Liouville–Bratu–Gelfand equation

For Liouville's equation in differential geometry, see Liouville's equation.

In mathematics, Liouville–Bratu–Gelfand equation or Lioville's equation is a non-linear eigenvalue Poisson equation, named after the mathematicians Joseph Liouville,^[1] G. Bratu^[2] and Israel Gelfand.^[3] The equation reads

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

The equation appears in thermal runaway as Frank-Kamenetskii theory, astrophysics for example, Chandrasekhar equation. This equation also describes space charge of electricity round a glowing wire^[4] and describes planetary nebula.

Liouville's solution^[5]

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 {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial u}{\partial y}}\right)^{2}\right]

where $f(z)=u+iv$ is an arbitrary analytic function with $z=x+iy$ . In 1915, G.W. Walker^[6] 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[\left({\frac {r}{a}}\right)^{n}+\left({\frac {a}{r}}\right)^{n}\right]^{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 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 '+\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 $\lambda \geq 0$ , the solution exists only for $\lambda \in [0,\lambda _{c}]$ , where $\lambda _{c}$ is the critical parameter. The critical parameter is $\lambda _{c}=0.88$ 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 $n=3$ is discovered by Israel Gelfand in 1963 and in later 1973 generalized for all $n$ by Daniel D. Joseph and Thomas S. Lundgren^[7].

References

↑ Liouville, J. "Sur l’équation aux différences partielles ${\frac {d^{2}\log \lambda }{dudv}}\pm {\frac {\lambda }{2a^{2}}}=0$ ." Journal de mathématiques pures et appliquées (1853): 71–72. http://sites.mathdoc.fr/JMPA/PDF/JMPA_1853_1_18_A3_0.pdf
↑ Bratu, G. "Sur les équations intégrales non linéaires." Bulletin de la Société Mathématique de France 42 (1914): 113–142.http://archive.numdam.org/article/BSMF_1914__42__113_0.pdf
↑ Gelfand, I. M. "Some problems in the theory of quasilinear equations." Amer. Math. Soc. Transl 29.2 (1963): 295–381. http://www.mathnet.ru/links/aa75c5d339030f17940afb64e17793d8/rm7290.pdf
↑ Richardson, Owen Willans. The emission of electricity from hot bodies. Longmans, Green and Company, 1921.
↑ Bateman, Harry. "Partial differential equations of mathematical physics." Partial Differential Equations of Mathematical Physics, by H. Bateman, Cambridge, UK: Cambridge University Press, 1932 (1932).
↑ Walker, George W. "Some problems illustrating the forms of nebulae." Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 91.631 (1915): 410-420.http://www.jstor.org/stable/pdf/93512.pdf?refreqid=excelsior%3Af4a4cc9656b8bbd9266f9d32587d02b1
↑ 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.

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Liouville–Bratu–Gelfand equation

Liouville's solution[5]

Radially symmetric forms

References

Liouville's solution^[5]