Liouville–Bratu–Gelfand Equation

: ''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, G. Bratu and Israel Gelfand. The equation reads

:$\nabla^2 \psi + \lambda e^\psi = 0$

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.

Liouville's solution

In two dimension with Cartesian Coordinates $(x,y)$, Joseph Liouville proposed a solution in 1853 as

:\lambda e^\psi (u^2 + v^2 + 1) ^2 = 2 \left left(\frac\right)^2 + \left(\frac\right)^2\right/math>

where $f(z)=u + i v$ 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 $r^2=x^2+y^2$, then Walker's solution is

:8 e^ = \lambda \left left(\frac\right)^n + \left(\frac\right)^n\right2

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

:$\psi'' + \frac\psi' + \lambda e^\psi=0$

where $r$ is the distance from the origin. With the boundary conditions

:$\psi'(0)=0, \quad \psi(1) = 0$

and for $\lambda\geq 0$, a real solution exists only for \lambda \in ,\lambda_c/math>, where $\lambda_c$ is the critical parameter called as Frank-Kamenetskii parameter. The critical parameter is $\lambda_c=0.8785$ for $n=1$, $\lambda_c=2$ for $n=2$ and $\lambda_c=3.32$ for $n=3$. For $n=1, \ 2$, two solution exists and for $3\leq n\leq 9$ infinitely many solution exists with solutions oscillating about the point $\lambda=2(n-2)$. For $n\geq 10$, the solution is unique and in these cases the critical parameter is given by $\lambda_c=2(n-2)$. 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.Joseph, D. D., and T. S. Lundgren. "Quasilinear Dirichlet problems driven by positive sources." Archive for Rational Mechanics and Analysis 49.4 (1973): 241-269.

The solution for $n=1$ that is valid in the range \lambda \in ,0.8785/math> is given by

:\psi = -2 \ln \left ^\cosh \left(\frace^r\right)\right/math>

where $\psi_m=\psi(0)$ is related to $\lambda$ as

:$e^ = \cosh \left(\frace^\right).$

The solution for $n=2$ that is valid in the range \lambda \in ,2/math> is given by

:\psi = \ln \left frac\right/math>

where $\psi_m=\psi(0)$ is related to $\lambda$ as

:$(\lambda e^+8)^2 - 64 e^ =0.$

References

{{DEFAULTSORT:Liouville-Bratu-Gelfand equation
Equations of physics
Differential equations