Flow (mathematics) - Examples - Solutions of Heat Equation

Solutions of Heat Equation

Let be a subdomain (bounded or not) of ( is an integer). We denote its boundary (supposed smooth). Let us consider the following Heat Equation on (for ):

$\begin{array}{rcll} u_t - \Delta u & = & 0 & \mbox{ in } \Omega \times (0,T), \\ u & = & 0 & \mbox{ on } \Gamma \times (0,T), \end{array}$

with the following initial boundary condition .

Equation corresponds to the Homogeneous Dirichlet boundary condition. The mathematical setting for this problem can be the semigroup approach. To use this tool, we introduce the unbounded operator defined on by its domain (see the classical Sobolev spaces with and is the closure of the infinitely differentiable functions with compact support in for the norm). For any, we have

$\Delta_D v = \Delta v = \sum_{i=1}^n \frac{\partial^2 }{\partial x_i^2} v.$

With this operator, the heat equation becomes and . Thus, the flow corresponding to this equation is (see notations above)

$\varphi(u^0,t) = \mbox{e}^{t\Delta_D}u^0$ where is the (analytic) semigroup generated by .

Read more about this topic: Flow (mathematics), Examples

Famous quotes containing the words solutions, heat and/or equation:

“Football strategy does not originate in a scrimmage: it is useless to expect solutions in a political compaign.”
—Walter Lippmann (1889–1974)

“Coal is a portable climate. It carries the heat of the tropics to Labrador and the polar circle; and it is the means of transporting itself whithersoever it is wanted. Watt and Stephenson whispered in the ear of mankind their secret, that a half-ounce of coal will draw two tons a mile, and coal carries coal, by rail and by boat, to make Canada as warm as Calcutta, and with its comfort brings its industrial power.”
—Ralph Waldo Emerson (1803–1882)

“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)