Rate of convergence of the method of lines for the nonlinear parabolic integrodifferential equation describing a photon recycling diode

We consider the Dirichlet problem and the mixed boundary-value problem for a nonlinear integro-differential equation of a parabolic type, which provides a mathematical model of the various modes in a photon-recycling semiconductor diode. The existence of the mixed derivative ²u/tx is proved for the solution of these problems. Method-of-lines schemes are constructed. An O(h) rate of convergence bound is established for the equation with nonlinear integral and differential components and an O(h²) bound for the equation with a nonlinear integral part.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 21–31, 1992;     

Spin correlations and a mesoscopic structure in Ni-Mn-Ga

The mesoscopic structures of the Heusler alloys Ni_49.1Mn_29.4Ga_21.5 and Ni₂MnGa are studied by small-angle polarized neutron scattering in the temperature range 15 < T < 400 K. The characteristic temperatures of phase transformations (ferromagnetic, martensitic, and premartensitic transformations) and the characteristic sizes of mesoscopic inhomogeneities in them have been determined. Differences in the spin dynamics of these phases and magnetic-nuclear interference upon neutron scattering have been revealed. The evolutions of the mesoscopic structures in the nonstoichiometric and stoichiometric alloys are found to be substantially different.     

Finite-difference solution of a nonlinear initial-boundary-value problem for a nonlinear parabolic equation of second order

An implicit finite-difference scheme is constructed for solving a nonlinear initial-boundary-value problem for a nonlinear homogeneous parabolic equation of second order with a nonlinear boundary condition that contains the time derivative of the sought function. The results are used for numerical solution of the mathematical model of internal-diffusion kinetics of adsorption from a constant bounded volume.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 34–46, 1988.     

Data-sparse approximation to the operator-valued functions of elliptic operator

In previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent

In the present paper, we consider various operator functions, the operator exponential negative fractional powers , the cosine operator function and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.