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:
- Numerical methods for interface problems
- 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
- R.Aitbayev, P. W. Bates, H. Lu, L. Zhang, and M. Zhang, J.. Mathematical studies of Poisson--Nernst--Planck systems: dynamics of ionic flows without electroneutrality conditions. Comput. Appl. Math., Volume 362, 15 December 2019, Pages 510-527, https://doi.org/10.1016/j.cam.2018.10.037
- R. Aitbayev and N. Yergaliyeva. A Fourth-Order Collocation Scheme for Two-Point Interface Boundary Value Problems. Advances in Numerical Analysis, vol. 2014, Article ID 875013, 8 pages, 2014. https://doi.org/10.1155/2014/875013
- R. Aitbayev, Existence and uniqueness for a two-point interface boundary value problem, Electron. J. Diff. Equ., Vol. 2013 (2013), No. 242, pp. 1-12.
- N. H. Ibragimov, R. Aitbayev, R. N. Ibragimov, Three-dimensional non-linear rotating surface waves in channels of variable depth in the presence of formation of a small perturbation of atmospheric pressure across the channel, Communications in Nonlinear Science and Numerical Simulation, 14(2009), pp. 3811-3820.
- R. Aitbayev, A quadrature finite element Galerkin scheme for a biharmonic problem on a rectangular polygon, Numerical Methods for Partial Differential Equations, 24(2008), pp. 518-534.
- R. Aitbayev, An error analysis and the mesh independence principle for a nonlinear collocation problem, Numerical Methods for Partial Differential Equations, 22(2006), pp. 1216-1237.
- R. Aitbayev, Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem, Numerical Methods for Partial Differential Equations, 22(2006), pp. 847-866.
- R. Aitbayev, Multilevel preconditioners for non-self-adjoint or indefinite orthogonal spline collocation problems, SIAM Journal on Numerical Analysis, 43(2005), pp. 686-706.
- R. Aitbayev and B. Bialecki, A preconditioned conjugate gradient method for non-selfadjoint or indefinite orthogonal spline collocation problems, SIAM Journal on Numerical Analysis, 41 (2003), pp. 589-604.
- R. Aitbayev, X.-C. Cai, and M. Paraschivoiu, Parallel two-level methods for three-dimensional transonic compressible flow simulations on unstructured meshes, Parallel Computational Fluid Dynamics: Towards Teraflops, Optimization and Novel Formulations, DE Keyes et al., eds., Elsevier, Amsterdam, pp. 89-96, 2000.
- R. Aitbayev and B. Bialecki, Orthogonal spline collocation for nonlinear Dirichlet problems, SIAM Journal on Numerical Analysis, 38 (2000), pp. 1582-1602.
- R. Aitbayev and Sh. Smagulov, Convergence of a finite difference scheme for a quasilinear differential equation with a solution in W^2_2 , Dynamics of Fluid with Free Boundaries (Continuum Dynamics), Issue 107, Hydrodynamics Institute of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 1993, pp. 3-10 (in Russian).
- 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).