Heat kernel

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 boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some auxiliary importance throughout mathematical physics. The heat kernel represents the evolution of temperature in a region whose boundary is held fixed at a particular temperature (typically zero), such that an initial unit of heat energy is placed at a point at time t = 0. K ( t , x , y ) = ∑ n = 0 ∞ e − λ n t ϕ n ( x ) ϕ n ( y ) . {displaystyle K(t,x,y)=sum _{n=0}^{infty }e^{-lambda _{n}t}phi _{n}(x)phi _{n}(y).}     (1) 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 boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some auxiliary importance throughout mathematical physics. The heat kernel represents the evolution of temperature in a region whose boundary is held fixed at a particular temperature (typically zero), such that an initial unit of heat energy is placed at a point at time t = 0. The most well-known heat kernel is the heat kernel of d-dimensional Euclidean space Rd, which has the form of a time-varying Gaussian function,