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If the medium is not the whole space, in order to solve the heat equation uniquely we also need to specify boundary conditions for u. To determine uniqueness of solutions in the whole space it is necessary to assume additional conditions, for example an exponential bound on the growth of solutions [ 1 ] or a sign condition (nonnegative ...
Periodic boundary conditions in 2D Unit cell with water molecules, used to simulate flowing water. Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a unit cell. PBCs are often used in computer simulations and mathematical models.
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 .
This series in fact is a general boundary condition which becomes a condition of the third kind in the first approximation. Each of those two expressions in the form of Duhamel's integral or in a series of derivatives reduces a conjugate problem to the solution of only the conduction equation for the body at given conjugate conditions.
The Born–von Karman boundary condition is important in solid state physics for analyzing many features of crystals, such as diffraction and the band gap. Modeling the potential of a crystal as a periodic function with the Born–von Karman boundary condition and plugging in Schrödinger's equation results in a proof of Bloch's theorem , which ...
As an example, number X11 denotes the Green's function that satisfies the heat equation in the domain (0 < x < L) for boundary conditions of type 1 at both boundaries x = 0 and x = L. Here X denotes the Cartesian coordinate and 11 denotes the type 1 boundary condition at both sides of the body.
Since the problems in magnetostatics involve solving Laplace's equation or Poisson's equation for the magnetic scalar potential, the boundary condition is a Neumann condition. In spatial ecology , a Neumann boundary condition on a reaction–diffusion system , such as Fisher's equation , can be interpreted as a reflecting boundary, such that ...
This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases (in the classical problem) the temperature is set to the phase change temperature. To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance ...