Tridiagonal matrix algorithm

Abstract: The NFC (negative factor counting) method ws extended to solve the eigenvalue problem of tridiagonal block matrices with elements corresponding to cross links which may be derived from the quantum-chemical calculation on a native protein molecule. The mathematical proof of the necessary theorem is given in detail.

Abstract: The purpose of this paper is to provide an efficient and robust second-order monotone hybrid scheme for singularly perturbed delay parabolic convection-diffusion initial boundary value problem.,The delay parabolic problem is solved numerically by a finite difference scheme consists of implicit Euler scheme for the time derivative and a monotone hybrid scheme with variable weights for the spatial derivative. The domain is discretized in the temporal direction using uniform mesh while the spatial direction is discretized using three types of non-uniform meshes mainly the standard Shishkin mesh, the Bakhvalov–Shishkin mesh and the Gartland Shishkin mesh.,The proposed scheme is shown to be a parameter-uniform convergent scheme, which is second-order convergent and optimal for the case. Also, the authors used the Thomas algorithm approach for the computational purposes, which took less time for the computation, and hence, more efficient than the other methods used in literature.,A singularly perturbed delay parabolic convection-diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a monotone hybrid scheme. The error analysis is carried out. It is shown to be parameter-uniform convergent and is of second-order accurate. Numerical results are shown to verify the theoretical estimates.

Abstract: This paper presents the mathematical model of the thermal power plant in cooling pond under different meteorological conditions, which is solved by three dimensional Navier-Stokes equations and temperature equation for an incompressible fluid in a stratified medium. A numerical method based on the projection method, which divides the problem into three stages. At the first stage it is assumed that the transfer of momentum occurs only by convection and diffusion. Intermediate velocity field is solved by method of fractional steps. At the second stage, three-dimensional Poisson equation is solved by the Fourier method in combination with tridiagonal matrix method (Thomas algorithm). At the third stage it is expected that the transfer is only due to the pressure gradient. The compact scheme was used to increase the order of approximation. Then the basic laws of the hydrothermal processes depending on different hydrometeorological conditions were determined qualitatively and quantitatively approximate.