Singularity Formation in the Inviscid Burgers Equation

We provide a lower bound for the blow up time of the H2 norm of the entropy solutions of the inviscid Burgers equation in terms of the H2 norm of the initial datum. This shows an interesting symmetry of the Burgers equation: the invariance of the space H2 under the action of such nonlinear equation. The argument is based on a priori estimates of energy and stability type for the (viscous) Burgers equation.

Lie Group Analysis of the Time-delayed Inviscid Burgers' Equation

In this paper, we discuss group analysis of rst-order delay partial di erential equations and use it to obtain symmetries of the Invis- cid Burgers' equation with delay, its kernel and extensions of the kernel. We obtain a Lie type invariance condition for rst-order delay partial di erential equations by using Taylor's theorem for a function of several variables. We obtain the symmetries admitted by this delay partial di er- ential equation. Further, we obtain representations of analytic solutions and the reduced equations from the symmetries.

Effects of a Logistic Reaction to Finite Difference Numerical Solutions of the Inviscid Burgers Equation

We solve the inviscid Burgers equation involving a logistic reaction. The goal is to investigate the effects of the logistic reaction to numerical solutions of the inviscid Burgers equation. We use standard and nonstandard finite difference methods. Based on our research results, the logistic reaction in the inviscid Burgers equation makes the shock wave propagates faster.