We present formulations of the Trotter-Kato theorem for approximation of linear C0-semigroups which provide very useful framework when convergence of numerical approximations to solutions of PDEs are studied. Applicability of our results is demonstrated using a first order hyperbolic equation, a wave equation and Stokes’ equation as illustrative examples.

This work uses the dual-phase-lag (DPL) model of heat conduction to demonstrate the effect of temperature gradient relaxation time on the result of non-Fourier hyperbolic conduction in a finite slab subjected to a periodic thermal disturbance. DPL model combines the wave features of hyperbolic conduction with a diffusion-like feature of the evidence not captured by the hyperbolic case. For the ...

The nonlocal boundary value problem for a hyperbolic-parabolic equation in a Hilbert space H is considered. The difference schemes approximately solving this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established. In applications, the stability estimates for the solutions of the difference schemes of the mixed type boundary val...

Numerical Solution of hyperbolic conservation laws is important in computational fluid dynamics. In this paper, we present a new numerical method to solve the hyperbolic conservation laws, which is constructed by using the derivative of the cubic B-spline quasi-interpolation to approximate the spatial derivative of the dependent variable and first order forward difference to approximate the tim...

In the paper [7] we defined a map, called the hyperbolic Schwarz map, from the one-dimensional projective space to the three-dimensional hyperbolic space by use of solutions of the hypergeometric differential equation, and thus obtained closed flat surfaces belonging to the class of flat fronts. We continue the study of such flat fronts in this paper. First, we introduce the notion of derived S...

A result of Beyn on the replication of the phase portrait of an ordinary diierential equation in the neighborhood of a hyperbolic steady state by a one-step numerical method is generalized under appropriate assumptions to random diierential equations with hyperbolicity deened in terms of the multiplicative ergodic theorem of Oseledets. The proof is complicated by the need to compare cocycles ra...

We are concerned with the hyperbolic Keller-Segel model with quorum sensing, a model describing the collective cell movement due to chemical signalling with a flux limitation for high cell densities. This is a first order quasilinear equation, its flux depends on space and time via the solution to an elliptic PDE in which the right-hand side is the solution to the hyperbolic equation. This mode...

The existence of a unique weak solution to the homogeneous closed Dirichlet problem is proven for an elliptic-hyperbolic equation. The equation arises in a model for electromagnetic wave propagation in cold plasma. A class of open boundary value problems for the equation is shown to possess strong solutions. MSC2000 : 35M10, 35D05, 82D10.

Random walks are studied on disordered cellular networks in 2-and 3dimensional spaces with arbitrary curvature. The coefficients of the evolution equation are calculated in term of the structural properties of the cellular system. The effects of disorder and space-curvature on the diffusion phenomena are investigated. In disordered systems the mean square displacement displays an enhancement at...