20.1 IMPORTANT PARTIAL DIFFERENTIAL EQUATIONS
The second term,f(r,t), represents a varying density of heat sources throughout
the material but is often not required in physical applications. In the most general
case,k,sandρmay depend on positionr, in which case the first term becomes
∇·(k∇u). However, in the simplest application the heat flow is one-dimensional
with no heat sources, and the equation becomes (in Cartesian coordinates)
∂^2 u
∂x^2=sρ
k∂u
∂t.20.1.3 Laplace’s equationLaplace’s equation,
∇^2 u=0, (20.5)may be obtained by setting∂u/∂t= 0 in the diffusion equation (20.2), and
describes (for example) thesteady-statetemperature distribution in a solid in
which there are no heat sources – i.e. the temperature distribution after a long
time has elapsed.
Laplace’s equation also describes the gravitational potential in a region con-taining no matter or the electrostatic potential in a charge-free region. Further, it
applies to the flow of an incompressible fluid with no sources, sinks or vortices;
in this caseuis the velocity potential, from which the velocity is given byv=∇u.
20.1.4 Poisson’s equationPoisson’s equation,
∇^2 u=ρ(r), (20.6)describes the same physical situations as Laplace’s equation, but in regions
containing matter, charges or sources of heat or fluid. The functionρ(r)is
called the source density and in physical applications usually contains some
multiplicative physical constants. For example, ifuis the electrostatic potential
in some region of space, in which caseρis the density of electric charge, then
∇^2 u=−ρ(r)/ 0 ,where 0 is the permittivity of free space. Alternatively,umight
represent the gravitational potential in some region where the matter density is
given byρ;then∇^2 u=4πGρ(r), whereGis the gravitational constant.
20.1.5 Schrodinger’s equation-The Schr ̈odinger equation
−^2
2 m∇^2 u+V(r)u=i∂u
∂t, (20.7)