On numerical solutions of partial differential equations by the decomposition method
On the Chebyshev penalty method for parabolic and hyperbolic equations
On the rate of convergence of a collocation projection of the KdV equation
Based on estimates for the KdV equation in analytic Gevrey classes, a spectral collocation approximation of the KdV equation is proved to converge exponentially fast.
Operational Methods in the Environment of a Computer Algebra System
This article presents the principal results of the doctoral thesis “Direct Operational Methods in the Environment of a Computer Algebra System” by Margarita Spiridonova (Institute of mathematics and Informatics, BAS), successfully defended before the Specialised Academic Council for Informatics and Mathematical Modelling on 23 March, 2009.The presented research is related to the operational calculus approach and its representative applications. Operational methods are considered, as well as their...
P-adaptive Hermite methods for initial value problems∗
We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems...
P-adaptive Hermite methods for initial value problems
We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems...
P-adaptive Hermite methods for initial value problems∗
We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems...
Prediction-correction Legendre spectral scheme for incompressible fluid flow
Prediction-correction legendre spectral scheme for incompressible fluid flow
The initial-boundary value problem of two-dimensional incompressible fluid flow in stream function form is considered. A prediction-correction Legendre spectral scheme is proposed, which is easy to be performed. The numerical solution possesses the accuracy of second-order in time and higher order in space. The numerical experiments show the high accuracy of this approach.
Random projection RBF nets for multidimensional density estimation
The dimensionality and the amount of data that need to be processed when intensive data streams are observed grow rapidly together with the development of sensors arrays, CCD and CMOS cameras and other devices. The aim of this paper is to propose an approach to dimensionality reduction as a first stage of training RBF nets. As a vehicle for presenting the ideas, the problem of estimating multivariate probability densities is chosen. The linear projection method is briefly surveyed. Using random...
Reduced basis solver for stochastic Galerkin formulation of Darcy flow with uncertain material parameters
In this contribution, we present a solution to the stochastic Galerkin (SG) matrix equations coming from the Darcy flow problem with uncertain material coefficients in the separable form. The SG system of equations is kept in the compressed tensor form and its solution is a very challenging task. Here, we present the reduced basis (RB) method as a solver which looks for a low-rank representation of the solution. The construction of the RB consists of iterative expanding of the basis using Monte...
Solution of singular and nonsingular initial and boundary value problems by modified variational iteration method.
Solving a class of Hamilton-Jacobi-Bellman equations using pseudospectral methods
This paper presents a numerical approach to solve the Hamilton-Jacobi-Bellman (HJB) problem which appears in feedback solution of the optimal control problems. In this method, first, by using Chebyshev pseudospectral spatial discretization, the HJB problem is converted to a system of ordinary differential equations with terminal conditions. Second, the time-marching Runge-Kutta method is used to solve the corresponding system of differential equations. Then, an approximate solution for the HJB problem...
Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift
This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential that is equal to along the boundary of the computational domain . Using a symmetrization...
Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift
This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential U that is equal to +∞ along the boundary ∂D of the computational domain D. Using a symmetrization...
Spectral/hp elements in fluid structure interaction
This work presents simulations of incompressible fluid flow interacting with a moving rigid body. A numerical algorithm for incompressible Navier-Stokes equations in a general coordinate system is applied to two types of body motion, prescribed and flow-induced. Discretization in spatial coordinates is based on the spectral/hp element method. Specific techniques of stabilisation, mesh design and approximation quality estimates are described and compared. Presented data show performance of the solver...
The eigenvalues of Hermite and rational spectral differentiation matrices.
The extended tanh method and the exp-function method to solve a kind of nonlinear heat equation.
The Fourier spectral method for the Sivashinsky equation.
The pseudospectral method for thermotropic primitive equation and its error estimation.