Liouville–Bratu–Gelfand Equation
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: ''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 Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
, 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 Thermal runaway describes a process that is accelerated by increased temperature, in turn releasing energy that further increases temperature. Thermal runaway occurs in situations where an increase in temperature changes the conditions in a way t ...
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Frank-Kamenetskii theory In combustion, Frank-Kamenetskii theory explains the Thermal runaway, thermal explosion of a homogeneous mixture of reactants, kept inside a closed vessel with constant temperature walls. It is named after a Russian scientist David A. Frank-Kamene ...
,
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the he ...
for example, Emden–Chandrasekhar equation. This equation also describes space charge of electricity around a glowing wire and describes
planetary nebula A planetary nebula (PN, plural PNe) is a type of emission nebula consisting of an expanding, glowing shell of ionized gas ejected from red giant stars late in their lives. The term "planetary nebula" is a misnomer because they are unrelate ...
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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 In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
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 Daniel Donald Joseph (March 26, 1929 – May 24, 2011) was an American mechanical engineer. He was the Regents Professor Emeritus and Russell J. Penrose Professor Emeritus of Department of Aerospace Engineering and Mechanics at the University of M ...
and
Thomas S. Lundgren Thomas S. Lundgren is an American fluid dynamicist and Professor Emeritus of Aerospace Engineering and Mechanics at the University of Minnesota He is known for his work in the field of theoretical fluid dynamics. In 2006, Lundgren received Fluid D ...
.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 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/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