: ''For Liouville's equation in differential geometry, see
Liouville's equation.''
In
mathematics, Liouville–Bratu–Gelfand equation or Liouville's equation is a non-linear
Poisson equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
, named after the mathematicians
Joseph Liouville, G. Bratu and
Israel Gelfand. The equation reads
:
The equation appears in
thermal runaway
Thermal runaway describes a process that is accelerated by increased temperature, in turn releasing energy that further increases temperature. Thermal runaway occurs in situations where an increase in temperature changes the conditions in a way t ...
as
Frank-Kamenetskii theory
In combustion, Frank-Kamenetskii theory explains the Thermal runaway, thermal explosion of a homogeneous mixture of reactants, kept inside a closed vessel with constant temperature walls. It is named after a Russian scientist David A. Frank-Kamene ...
,
astrophysics
Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the he ...
for example,
Emden–Chandrasekhar equation. This equation also describes space charge of electricity around a glowing wire and describes
planetary nebula
A planetary nebula (PN, plural PNe) is a type of emission nebula consisting of an expanding, glowing shell of ionized gas ejected from red giant stars late in their lives.
The term "planetary nebula" is a misnomer because they are unrelate ...
.
Liouville's solution
In two dimension with Cartesian Coordinates
,
Joseph Liouville proposed a solution in 1853 as
: