In this paper we construct fundamental solutions of a scalar conservation law in one space dimension. These source-type solutions are well known for a convex case and hence our focus is on a general non-convex case which may have a finite number of inflection points. Signed fundamental solutions are constructed first and then under an extra hypothesis on nonnegativity of the flux two parameter ...

An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An e...

This paper is concerned with the asymptotic convergence of numerical solutions toward discrete travelling waves for a class of relaxation numerical schemes, approximating the scalar conservation law. It is shown that if the initial perturbations possess some algebraic decay in space, then the numerical solutions converge to the discrete travelling wave at a corresponding algebraic rate in time,...

This paper deals with the Burgers equation which is most common model used in nonlinear conservation laws. Here theoretical aspect of law discussed by using inviscid equation. At first, we introduce general non-linear as a partial differential and its solution procedure method characteristic. Next, present weak problem entropy condition. Taking into account shock wave rarefaction wave, Riemann ...

The unsmooth boundary will greatly affect motion morphology of a shallow water wave, and a fractal space is introduced to establish a generalized KdV-Burgers equation with fractal derivatives. The semi-inverse method is used to establish a fractal variational formulation of the problem, which provides conservation laws in an energy form in the fractal space and possible solution structures of t...