In mathematics, Liouville–Bratu–Gelfand equation or Liouville's equation is a non-linear Poisson equation, named after the mathematicians Joseph Liouville,[1] Gheorghe Bratu[2] and Israel Gelfand.
[3] The equation reads The equation appears in thermal runaway as Frank-Kamenetskii theory, astrophysics for example, Emden–Chandrasekhar equation.
This equation also describes space charge of electricity around a glowing wire[4] and describes planetary nebula.
Source:[5] In two dimension with Cartesian Coordinates
, Joseph Liouville proposed a solution in 1853 as where
is an arbitrary analytic function with
Walker[6] found a solution by assuming a form for
, then Walker's solution is where
is some finite radius.
This solution decays at infinity for any
, but becomes infinite at the origin for
, becomes finite at the origin for
Walker also proposed two more solutions in his 1915 paper.
If the system to be studied is radially symmetric, then the equation in
is the distance from the origin.
With the boundary conditions and for
λ ≥ 0
, a real solution exists only for
λ ∈ [ 0 ,
is the critical parameter called as Frank-Kamenetskii parameter.
The critical parameter is
, two solution exists and for
infinitely many solution exists with solutions oscillating about the point
, the solution is unique and in these cases the critical parameter is given by
Multiplicity of solution for
was discovered by Israel Gelfand in 1963 and in later 1973 generalized for all
by Daniel D. Joseph and Thomas S.
that is valid in the range
that is valid in the range