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.     

Statistical description of a cloud of compressible bubbles

We use the methods of statistical mechanics to describe the interaction of N compressible gas bubbles in an incompressible, inviscid and irrotational liquid. The governing equations for bubble positions, radii and corresponding momenta form a Hamiltonian system depending on the virtual mass matrix. An explicit expression of the virtual mass matrix is presented, which is calculated with accuracy (b/d)³, where b and d are respectively the mean bubble radius and the mean inter-bubble distance. We study two limit cases: the limit of moving rigid spheres and the limit of immobile oscillating bubbles. In each case, we construct a canonical ensemble partition function. In the limit of rigid spheres, we improve results by Yurkovetsky and Brady (phys Fluids 8(4): 881–895, 1996). In particular, we derive an analytic expression for the “attractive” potential which may be responsible for the clustering effect, and show why the accuracy (b/d)³ is not sufficient to characterize the “repulsive potential” . In the limit of immobile oscillating bubbles, we prove the existence of a long range repulsive potential.     

Amply regular graphs and block designs

We study the amply regular diameter d graphs Γ such that for some vertex a the set of vertices at distance d from a is the set of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set of vertices at distance d-1 from a. We prove that the subgraph induced by the set of points is a clique, a coclique, or a strongly regular diameter 2 graph. For diameter 3 graphs we establish that this construction is a 2-design for each vertex a if and only if the graph is distance-regular and for each vertex a the subgraph Γ₃(a) is a clique, a coclique, or a strongly regular graph. We obtain the list of admissible parameters for designs and diameter 3 graphs under the assumption that the subgraph induced by the set of points is a Seidel graph. We show that some of the parameters found cannot correspond to distance-regular graphs.     

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.

     

Linear and nonlinear sloshing in a circular conical tank

Linear and nonlinear fluid sloshing problems in a circular conical tank are studied in a curvilinear coordinate system. The linear sloshing modes are approximated by a series of the solid spheric harmonics. These modes are used to derive a new nonlinear modal theory based on the Moiseyev asymptotics. The theory makes it possible to both classify steady-state waves occurring due to horizontal resonant excitation and visualise nonlinear wave patterns. Secondary (internal) resonances and shallow fluid sloshing (predicted for the semi-apex angles >60) are extensively discussed.