Analytic solution of boundary-value problems for nonstationary model kinetic equations

A theory for constructing the solutions of boundary-value problems for nonstationary model kinetic equations is constructed. This theory was incorrectly presented in the recent well-known monographs of Cercignani [1,2]. After application of a Laplace transformation to the studied equation, separation of the variables is used, this leading to a characteristic equation. Eigenfunctions are found in the space of generalized functions, and the eigenvalue spectrum is investigated. An existence and uniqueness theorem for the expansion of the Laplace transform of the solution with respect to the eigenfunctions is proved. The proof is constructive and gives explicit expressions for the expansion coefficients. An application to the Rayleight problem is obtained, and the corresponding result of Cercignani is corrected.Moscow Pedagogical University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 127–138, July, 1992.     

A Lipschitz constant formula for vector addition in cones with applications to Stein-like equations

For a cone C equipped with the Thompson metric and a ? C{ain C}, we show that the translation map x? x+a{xmapsto x+a} is a strict contraction on any lower (or initial) set C?x-C{Ccap x-C} of the cone and derive an explicit formula for the Lipschitz constant. We apply our results to Stein equations, Riccati equations, and Ferrante–Levy equations on normal cones of Banach spaces to establish the existence, uniqueness and continuous dependence on parameters of positive solutions.     

Regularity of the solution of Hammerstein equations with weakly singular kernel

Some regularity properties of the solution to a class of weakly singular Hammerstein equations are derived. The results obtained in this paper extend the results of C. Schneider [4], where he obtains similar properties for the solution to weakly singular Fredholm equations of the second kind.     

Adjoint equations and their applications

Proceedings of the Steklov Institute of Mathematics -     

Some vector sequence transformations with applications to systems of equations

First, recursive algorithms for implementing some vector sequence transformations are given. In a particular case, these transformations are generalizations of Shanks transformation and the G-transformation. When the sequence of vectors under transformation is generated by linear fixed point iterations, Lanczos' method and the CGS are recovered respectively. In the case of a sequence generated by nonlinear fixed point iterations, a quadratically convergent method based on the -algorithm is recovered and a nonlinear analog of the CGS method is obtained.     

On the solution of systems of equations with constant rank derivatives

The famous for its simplicity and clarity Newton–Kantorovich hypothesis of Newton’s method has been used for a long time as the sufficient convergence condition for solving nonlinear equations. Recently, in the elegant study by Hu et al. (J Comput Appl Math 219:110–122, 2008), a Kantorovich-type convergence analysis for the Gauss–Newton method (GNM) was given improving earlier results by Häubler (Numer Math 48:119–125, 1986), and extending some results by Argyros (Adv Nonlinear Var Inequal 8:93–99, 2005, 2007) to hold for systems of equations with constant rank derivatives. In this study, we use our new idea of recurrent functions to extend the applicability of (GNM) by replacing existing conditions by weaker ones. Finally, we provide numerical examples to solve equations in cases not covered before (Häubler, Numer Math 48:119–125, 1986; Hu et al., J Comput Appl Math 219:110–122, 2008; Kontorovich and Akilov 2004).     

Direct methods for the approximate solution of operator equations with nonzero kernel

We justify direct methods for the approximate solution of linear operator equations with nonzero kernels and apply these methods to the justification of projective methods for the approximate solution of standard singular integral equations with Cauchy kernels and positive index on the unit disk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1202–1213, September, 1998.     

An analytical solution of the S-model kinetic equations

An analytical variation of the discrete ordinates method is used to establish a concise and accurate solution to the plane Poiseuille and thermal creep problems based on the S-model kinetic equations. The results are of high accuracy and include the flow rates and the heat fluxes of both problems for a wide range of Knudsen numbers and various values of the accommodation coefficient at the boundaries.     

Systems with constant eigenvectors with applications to exact and numerical solutions of ordinary differential equations

Necessary and sufficient conditions are given for an n × n diagonable matrix with entries depending on a parameter to have a set of constant eigenvectors.