Liouville–Bratu–Gelfand equation

Short description: Non-linear partial differential 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, Gheorghe Bratu and Israel Gelfand. The equation reads

${\mathrm{\nabla }}^2\psi +\lambda e^{\psi }=0$

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

Source:

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

$\lambda e^{\psi }(u^2+v^2+1{)}^2=2[{\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial u}{\partial y}\right)}^2]$

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 $r^2=x^2+y^2$, then Walker's solution is

$8e^{-\psi }=\lambda {[{\left(\frac{r}{a}\right)}^n+{\left(\frac{a}{r}\right)}^n]}^2$

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

${\psi }^{″}+\frac{n-1}{r}{\psi }^{\prime }+\lambda e^{\psi }=0$

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

${\psi }^{\prime }(0)=0,\psi (1)=0$

and for $\lambda \ge 0$, a real solution exists only for $\lambda \in [0,{\lambda }_c]$, 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 $3\le n\le 9$ infinitely many solution exists with solutions oscillating about the point $\lambda =2(n-2)$. For $n\ge 10$, 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.

The solution for $n=1$ that is valid in the range $\lambda \in [0,0.8785]$ is given by

$\psi =-2\ln [e^{-{\psi }_m/2}\cosh \left(\frac{\sqrt{\lambda }}{\sqrt{2}}{e}^{-{\psi }_m/2}r\right)]$

where ${\psi }_m=\psi (0)$ is related to $\lambda$ as

$e^{{\psi }_m/2}=\cosh \left(\frac{\sqrt{\lambda }}{\sqrt{2}}{e}^{-{\psi }_m/2}\right).$

The solution for $n=2$ that is valid in the range $\lambda \in [0,2]$ is given by

$\psi =\ln \left[\frac{64e^{{\psi }_m}}{(\lambda e^{{\psi }_m}{r}^2+8{)}^2}\right]$

where ${\psi }_m=\psi (0)$ is related to $\lambda$ as

$(\lambda e^{{\psi }_m}+8{)}^2-64e^{{\psi }_m}=0.$