Rakhim Aitbayev's home page

Research Interests of Rakhim Aitbayev

Associate Professor of Mathematics
New Mexico Institute of Mining and Technology
Socorro, New Mexico

Many phenomena in sciences, engineering, economics, and other areas are described by partial differential equations or PDEs. Exact (analytical) solutions of most of real world problems are often very difficult or impossible to obtain. Solutions to such problems can be approximated using numerical methods. At present, Numerical Methods for Solving Partial Differential Equations is a vast area which deals with numerical errors, stability, parallel algorithms, efficient computation, numerical solution of challenging multiphysics problems. I am carrying out research in the following areas of Numerical PDEs:

Orthogonal spline collocation method
Iterative methods for solving large systems of linear and nonlinear equations
Numerical solution of nonlinear elliptic PDEs
Numerical solution of non-self-adjoint or indefinite problems
Multigrid/multilevel methods
Domain decomposition methods

The following is a list of my publications.

List of Publications

A quadrature finite element Galerkin Scheme for a biharmonic problem on a rectangular polygon, Numerical Methods for Partial Differential Equations, to appear, available online at Wiley Interscience, DOI: 10.1002/num.20278.
An error analysis and the mesh independence principle for a nonlinear collocation problem, Numerical Methods for PDEs, Wiley Periodicals, Inc., 22(2006), 1216--1237.
Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem, Numerical Methods for PDEs, published online in Wiley InterScience, DOI 10.1002/num.20122, 2005.
Multilevel Preconditioners for Nonselfadjoint or Indefinite Orthogonal Spline Collocation Problem s, SIAM Journal on Numerical Analysis, 43 (2005), pp. 686-706.
A preconditioned conjugate gradient method for non-self-adjoint or indefinite orthogonal spline collocation problems. SIAM Journal on Numerical Analysis, 41 (2003), pp. 589-604.
Orthogonal spline collocation for nonlinear Dirichlet boundary value problems. SIAM Journal on Numerical Analysis, 38 (2000), pp. 1582-1602.
Parallel two-level methods for three-dimensional transonic compressible flow simulations on unstructured meshes. In Parallel Computational Fluid Dynamics: Towards Teraflops, Optimization and Novel Formulations, D. Keyes, A. Ecer, J. Periaux, N. Satofuka and P. Fox (Editors), Elsevier, 2000, pp. 89 -- 96.
Orthogonal spline collocation for nonlinear elliptic boundary value problems. Ph.D. thesis. University of Kentucky, 1998. Advisor: Prof. Bernard Bialecki.
Convergence of a finite difference scheme for a quasilinear differential equation with a solution in $W^{2}_{2}$. In Dynamics of the Fluid with Free Boundaries (The Continuum Dynamics), Issue 107, Hydrodynamics Institute of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 1993, pp. 3 -- 10 (in Russian).
Finite difference schemes for nonlinear differential equations modeling filtration and hydraulic processes. Ph. D. thesis, Kazakh State University, 1992. Advisor: Prof. Shaltay Smagulov.
Study of a finite difference scheme for the diffusion approximation of Saint Venant's equations (stability and convergence). In proceedings of the XXVI USSR National Student Scientific Conference (Mathematics), Novosibirsk State University, Novosibirsk, 1988 (in Russian).
Study of a finite difference scheme for the diffusion analogy of Saint Venant's equations ( a priori estimates). In proceedings of the conference of young scientists and specialists of the Kazakh State University, Alma-Ata, 1988 (in Russian).

Rakhim Aitbayev aitbayev@nmt.edu