Main purpose of this paper is to study the local propagation singularities viscosity solution contact type evolutionary Hamilton-Jacobi equationDtu(t,x)+H(t,x,Dxu(t,x),u(t,x))=0. An important issue topic existence, uniqueness and regularity strict singular characteristic. We apply recent existence results on Herglotz' variational problem aforementioned equation with smooth initial data. obtain ...

This paper is concerned with solving the Cauchy problem for an elliptic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, numerical convergence analysis of the energy-like minimization method is carried out and leads to adapted stopping criteria for the minimization process depending on the noise rate. Num...

A semi-spectral Chebyshev method for solving numerically singular integral equations is presented and applied in the quarkonium bound-state problem in momentum space. The integrals containing both, logarithmic and Cauchy singular kernels, can be evaluated without subtractions by dedicated automatic quadratures. By introducing a Chebyshev mesh and using the Nystrom algorithm the singular integra...

In this paper, we present Legendre wavelet method to obtain numerical solution of a singular integro-differential equation. The singularity is assumed to be of the Cauchy type. The numerical results obtained by the present method compare favorably with those obtained by various Galerkin methods earlier in the literature.

Using the definitions of δand δ′-shocks for the systems of conservation laws [12], [13], [39], the Rankine–Hugoniot conditions for δand δ′-shocks are derived. We present a construction of solutions to the Cauchy problems admitting δand δ′-shocks. In particular, the Riemann problem admitting shocks, δ-shocks, δ′-shocks, and vacuum states is considered. The geometric aspects of δand δ′-shocks are...

This paper presents a general solution of hyperbolic type equation with second-order singular coefficient and to the Cauchy problem posed for this equation.