Narrow Results

This paper is concerned with a class of nonlinear reaction–hyperbolic systems as models for axonal transport in neuroscience. We show the global existence of entropy-satisfying BV-solutions to the initial-value problems by using hyperbolic-type methods. Moreover, we rigorously justify the limit as the biochemical processes are much faster than the transport ones.

We study the compressible Euler equations in spherical symmetry with a gravitational source surrounding a solid ball. Despite its physical importance, this problem has not received much attention until now. We prove the global existence of weak solutions by constructing approximate solutions and using a modified Godunov scheme. The main point is to obtain a sup-norm bound for the approximate solutions.

The general finite difference schemes with intrinsic parallelism for the boundary value problem of the semilinear parabolic system of divergence type with bounded measurable coefficients is studied. By the approach of the discrete functional analysis, the existence and uniqueness of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. Moreover the unconditional stability of the general difference schemes with intrinsic parallelism justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete initial data of the original problems in the discrete norms. Finally the convergence of the discrete vector solutions of the certain difference schemes with intrinsic parallelism to the unique generalized solution of the original semilinear parabolic problem is proved.

No abstract received.

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a quasilinear Sobolev or pseudo-parabolic equation with initial jump. We have derived a method based on using finite elements with piecewise linear functions in space and with exponential functions in time and appropriate quadrature formulae with remainder term in integral form. In the initial layer, we introduce a special non-uniform mesh which is constructed by using estimates of derivatives of the exact solution and the analysis of the local truncation error. For the time integration we use the implicit rule. The fully discrete scheme is shown to be convergent of order 2 in space and of order one in time, uniformly in the singular perturbation parameter. Numerical results supporting the theory are presented.

The method proposed presently replaces the propagating rupture on the fault surface by a fictitious focal point and a seismograph station in the vicinity of the given soil site. Infinite elements are adopted in the far field and finite elements in the near field. A fictitious focal point and seismograph station scheme is used to calibrate the free field ground motion of the soil site. The seismic analysis of an embedded body, which is finite, uses the difference scheme to solve the problem. The impedance equations, governing the difference between the embedded body and the seismic free field, contain the difference displacements and the already known free-field quantities. No infinite element free-field node is involved in the analysis of the difference system. For an embedded long and slender body, the part of interest of the body should be discretized into finite elements in the near field, and the remaining part of the body into infinite elements in the far field. The analysis described for a finite body is followed; and no infinite element free-field node, beside those inside the region where the actual long body will be embedded, is involved.

The interaction between diffusion and absorption gives several interesting phenomena in the flow through porous media with some kind of evaporation effect. In particular, it can be observed from numerical computations that the effect of strong absorption causes support splitting or non-splitting phenomena. The aim of this paper is to find some sufficient conditions under which such phenomena appear.

A numerical method is proposed for solving Bitsadze-Samarskii-type nonlocal boundary value problem for multidimensional elliptic partial differential equations. The second and fourth orders of accuracy stable difference schemes are presented. The stability and almost coercive stability of these difference schemes are established. The method is illustrated by numerical examples.