Читать книгу An Introduction to the Finite Element Method for Differential Equations - Mohammad Asadzadeh - Страница 24

Below we model the heat conduction in a thin heat‐conducting wire stretched between the two endpoints of an interval that is subject to a heat source of intensity , as in Figure 1.3. We are interested in the stationary distribution of temperature in the wire.

Figure 1.3 A heat‐conducting one‐dimensional wire.

To this end, let denote the heat flux in the direction of the positive ‐axis in the wire . Conservation of energy in the stationary case requires that the net heat through the endpoints of an arbitrary subinterval of is equal to the heat produced in per unit time:

By the Fundamental Theorem of Calculus,

Hence, we conclude that

Since and are arbitrary, assuming that the integrands are continuous, yields

(1.5.4)

which expresses conservation of energy in differential equation form. We need an additional equation that relates the heat flux to the temperature gradient called a constitutive equation. The simplest constitutive equation for heat flow is Fourier's law:

(1.5.5)

which states that heat flows from warm regions to cold regions at a rate proportional to the temperature gradient . The constant of proportionality is the coefficient of heat conductivity , which we assume to be a positive function in . Combining (1.5.4) and (1.5.5) gives the stationary heat equation in one dimension:

(1.5.6)

To define a solution uniquely, the differential equation is complemented by boundary conditions imposed at the boundary points and . A common example is the homogeneous Dirichlet conditions , corresponding to keeping the temperature at zero at the endpoints of the wire. The result is a two‐point BVP:

(1.5.7)

The boundary condition may be replaced by , corresponding to prescribing zero heat flux, or insulating the wire, at . Later, we also consider nonhomogeneous boundary conditions of the form or where and may be different from zero. For other types of boundary conditions, see Trinities (Section 1.2).

The time‐dependent heat equation in (1.5.2) describes the diffusion of thermal energy in a homogeneous material, where is the temperature at a position at time and is called thermal diffusivity or heat conductivity (corresponding to in (1.5.5)–(1.5.7)) of the material.