Introducing the notion of thermal entropy density via the first law of thermodynamics and assuming the Einstein equation as an equation of thermal state, we obtain the thermal entropy density of any arbitrary spacetime without assuming a temperature or a horizon. The results confirm that there is a profound connection between gravity and thermodynamics.

The coupled viscous Burgers’ equation which was derived by Esipov in 1995 to study the model of poly-dispersive sedimentation. The system of coupled viscous Burgers’ equation is a simple model of sedimentation or evolution of scaled volume concentrations of two kinds of particles in fluid suspensions or colloids under the effect of gravity. Coupled Burgers’ equation can be written in the follow...

The Benjamin-Bona-Mahony equation has improved short-wavelength behaviour, as compared to the Korteweg-de Vries equation, and is another uni-directional wave equation with cnoidal wave solutions. Cnoidal wave solutions can appear in other applications than surface gravity waves as well, for instance to describe ion acoustic waves in plasma physics. Using a computerized symbolic computation tech...

Equivalent Lagrangians are used to find, via transformations, solutions and conservation law of a given differential equation by exploiting the possible existence of an isomorphic algebra of Lie point symmetries and, more particularly, an isomorphic Noether point symmetry algebra. Applications include ordinary differential equations such as the Kummer equation and the combined gravity-inertial-...

1. Poisson Equation 1 2. Outline of Topics 3 2.1. Finite Difference Method 3 2.2. Finite Element Method 3 2.3. Finite Volume Method 3 2.4. Sobolev Spaces and Theory on Elliptic Equations 3 2.5. Iterative Methods: Conjugate Gradient and Multigrid Methods 3 2.6. Nonlinear Elliptic Equations 3 2.7. Fast Multiple Method 3 3. Physical Examples 4 3.1. Gauss’ law and Newtonian gravity 4 3.2. Electrost...

We show that Fokker-Planck equation for chordal SLE process under a simple rescaling of the probability density can be traced to the minisuperspace Wheeler-de Witt equation for boundary operator in 2d Liouville gravity. Insertion of an operator, calculating SLE critical exponent, corresponds to adding matter contribution to WdW equation. This observation may be useful for understanding of why S...