parabolic_partial_differential_equation
Type of functional equation (mathematics)
book was Fourier's proposal of his heat equation for conductive diffusion of heat. This partial differential equation is now a common part of mathematical
Differential_equation
Physical property of matter
Heat equation – Partial differential equation describing the evolution of temperature in a region Heat transfer coefficient – Quantity relating heat flux
Heat_capacity
Partial differential equations with random force terms and coefficients
and spatial modeling. One of the most studied SPDEs is the stochastic heat equation, which may formally be written as ∂ t u = Δ u + ξ , {\displaystyle \partial
Stochastic partial differential equation
Stochastic_partial_differential_equation
Model compatible with special relativity
usual heat equation for non-relativistic heat conduction must be modified, as it leads to faster-than-light signal propagation. Relativistic heat conduction
Relativistic_heat_conduction
Motion of particles in a fluid
approach as in the case of the Heat Equation above. We write the wave equation as a first order in time partial differential equation by introducing the following
Flow_(mathematics)
American mathematician (1943–2024)
for applying the maximum principle to control the solutions of the heat equation. Their results take the form of asserting the nonnegativity of certain
Richard_S._Hamilton
Method in numerical analysis
difference method used for numerically solving the heat equation and similar parabolic partial differential equations. It is a first-order method in time, explicit
FTCS_scheme
Thermal engineering discipline concerning transfer of heat in physical systems
Heat transfer can be modeled in various ways. The heat equation is an important partial differential equation that describes the distribution of heat
Heat_transfer
Mathematical relationship describing the flow of groundwater through an aquifer
by a form of the diffusion equation, similar to that used in heat transfer to describe the flow of heat in a solid (heat conduction). The steady-state
Groundwater_flow_equation
Elliptic partial differential equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the
Poisson's_equation
Equation in convective heat transfer
In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at
Churchill–Bernstein_equation
Class of numerical techniques
methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial
Finite_difference_method
Method for solving partial differential equations
differential equations, Duhamel's principle is a general method for obtaining solutions to inhomogeneous linear evolution equations like the heat equation, wave
Duhamel's_principle
Partial differential equation
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas
Burgers'_equation
Equation describing the transport of some quantity
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when
Continuity_equation
Partial differential equation
kind of parabolic partial differential equations (PDEs) with a quadratic nonlinearity into a linear heat equation. In particular, it provides an explicit
Cole–Hopf_transformation
Mathematical descriptions of transmission line voltage and current
The telegrapher's equations (or telegraph equations) are a set of two coupled, linear partial differential equations that model voltage and current along
Telegrapher's_equations
Dimensionless quantity related to transient heat conduction
The Fourier number arises naturally in nondimensionalization of the heat equation. The general definition of the Fourier number, Fo, is: F o = time time
Fourier_number
Functional analysis theorem
Lax–Milgram theory. To illustrate the power of Lions's theorem, consider the heat equation in n spatial dimensions (x) and one time dimension (t): ∂ t u ( t ,
Lions–Lax–Milgram_theorem
Nonlinear partial differential equation
The porous medium equation, also called the nonlinear heat equation, is a nonlinear partial differential equation taking the form: ∂ u ∂ t = Δ ( u m )
Porous_medium_equation
Decomposition of periodic functions
Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric
Fourier_series
Partial differential equation
partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities
Ricci_flow
Physical law relating heat loss to temperature difference
heat capacity) results in a simple differential equation expressing temperature-difference as a function of time. The solution to that equation describes
Newton's_law_of_cooling
the generated heat per unit volume in [W·m−3]. The governing equation of the physics of the heat transfer problem relates the flux of heat in space, its
Thermal simulations for integrated circuits
Thermal_simulations_for_integrated_circuits
French mathematician and physicist (1768–1830)
proposal of his partial differential equation for conductive diffusion of heat, often called the heat equation. This equation is now taught to every student
Joseph_Fourier
Image processing technique
Gaussian smoothed image is a single time slice of the solution to the heat equation, that has the original image as its initial conditions. Anisotropic
Edge-preserving_smoothing
Technique for solving differential equations
differential equations with boundary and initial conditions, such as the heat equation, wave equation, Laplace equation, Helmholtz equation and biharmonic
Separation_of_variables
Mathematical function
are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform. They are also abundantly
Gaussian_function
θ ) {\displaystyle u_{S}(\theta ,t)=\exp(-t)\sin(\theta )} for the heat equation. Forward Euler time-stepping is used with relation Δ t = 0.1 Δ x 2 {\displaystyle
Closest_point_method
Mathematical transform that expresses a function of time as a function of frequency
Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can be formally
Fourier_transform
Russian mathematician (born 1966)
partial differential equation formally analogous to the heat equation, for how to deform a Riemannian metric on a manifold. The heat equation, such as when applied
Grigori_Perelman
Special functions of several complex variables
descent. One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution
Theta_function
Ability of a material to exchange energy with surroundings
effusivity is a parameter that emerges upon applying solutions of the heat equation to heat flow through a thin surface-like region. It becomes particularly
Thermal_effusivity
Thermodynamic cycle for spark ignition piston engines
u=(C_{\text{v}})(\delta T)} Inserting the specific heat equation into the thermal efficiency equation (Equation 2) yields. η = 1 − ( C v ( T 4 − T 1 ) C v (
Otto_cycle
Non-linear stochastic partial differential equation
That equation is the Edwards–Wilkinson (EW) equation, also known descriptively as the stochastic heat equation with additive noise. The EW equation is mathematically
Kardar–Parisi–Zhang_equation
Type of energy transfer
Heat death of the Universe Heat diffusion Heat equation Heat exchanger Heat flux sensor Heat recovery steam generator Heat recovery ventilation Heat transfer
Heat
Equation of statistical mechanics
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium;
Boltzmann_equation
conjecture Tzitzeica equation Lorenz equations Rabinovich–Fabrikant equations General Legendre equation Heat equation Ishimori equation, an integrable nonlinear
List of named differential equations
List_of_named_differential_equations
Procedure for solving differential equations
inhomogeneous problems for linear evolution equations like the heat equation, wave equation, and vibrating plate equation. In this setting, the method is more
Variation_of_parameters
Property of differential equations describing physical phenomena
well-posed problems include the Dirichlet problem for Laplace's equation and the heat equation with specified initial conditions. These might be regarded as
Well-posed_problem
Quantity relating heat flux and temperature difference
\textstyle {\rm {\frac {W}{mK}}}} . Convective heat transfer Heat sink Convection Churchill–Bernstein equation Heat Heat pump Heisler Chart Thermal conductivity
Heat_transfer_coefficient
Relation between vapour pressure and temperature
and n is the number density. The equation expresses this in a more convenient form just in terms of the latent heat, for moderate temperatures and pressures
Clausius–Clapeyron_relation
Polynomial sequence
quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term x u x {\displaystyle {\begin{aligned}xu_{x}\end{aligned}}}
Hermite_polynomials
Concept in mathematics
melting of a solid, such as ice to water. This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At
Stefan_problem
"Smoothing" integral transform
Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient). If the function
Weierstrass_transform
Topics referred to by the same term
variables that defines an integral transform Heat kernel, the fundamental solution to the heat equation on a specified domain Convolution kernel Stochastic
Kernel
Surface area per unit volume
examples for such processes are processes governed by the heat equation, that is, diffusion and heat transfer by thermal conduction. SA:V is used to explain
Surface-area-to-volume_ratio
Theorem in geometric topology
the heat equation, which describes the way heat flows in a solid. Like the heat flow, Ricci flow tends towards uniform behavior. Unlike the heat flow
Poincaré_conjecture
Materials' resistance to heat transfer
Assuming that the temperature distribution, equation 7, is used with Fourier's law in equation 5, the heat transfer rate can be expressed in the following
Thermal conductance and resistance
Thermal_conductance_and_resistance
Common thermodynamic equations and quantities in thermodynamics, using mathematical notation, are as follows: Many of the definitions below are also used
Table of thermodynamic equations
Table_of_thermodynamic_equations
mathematical heat conduction, the Green's function number is used to uniquely categorize certain fundamental solutions of the heat equation to make existing
Green's_function_number
Equations in thermodynamics
Thermodynamics is expressed by a mathematical framework of thermodynamic equations which relate various thermodynamic quantities and physical properties
Thermodynamic_equations
Part of mathematics that addresses the stability of solutions
of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is
Stability_theory
Thermodynamic quantity
Relations between heat capacities Heat capacity Specific heat capacity Speed of sound Thermodynamic equations Thermodynamics Volumetric heat capacity γ first
Heat_capacity_ratio
Lie group of complex numbers of unit modulus; topologically a circle
the heat equation is ∂ u ∂ t = ∂ 2 u ∂ x 2 . {\displaystyle {\frac {\partial u}{\partial t}}={\frac {\partial ^{2}u}{\partial x^{2}}}.} This equation can
Circle_group
Mathematical formula expressing equality
an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and
Equation
Mathematical result in differential geometry
Bott, and Vijay Patodi gave a new proof of the index theorem using the heat equation, described in a paper by Melrose. 1977: Dennis Sullivan establishes
Atiyah–Singer_index_theorem
Analog of the continuous Laplace operator
{\frac {d\phi }{dt}}+kL\phi =0.} Notice that this equation takes the same form as the heat equation, where the matrix −L is replacing the Laplacian operator
Discrete_Laplace_operator
Numerical analysis procedure
Fourier series. To illustrate the procedure, consider the one-dimensional heat equation ∂ u ∂ t = α ∂ 2 u ∂ x 2 {\displaystyle {\frac {\partial u}{\partial
Von Neumann stability analysis
Von_Neumann_stability_analysis
equation Heat equation Laplace's equation Laplace operator Harmonic function Spherical harmonic Poisson integral formula Klein–Gordon equation Korteweg–de
List of partial differential equation topics
List_of_partial_differential_equation_topics
Technique to solve geological problems by computational simulation
Then, governing equations that describe the geological problems are written, for example, the heat equations describe the flow of heat in a system. Since
Numerical_modeling_(geology)
Equations of motion for viscous fluids
Navier–Stokes equations (/nævˈjeɪ ˈstoʊks/ nav-YAY STOHKS) describe the motion of viscous fluids. This system of partial differential equations was named
Navier–Stokes_equations
Finite difference method for numerically solving parabolic differential equations
difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time. It is implicit
Crank–Nicolson_method
Equation describing a state of matter under a given set of conditions
In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given
Equation_of_state
Direct conversion of temperature differences to electric voltage and vice versa
heating and ordinary heat conduction. As stated above, the Seebeck effect generates an electromotive force, leading to the current equation J = σ ( − ∇ V −
Thermoelectric_effect
Curve from a cone intersecting a plane
representative examples is that the Poisson equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic. Eccentricity classifications
Conic_section
Ovsjanikov and Leonidas Guibas. It is based on the heat kernel, which is a fundamental solution to the heat equation. HKS is one of the many recently introduced
Heat_kernel_signature
Ratio of the thermal resistances of a body's interior to its surface
this temperature may be changing with time as heat passes into the sphere from the surface. The equation to describe this change in (relatively uniform)
Biot_number
Mathematical form
derivatives. This allows mathematicians to study the Laplace equation and heat equation on spaces that are not manifolds, for example, fractals. The benefit
Dirichlet_form
Type of partial differential equation
example is the melting of ice: Given a block of ice, one can solve the heat equation given appropriate initial and boundary conditions to determine its temperature
Free_boundary_problem
differential equations, the mth-degree caloric polynomial (or heat polynomial) is a "parabolically m-homogeneous" polynomial Pm(x, t) that satisfies the heat equation
Caloric_polynomial
flows as well as to other systems such as the Navier–Stokes equations and heat equation. Perelman, Grigori (2002), The entropy formula for the Ricci
Ancient_solution
Concept in science
conductivity in the heat equation, stimulating the heat concentration at the graph vertices connected by the graph edge, rather than the normal heat dissipation
Negative_probability
certain partial differential equations: Gevrey originally formulated the definition while investigating the homogeneous heat equation, whose solutions are in
Gevrey_class
Computational materials scientist and condensed-matter physicist
of Fourier's law into viscous heat equations, introducing the notion of thermal viscosity that governs fluid-like heat flow in the hydrodynamic regime
Nicola_Marzari
Probability distribution
absolute value. In the physics of heat conduction, the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds
Folded_normal_distribution
Mathematical descriptions of molecular diffusion
law has the same mathematical form as the Heat equation and its fundamental solution is the same as the Heat kernel, except switching thermal conductivity
Fick's_laws_of_diffusion
Idealized thermodynamic cycle
same heat reservoirs are equally efficient. Rearranging the right side of the equation gives what may be a more easily understood form of the equation, namely
Carnot_cycle
Temperature index that accounts for the effects of humidity
whereas the heat index uses a dew point base of 14 °C (57 °F).[further explanation needed] Further, the heat index uses heat balance equations which account
Heat_index
Partial differential operator
also analytically hypoelliptic). In addition, the operator for the heat equation ( P ( u ) = u t − k Δ u {\displaystyle P(u)=u_{t}-k\,\Delta u\,} ) P
Hypoelliptic_operator
French mathematician (born 1948)
manifolds. Since 1984, Bismut works on differential geometry. He found a heat equation proof for the Atiyah–Singer index theorem. And he established a local
Jean-Michel_Bismut
Equipment used to transfer heat between fluids
A heat exchanger is a system used to transfer heat between a source and a working fluid. Heat exchangers are used in both cooling and heating processes
Heat_exchanger
Combination of the diffusion and convection (advection) equations
convection–diffusion equation is a parabolic partial differential equation that combines the diffusion and convection (advection) equations. It describes physical
Convection–diffusion_equation
Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
In fluid dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard
Euler equations (fluid dynamics)
Euler_equations_(fluid_dynamics)
Capacity of a material to conduct heat
defining equation for thermal conductivity is q = − k ∇ T {\displaystyle \mathbf {q} =-k\nabla T} , where q {\displaystyle \mathbf {q} } is the heat flux
Thermal conductivity and resistivity
Thermal_conductivity_and_resistivity
Equation in fluid dynamics
In fluid dynamics, the Darcy–Weisbach equation is an empirical equation that relates the head loss, or pressure loss, due to viscous shear forces along
Darcy–Weisbach_equation
English mathematical physicist
differential equations and, in particular, the solution of heat-conduction problems. He is best known for his work with Phyllis Nicolson on the heat equation, which
John_Crank
Equation describing evaporation
mm/day, because it is flux m3/s per m2=m/s. This equation assumes a daily time step so that net heat exchange with the ground is insignificant, and a
Penman_equation
Differential operator in mathematics
differential equations describing physical phenomena. Poisson's equation describes electric and gravitational potentials; the diffusion equation describes heat and
Laplace_operator
Fluid dynamics equation
Rayleigh number. For R a ≪ 1 {\displaystyle Ra\ll 1} , the equation reduces to the linear heat equation, φ τ = φ ξ ξ {\displaystyle \varphi _{\tau }=\varphi
Erdogan–Chatwin_equation
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