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, named after the mathematicians Joseph Liouville, Gheorghe Bratu and Israel Gelfand. The equation reads
∇2ψ+λeψ=0
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 and describes planetary nebula.
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Liouville's solution
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In two dimension with Cartesian Coordinates (x,y), Joseph Liouville proposed a solution in 1853 as
λeψ(u2+v2+1)2=2[(∂u∂x)2+(∂u∂y)2]
where f(z)=u+iv is an arbitrary analytic function with z=x+iy. In 1915, G.W. Walker found a solution by assuming a form for f(z). If r2=x2+y2, then Walker's solution is
8e−ψ=λ[(ra)n+(ar)n]2
where a is some finite radius. This solution decays at infinity for any n, but becomes infinite at the origin for n<1 , becomes finite at the origin for n=1 and becomes zero at the origin for n>1. Walker also proposed two more solutions in his 1915 paper.
Radially symmetric forms
If the system to be studied is radially symmetric, then the equation in n dimension becomes
ψ″+n−1rψ′+λeψ=0
where r is the distance from the origin. With the boundary conditions
ψ′(0)=0,ψ(1)=0
and for λ≥0, a real solution exists only for λ∈[0,λc], where λc is the critical parameter called as Frank-Kamenetskii parameter. The critical parameter is λc=0.8785 for n=1, λc=2 for n=2 and λc=3.32 for n=3. For n=1, 2, two solution exists and for 3≤n≤9 infinitely many solution exists with solutions oscillating about the point λ=2(n−2). For n≥10, the solution is unique and in these cases the critical parameter is given by λc=2(n−2). Multiplicity of solution for n=3 was discovered by Israel Gelfand in 1963 and in later 1973 generalized for all n by Daniel D. Joseph and Thomas S. Lundgren.
The solution for n=1 that is valid in the range λ∈[0,0.8785] is given by
ψ=−2ln[e−ψm/2cosh(λ2e−ψm/2r)]
where ψm=ψ(0) is related to λ as
eψm/2=cosh(λ2e−ψm/2).
The solution for n=2 that is valid in the range λ∈[0,2] is given by
ψ=ln[64eψm(λeψmr2+8)2]
where ψm=ψ(0) is related to λ as
(λeψm+8)2−64eψm=0.