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We consider self-similar approximations of nonlinear hyperbolic systems in one space dimension with Riemann initial data and general diffusion matrix. We assume that the matrix of the system is strictly hyperbolic and the diffusion matrix is close to the identity. No genuine nonlinearity assumption is required. We show the existence of a smooth, self-similar solution which has bounded total variation, uniformly in the diffusion parameter. In the zero-diffusion limit, the solutions converge to a solution of the Riemann problem associated with the hyperbolic system. A similar result is established for the relaxation approximation and the boundary-value problem in a half-space for the same regularizations.
This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solu
In this paper we develop an existence theory for the nonlinear initial-boundary value problem with singular diffusion $partial_t u = text{div}(k(x) abla G(u))$, $u|_{t=0}=u_0$ with Neumann boundary conditions $k(x) abla G(u)cdot u = 0$. Here $xin Bs
In this paper, we consider artificial boundary conditions for the linearized mixed Korteweg-de Vries (KDV) Benjamin-Bona-Mahoney (BBM) equation which models water waves in the small amplitude, large wavelength regime. Continuous (respectively discret
We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posednes
We consider a linear size-structured population model with diffusion in the size-space. Individuals are recruited into the population at arbitrary sizes. The model is equipped with generalized Wentzell-Robin (or dynamic) boundary conditions. This all