Class Meeting # 24: Transport Equations and Burger’s Equation In these notes, we introduce a class of evolution PDEs known as transport equations. Such equations arise in a physical context whenever a quantity is “transported” in a certain direction. Some important physical examples include the mass density flow for an incompressible fluid, and the Boltzmann equation of kinetic theory. We discu...

in this paper. (1) we determine the complex-valued solutions of the following variant of van vleck's functional equation $$int_{s}f(sigma(y)xt)dmu(t)-int_{s}f(xyt)dmu(t) = 2f(x)f(y), ;x,yin s,$$ where $s$ is a semigroup, $sigma$ is an involutive morphism of $s$, and $mu$ is a complex measure that is linear combinations of dirac measures $(delta_{z_{i}})_{iin i}$, such that for all $iin i$,...

In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation $$int_{S}f(sigma(y)xt)dmu(t)-int_{S}f(xyt)dmu(t) = 2f(x)f(y), ;x,yin S,$$ where $S$ is a semigroup, $sigma$ is an involutive morphism of $S$, and $mu$ is a complex measure that is linear combinations of Dirac measures $(delta_{z_{i}})_{iin I}$, such that for all $iin I$, $z_{...

let $rin textbf{c}^{mtimes m}$ and $sin textbf{c}^{ntimes n}$ be nontrivial involution matrices; i.e., $r=r^{-1}neq pm~i$ and $s=s^{-1}neq pm~i$. an $mtimes n$ complex matrix $a$ is said to be an $(r, s)$-symmetric ($(r, s)$-skew symmetric) matrix if $ras =a$ ($ ras =-a$). the $(r, s)$-symmetric and $(r, s)$-skew symmetric matrices have a number of special properties and widely used in engi...

We consider the solution to Burger’s equation coupled to a stochastic noise in Ito’s sense. The main random properties of the wave are determined. The solution is related to a deterministic problem with a rescaled diffusion coefficient. Depending on the value of a parameter, the initial value problem may be ill posed, well posed up to an explosion time, or well posed for all time. Traveling wav...