Models of a linearized Boltzmann collision integral

For rarefied gas flows at moderate and low Knudsen numbers, model equations are derived that approximate the Boltzmann equation with a linearized collision integral. The new kinetic models generalize and refine the S-model kinetic equation.     

Reliable computation of eigenvalues of the magnetostatic integral operator

Some variational problems in magnetostatics can be reformulated as eigenvalue problems for vector surface integral operators in appropriate function spaces, e.g., the magnetostatic integral operator is of considerable interest in the theory of permanent magnetization of compact bodies. In the case that the underlying surface is either a sphere, a spheroid, or a triaxial ellipsoid, explicit expressions for eigenvalues and eigenfunctions are well known. For the ellipsoid, these quantities are given in terms of Lamé functions and surface ellipsoidal harmonics. Since there is an apparent lack in literature we provide an new effective scheme for the reliable computation of these functions and of the corresponding eigenvalues of the magnetostatic operator.     

On the numerical method for solving a hypersingular integral equation with the computation of the solution gradient

We consider a linear integral equation, which arises when solving the Neumann boundary value problem for the Laplace equation with the representation of the solution in the form of a double layer potential, with a hypersingular integral treated in the sense of Hadamard finite value. We consider the case in which the exterior or interior problem is solved in a domain whose boundary is a closed smooth surface and the integral equation is written over that surface. A numerical scheme for solving the integral equation is constructed with the use of quadrature formulas of the type of the method of discrete singularities with a regularization for the use of an irregular grid. We prove the convergence, uniform over the grid points, of the numerical solutions to the exact solution of the hypersingular equation and, in addition, the uniform convergence of the values of the approximate finite-difference derivative operator on the numerical solution to the values on the projection of the exact solution onto the subspace of grid functions with nodes at the collocation points.     

On a formula for the computation of uniform integral means of piecewise continuous functions on the half-line

For the computation of lower and upper limits of integral means of piecewise continuous functions as the length of the integration interval tends to +∞, we obtain a formula that permits one, without loss of generality, to assume that the endpoints of the integration interval belong to closed intervals into which the half-line is divided by any sequence in which the difference between neighboring terms tends to +∞.     

Numerical computation of the projection of a point onto a polyhedron

Some active-set methods are studied for the computation of the projection of a point onto a polyhedron. Special attention is given to the study of the propagation of computation errors. Error bounds for the solution due to the propagation of the data perturbations (inherent errors) are given. Then, an extensive numerical experimentation on test problems is performed. Finally, the errors of the computed solutions are compared with the inherent errors.This research was partially supported by the Progetto Finalizzato Informatica (Sottoprogetto P1, Sofmat), CNR, Rome, Italy.     

Increasing the accuracy of eigenvalue computation in the spectral problem

A simple method is proposed for increasing the accuracy of computation of the eigenvalues of elliptic operators, which does not require high-order accurate schemes.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 63–68, 1987.     

Numerical computation of the Tau approximation for the Volterra-Hammerstein integral equations

In this work, we propose an extension of the algebraic formulation of the Tau method for the numerical solution of the nonlinear Volterra-Hammerstein integral equations. This extension is based on the operational Tau method with arbitrary polynomial basis functions for constructing the algebraic equivalent representation of the problem. This representation is an special semi lower triangular system whose solution gives the components of the vector solution. We will show that the operational Tau matrix representation for the integration of the nonlinear function can be represented by an upper triangular Toeplitz matrix. Finally, numerical results are included to demonstrate the validity and applicability of the method and some comparisons are made with existing results. Our numerical experiments show that the accuracy of the Tau approximate solution is independent of the selection of the basis functions.     

Local principle of computation of the factor-norms of singular integral operators

V. I. Lenin Kishinev State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 2, pp. 57–58, April–June, 1988.     

The Bergman projection as a singular integral operator

We show that the Bergman projection operator, associated to one of three classes of domains (all smoothly bounded)-a finite type domain ?²; a decoupled, finite type domain in ?ⁿ; or a convex, finite type domain in wfⁿ-may be viewed as a generalized Calderón-Zygmund operator. As an application of this observation, we show that the Bergman projector on any of these domains preserves the Lebesgue classesL ^p , 1 <p < ∞.     

On optimal high accuracy linear multistep methods for first kind volterra integral equations

General linear multistep methods which include those of Holyhead et al. [9] and Gladwin et al. [8] are introduced. A new simple root condition for stability is deduced. A theorem relating the coefficients of this new associated stability polynomial to those of the method is proved. This permits a constructive approach for obtaining high accuracy convergent methods. Two such methods are derived and numerical results are presented.This work was done under the financial support of: Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), and Financiadora Nacional de Empreendimentos e Projetos (FINEP)—Ministério do Planejamento (Brazil).     

On the computation of a-invariants

Thea-invariant of a graded Cohen-Macaulay ring is the least degree of a generator of its graded canonical module. We compute thea-invariants of (i) graded algebras with straightening laws on upper semi-modular lattices and (ii) the Stanley-Reisner rings of shellable weighted simplicial complexes. The formulas obtained are applied to rings defined by determinantal and pfaffian ideals.     

On the collision of rigid bodies

The problem of collision of two rigid bodies has been reexamined and several alternative conditions, which extend the validity of the so-called Newton's law of impact, have been derived. The obtained results concern, among others, cases of constrained impulsive motion of rigid bodies.     

On accuracy and efficiency of the CESE-method of C. S. Chang for one-dimensional unstaedy flow computation

About twenty years ago a NASA-group around S. C. Chang began to develop a method using space-time domains as elements for unstaedy flow computation. There are two kinds of elements. One within which the linearized form of the underlying differential equation is satisfied and one whithin which space-time flux is conserved. Therefore the method is called conservation-element solution-element or shortly CESE. Applied to one-dimensional flow equations we show conditions under which the method in the version of Jerez et al. [1] is - second order accurate and - more efficient than Lax-Wendroff type methods. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary integral equation method of higher computational accuracy

The boundary integral equation method presented in the paper features the following: (1) no singular kernels, strong or weak, are involved, and computationally no local “element” approximations are needed; (2) the integral equations are well conditioned, including the cases of bounded and multiply connected regions, and no iterative approximations are involved; (3) no post-solution differentiation is involved. These features provide for a higher computational efficiency. The method solves in full a number of engineering problems, and can be used for the stiffness matrix formulation in more complex situations.     

On the nontrivial projection problem

The nontrivial projection problem asks whether every finite-dimensional normed space admits a well-bounded projection of nontrivial rank and corank or, equivalently, whether every centrally symmetric convex body (of arbitrary dimension) is approximately affinely equivalent to a direct product of two bodies of nontrivial dimensions. We show that this is true “up to a logarithmic factor.”