On the Effective Implementation and Capabilities of the Least-Squares Collocation Method for Solving Second-Order Elliptic Equations

Исследованы возможности численного метода коллокации и наименьших квадратов (КНК) на примерах кусочно-полиномиального решения задачи Дирихле для уравнений Пуассона и типа диффузии-конвекции с особенностями в виде больших градиентов и разрыва решения на границах раздела двух подобластей. Предложены и реализованы новые hp-варианты метода КНК, основанные на присоединении внутри области малых и/или вытянутых нерегулярных ячеек, отсекаемых криволинейной границей раздела от исходных прямоугольных ячеек сетки, к соседним самостоятельным ячейкам. Выписываются с учетом особенности условия согласования между собой кусков решения в ячейках, примыкающих с разных сторон к границе раздела. Проведено сравнение результатов, полученных методом КНК и другими высокоточными методами. Показаны преимущества и достоинства метода КНК. Для ускорения итерационного процесса применены современные алгоритмы и методы: предобуславливание; свойства локальной системы координат в методе КНК; ускорение, основанное на подпространствах Крылова; операция продолжения на многосеточном комплексе; распараллеливание. Исследовано влияние этих способов на количество итераций и время расчетов при аппроксимации полиномами различных степеней. The capabilities of the numerical least-squares collocation (LSC) method of the piecewise polynomial solution of the Dirichlet problem for the Poisson and diffusion-convection equations are investigated. Examples of problems with singularities such as large gradients and discontinuity of the solution at interfaces between two subdomains are considered. New hp-versions of the LSC method based on the merging of small and/or elongated irregular cells to neighboring independent cells inside the domain are proposed and implemented. They cut off by a curvilinear interface from the original rectangular grid cells. Taking into account the problem singularity the matching conditions between the pieces of the solution in cells adjacent from different sides to the interface are written out. The results obtained by the LSC method are compared with other high-accuracy methods. Advantages of the LSC method are shown. For acceleration of an iterative process modern algorithms and methods are applied: preconditioning, properties of the local coordinate system in the LSC method, Krylov subspaces; prolongation operation on a multigrid complex; parallelization. The influence of these methods on iteration numbers and computation time at approximation by polynomials of various degrees is investigated.

The Polynomial Solutions of Quadratic Diophantine Equation X 2 − p t Y 2 + 2 K t X + 2 p t L t Y = 0

In this study, we consider the number of polynomial solutions of the Pell equation x 2 − p t y 2 = 2 is formulated for a nonsquare polynomial p t using the polynomial solutions of the Pell equation x 2 − p t y 2 = 1 . Moreover, a recurrence relation on the polynomial solutions of the Pell equation x 2 − p t y 2 = 2 . Then, we consider the number of polynomial solutions of Diophantine equation E : X 2 − p t Y 2 + 2 K t X + 2 p t L t Y = 0 . We also obtain some formulas and recurrence relations on the polynomial solution X n , Y n of E .

Dual exponential polynomials and a problem of Ozawa

Complex linear differential equations with entire coefficients are studied in the situation where one of the coefficients is an exponential polynomial and dominates the growth of all the other coefficients. If such an equation has an exponential polynomial solution $f$ , then the order of $f$ and of the dominant coefficient are equal, and the two functions possess a certain duality property. The results presented in this paper improve earlier results by some of the present authors, and the paper adjoins with two open problems.

On Markov Moment Problem and Related Results

We prove new results and complete our recently published theorems on the vector-valued Markov moment problem, by means of polynomial approximation on unbounded subsets, also applying an extension of the positive linear operators’ result. The domain is the Banach lattice of continuous real-valued functions on a compact subset or an space, where is a positive moment determinate measure on a closed unbounded set. The existence and uniqueness of the operator solution are proved. Our solutions satisfy the interpolation moment conditions and are between two given linear operators on the positive cone of the domain space. The norm controlling of the solution is emphasized. The most part of the results are stated and proved in terms of quadratic forms. This type of result represents the first aim of the paper. Secondly, we construct a polynomial solution for a truncated multidimensional moment problem.

Time-Varying Semidefinite Programs

We study time-varying semidefinite programs (TV-SDPs), which are semidefinite programs whose data (and solutions) are functions of time. Our focus is on the setting where the data vary polynomially with time. We show that under a strict feasibility assumption, restricting the solutions to also be polynomial functions of time does not change the optimal value of the TV-SDP. Moreover, by using a Positivstellensatz (positive locus theorem) on univariate polynomial matrices, we show that the best polynomial solution of a given degree to a TV-SDP can be found by solving a semidefinite program of tractable size. We also provide a sequence of dual problems that can be cast as SDPs and that give upper bounds on the optimal value of a TV-SDP (in maximization form). We prove that under a boundedness assumption, this sequence of upper bounds converges to the optimal value of the TV-SDP. Under the same assumption, we also show that the optimal value of the TV-SDP is attained. We demonstrate the efficacy of our algorithms on a maximum-flow problem with time-varying edge capacities, a wireless coverage problem with time-varying coverage requirements, and on biobjective semidefinite optimization where the goal is to approximate the Pareto curve in one shot.

Convective heat transfer for Peristaltic flow of SWCNT inside a sinusoidal elliptic duct

A mathematical model is presented to analyse the flow characteristics and heat transfer aspects of a heated Newtonian viscous fluid with single wall carbon nanotubes inside a vertical duct having elliptic cross section and sinusoidally fluctuating walls. Exact mathematical computations are performed to get temperature, velocity and pressure gradient expressions. A polynomial solution technique is utilized to obtain these mathematical solutions. Finally, these computational results are presented graphically and different characteristics of peristaltic flow phenomenon are examined in detail through these graphs. The velocity declines as the volume fraction of carbon nanotubes increases in the base fluid. Since the velocity of fluid is dependent on its temperature in this study case and temperature decreases with increasing volumetric fraction of carbon nanotubes. Thus velocity also declines for increasing volumetric fraction of nanoparticles.

Simulation of velocity profile inside turbulent boundary layer

The second-order differential equation with a general polynomial solution [1], is adapted for simulation of complex velocity profile inside the turbulent boundary layer. Consequently, the simulation strategy is suggested.

Exponential Polynomials and Nonlinear Differential-Difference Equations

In this paper, we study finite-order entire solutions of nonlinear differential-difference equations and solve a conjecture proposed by Chen, Gao, and Zhang when the solution is an exponential polynomial. We also find that any exponential polynomial solution of a nonlinear difference equation should have special forms.