Equação de Liouville-Bratu-Gelfand

Na matemática, a Equação Liouville–Bratu–Gelfand or Equação de Liouville é uma equação de Poisson não linear, nomeada em homenagem aos matemáticos Joseph Liouville, Gheorghe Bratu e Israel Gelfand, que é descrita da seguinte forma

∇2ψ+λeψ=0

Essa equação aparece em problemas de fuga térmica, como na teoria de Frank-Kamenetskii, e na astrofísica, por exemplo, na equação Emden–Chandrasekhar. Esta equação pode descrever a carga espacial de eletricidade em torno de um fio brilhante ou até mesmo uma nebulosa planetária.

A solução de Liouville

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. </source>

Referências

1. https://en.wikipedia.org/wiki/Liouville%E2%80%93Bratu%E2%80%93Gelfand_equation
2. Scherer, CLÁUDIO. Métodos Computacionais da Física. 2010.