Heat equation

In physics and mathematics, the heat equation is a partial differential equation that describes how the distribution of some quantity (such as heat) evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower. It is a special case of the diffusion equation. u ˙ = α ∇ 2 u {displaystyle {dot {u}}=alpha abla ^{2}u} u t = α u x x {displaystyle displaystyle u_{t}=alpha u_{xx}} (1) u ( x , 0 ) = f ( x ) ∀ x ∈ [ 0 , L ] {displaystyle u(x,0)=f(x)quad forall xin } (2) u ( 0 , t ) = 0 = u ( L , t ) ∀ t > 0 {displaystyle u(0,t)=0=u(L,t)quad forall t>0} . (3) u ( x , t ) = X ( x ) T ( t ) . {displaystyle displaystyle u(x,t)=X(x)T(t).} (4) T ′ ( t ) = − λ α T ( t ) {displaystyle T'(t)=-lambda alpha T(t)} (5) X ″ ( x ) = − λ X ( x ) . {displaystyle X''(x)=-lambda X(x).} (6) In physics and mathematics, the heat equation is a partial differential equation that describes how the distribution of some quantity (such as heat) evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower. It is a special case of the diffusion equation. This equation was first developed and solved by Joseph Fourier in 1822 to describe heat flow. However, it is of fundamental importance in diverse scientific fields. In probability theory, the heat equation is connected with the study of random walks and Brownian motion, via the Fokker–Planck equation. In financial mathematics it is used to solve the Black–Scholes partial differential equation. A variant was also instrumental in the solution of the longstanding Poincaré conjecture of topology. For a function u ( x , y , z , t ) {displaystyle u(x,y,z,t)} of three spatial variables ( x , y , z ) {displaystyle (x,y,z)} (see Cartesian coordinate system) and the time variable t {displaystyle t} , the heat equation is where α {displaystyle alpha } is a real coefficient called the diffusivity of the medium. Using Newton's notation for derivatives, and the notation of vector calculus, the heat equation can be written in compact form as Here ∇ 2 {displaystyle abla ^{2}} denotes the Laplace operator, and u ˙ {displaystyle {dot {u}}} is the time derivative of u {displaystyle u} . One advantage of this formula is that the operator ∇ 2 {displaystyle abla ^{2}} can usually be defined in purely physical terms, independently of the choice of coordinate system. This equation describes the flow of heat in a homogeneous and isotropic medium, with u ( x , y , z , t ) {displaystyle u(x,y,z,t)} being the temperature at the point ( x , y , z ) {displaystyle (x,y,z)} and time t {displaystyle t} . However, it also describes many other physical phenomena as well. The value of α {displaystyle alpha } affects the speed and spatial scale of the process; changing it has the same effect as changing the unit of measure for time (which affects the value of u ˙ {displaystyle {dot {u}}} ), and/or the unit of measure of length (that affects the value of ∇ 2 u {displaystyle abla ^{2}u} ). Therefore, in mathematical studies of this equation, one often sets α = 1 {displaystyle alpha =1} . With this simplification, the heat equation is the prototypical parabolic partial differential equation. Informally, the Laplacian operator ∇ 2 {displaystyle abla ^{2}} gives the difference between the average value of a function in the neighboring of a point, and its value at that point. Thus, if u {displaystyle u} is the temperature, ∇ 2 u {displaystyle abla ^{2}u} tells whether (and by how much) the material surrounding each point is hotter or colder, on the average, than the material at that point. By the second law of thermodynamics, heat will flow from hotter bodies to adjacent colder bodies, in proportion to the difference of temperature and of the thermal conductivity of the material between them. When heat flows into (or out of) a material, its temperature increases (respectively, decreases), in proportion to the amount of heat divided by the amount (mass) of material, with a proportionality factor called the specific heat capacity of the material.