Continuous Polynomials Research Articles

Generalized 2D Extension of the q-Bessel polynomials

The article aims at introducing a generalized 2D extension of the q-Bessel polynomials. The generating equation, series expansion and determinant form for the generalized 2D q-Bessel polynomials are established. An example providing the continuous 2D q-Hermite--Bessel polynomials is considered. The recurrence relation and other properties for the continuous 2D q-Hermite--Bessel polynomials are derived. The graphs and plots depicting the behaviour of the continuous 2D Hermite polynomials, continuous 2D Hermite--Bessel polynomials and continuous 2D q-Hermite--Bessel polynomials are drawn. The distribution of zeros for these polynomials is also presented.

Iterative Continuous Collocation Method for Solving Nonlinear Volterra Integral Equations

This paper is concerned with the numerical solution of nonlinear Volterra integral equations. The main purpose of this work is to provide a new numerical approach based on the use of continuous collocation Lagrange polynomials for the numerical solution of nonlinear Volterra integral equations. It is shown that this method is convergent. The results are compared with the results obtained by other well-known numerical methods to prove the effectiveness of the presented algorithm.

Implementation and Analyses of an Eco-Driving Algorithm for Different Battery Electric Powertrain Topologies Based on a Split Loss Integration Approach

Eco-driving algorithms optimize the speed profile to reduce the energy consumption of a vehicle. This paper presents an eco-driving algorithm for battery electric powertrains that applies a split loss integration approach to incorporate the component losses. The algorithm consistently uses loss models to overcome the drawbacks of efficiency maps, which cannot represent no-load losses at zero torque. The use of loss models is crucial since the optimal solution includes gliding, during which there are no-load losses. An analysis shows, that state-of-the-art nonlinear programming algorithms cannot represent these no-load losses at zero torque with a small modeling error. To effectively compute the powertrain losses with only a small error in comparison to the measurement data, we introduce a tailored combination of nonlinear inequality constraints that interleave two polynomial fits. This approach can properly represent reality. We parameterize the algorithm and validate the vehicle model used with real-world measurement data. Furthermore, we investigate the influence of the proposed interleaved fits by comparing them to a single continuous high-order polynomial fit and to the state of the art. The algorithm is published open source.

A High-Order Residual-Based Viscosity Finite Element Method for the Ideal MHD Equations

We present a high order, robust, and stable shock-capturing technique for finite element approximations of ideal MHD. The method uses continuous Lagrange polynomials in space and explicit Runge-Kutta schemes in time. The shock-capturing term is based on the residual of MHD which tracks the shock and discontinuity positions, and adds sufficient amount of viscosity to stabilize them. The method is tested up to third order polynomial spaces and an expected fourth-order convergence rate is obtained for smooth problems. Several discontinuous benchmarks such as Orszag-Tang, MHD rotor, Brio-Wu problems are solved in one, two, and three spacial dimensions. Sharp shocks and discontinuity resolutions are obtained.

Representable and Radon-Nikodým polynomials

Every k-homogeneous (continuous) polynomial between Banach spaces admits a unique Aron-Berner extension to the biduals. Our main result states that, for every σ-finite measure µ, every polynomial P ∈ with Y -valued Aron-Berner extension is representable, in other words, it factors through ℓ 1 in the form P = A◦Q where Q is a polynomial and A is an operator. This may be viewed as a polynomial strengthening of the Dunford-Pettis-Phillips theorem stating that every weakly compact operator on L 1(µ) is representable. We introduce the Radon-Nikodým polynomials and show that every polynomial with Y -valued Aron-Berner extension is Radon-Nikodým. Finally, we prove that every Radon-Nikodým polynomial is unconditionally converging.

Central part interpolation schemes for a class of fractional initial value problems

We consider an initial value problem for linear fractional integro-differential equations with weakly singular kernels. Using an integral equation reformulation of the underlying problem, a collocation method based on the central part interpolation by continuous piecewise polynomials on the uniform grid is constructed and analysed. Optimal convergence order of the proposed method is established and confirmed by numerical experiments.

A stabilizer-free C0 weak Galerkin method for the biharmonic equations

In this article, we present and analyze a stabilizer-free C0 weak Galerkin (SF-C0WG) method for solving the biharmonic problem. The SF-C0WG method is formulated in terms of cell unknowns which are C0 continuous piecewise polynomials of degree k + 2 with k ≽ 0 and in terms of face unknowns which are discontinuous piecewise polynomials of degree k + 1. The formulation of this SF-C0WG method is without the stabilized or penalty term and is as simple as the C1 conforming finite element scheme of the biharmonic problem. Optimal order error estimates in a discrete H2-like norm and the H1 norm for k ≽ 0 are established for the corresponding WG finite element solutions. Error estimates in the L2 norm are also derived with an optimal order of convergence for k > 0 and sub-optimal order of convergence for k = 0. Numerical experiments are shown to confirm the theoretical results.

Polynomial Annuities

We use a payment pattern of the type {1k,2k,3k,…} to generalize the standard level payment and increasing annuity to polynomial payment patterns. We derive explicit formulas for the present value of an n-year polynomial annuity, the present value of an m-monthly n-year polynomial annuity, and the present value of an n-year continuous polynomial annuity. We also use the idea to extend the annuities to payment patterns derived from analytic functions, as well as to payment patterns of the type {1r,2r,3r,…}, with r being an arbitrary real number. In the process, we develop possible approximations to k! and for the gamma function evaluated at real numbers.

A divergence-free finite element method for the Stokes problem with boundary correction

Abstract This paper constructs and analyzes a boundary correction finite element method for the Stokes problem based on the Scott–Vogelius pair on Clough–Tocher splits. The velocity space consists of continuous piecewise polynomials of degree k, and the pressure space consists of piecewise polynomials of degree (k – 1) without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise polynomials with respect to the boundary partition is introduced to enforce boundary conditions and to mitigate the lack of pressure-robustness. We prove several inf-sup conditions, leading to the well-posedness of the method. In addition, we show that the method converges with optimal order and the velocity approximation is divergence-free.

A weak divergence CDG method for the biharmonic equation on triangular and tetrahedral meshes

A conforming discontinuous Galerkin (CDG) C0-Pk finite element method is introduced for solving the biharmonic equation on triangular and tetrahedral meshes. A C0-Pk finite element function is a continuous and piecewise polynomial of degree k on a triangular or tetrahedral mesh. The CDG method is based on taking weak divergence on the gradient of C0-Pk finite elements. Optimal order error estimates in both a discrete H2 norm and the L2 norm are established. Numerical results are presented to verify the theory.

Moments of orthogonal polynomials and exponential generating functions

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ( $$q=1$$ ) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont–Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally, we show that one of our methods can be applied to deal with the moments of Askey–Wilson polynomials.

Time-optimal trajectory optimization of serial robotic manipulator with kinematic and dynamic limits based on improved particle swarm optimization

Effective motion control could achieve the accurate positioning and fast movement of industrial robotics to improve industrial productivity significantly. Time-optimal trajectory optimization (TO) is a great concern in the motion control of robotics, which could improve motion efficiency by providing high-speed and reasonable motion references to motion controllers. In this study, a new general time-optimal TO strategy, the second-order continuous polynomial interpolation function (SCPIF) combined with the particle swarm optimization with cosine-decreasing weight (CDW-PSO) subject to kinematic and dynamic limits, successfully optimizes the movement time of the PUMA 560 serial manipulator. The SCPIF could be used to generate the second-order continuous movement trajectories of six joints in joint space based on the assigned positions and time intervals. The CDW-PSO algorithm could further search for the optimal movement time subject to the limits of the angular displacement, angular velocity, angular acceleration, angular jerk, and joint torque of the manipulator. Two numerical experiments are conducted to illustrate the generalization ability of the CDW-PSO algorithm. The advantage of the CDW would be reflected by comparing with the random weight (RW), the constant weight (CW), and the linearly decreasing weight (LDW), respectively, in each experiment. The experimental results show that the CDW-PSO algorithm would perform better than the RW-PSO, CW-PSO, and LDW-PSO algorithms in terms of the convergence rate and quality of the convergent solution. The proposed time-optimal TO strategy would be applied to all types of manipulators while the optimized trajectories could be incorporated in the motion controllers of the actual manipulators due to considering the kinematic and dynamic limits.

Algebraic Basis of the Algebra of All Symmetric Continuous Polynomials on the Cartesian Product of ℓp-Spaces

We construct a countable algebraic basis of the algebra of all symmetric continuous polynomials on the Cartesian product ℓp1×…×ℓpn, where p1,…,pn∈[1,+∞), and ℓp is the complex Banach space of all p-power summable sequences of complex numbers for p∈[1,+∞).

Continuous orthosymmetric multilinear maps and homogeneous polynomials on Riesz spaces revisited

Continuous orthosymmetric multilinear maps and homogeneous polynomials on Riesz spaces revisited

PSO-Based Time Optimal Rapid Orientation for Micro-Nano Space Robot

Simultaneous multi-directional orientation tasks on space robots exhibits strong dynamic coupling between the manipulators and the base-spacecraft, and this phenomenon is more obvious in the micro-nano space robot. Regarding to the highly coupled dynamic system of micro-nano space robot and its nonholonomic constraints, a multi-directional rapid orientation method of micro-nano space robot based on optimization theory is proposed in this paper. Dynamic model of satellite-manipulators cooperative operation micro-nano space robot is established by the combination of virtual manipulator (VM) and Newton-Euler method. Then, taking the acceleration continuous cubic polynomial trajectory planning method and the dynamics-based feedback linearization controller as an example, the particle swarm optimization (PSO) method is used to solve the time and path nodes of the orientation process of the micro-nano space robot. Finally, three typical spatial orientation problems are simulated to verify the feasibility and rapidity of this orientation method, which are simultaneous orientation, assisted orientation and cooperative braking, respectively. Furthermore, the paper presents the structure of the objective function for optimization in different multi-component and multi-objective co-orientation tasks.

Piecewise discretization of monodromy operators of delay equations on adapted meshes

<p style='text-indent:20px;'>Periodic solutions of delay equations are usually approximated as continuous piecewise polynomials on meshes adapted to the solutions' profile. In practical computations this affects the regularity of the (coefficients of the) linearized system and, in turn, the effectiveness of assessing local stability by approximating the Floquet multipliers. To overcome this problem when computing multipliers by collocation, the discretization grid should include the piecewise adapted mesh of the computed periodic solution. By introducing a piecewise version of existing pseudospectral techniques, we explain why and show experimentally that this choice is essential in presence of either strong mesh adaptation or nontrivial multipliers whose eigenfunctions' profile is unrelated to that of the periodic solution.</p>

Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree N inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor. The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme.

Hidden positivity and a new approach to numerical computation of Hausdorff dimension: higher order methods

In 2018, the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods. We again rely on the fact that we can associate to the IFS a parametrized family of positive, linear, Perron–Frobenius operators L_s , an idea known in varying degrees of generality for many years. Although L_s is not compact in the setting we consider, it possesses a strictly positive C^m eigenfunction v_s with eigenvalue R(L_s) for arbitrary m and all other points z in the spectrum of L_s satisfy |z| \le b for some constant b < R(L_s) . Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value s=s_* for which R(L_s) =1 . This eigenvalue problem is then approximated by a collocation method at the extended Chebyshev points of each subinterval using continuous piecewise polynomials of arbitrary degree r . Using an extension of the Perron theory of positive matrices to matrices that map a cone K to its interior and explicit a priori bounds on the derivatives of the strictly positive eigenfunction v_s , we give rigorous upper and lower bounds for the Hausdorff dimension s_* , and these bounds converge rapidly to s_* as the mesh size decreases and/or the polynomial degree increases.

A spline-based FE approach to modelling of high frequency dynamics of 1-D structures

In this paper a computational methodology leading to the development of a new class of FEs, based on the application of continuous and smooth approximation polynomials, being splines, has been presented. Application of the splines as appropriately defined piecewise elemental shape functions led the authors to the formulation of a new approach for FEM, named as spFEM, where contrary to the well-known NURBS approach, the boundaries of spFEs are well-defined, exactly as it is in the case of the traditional FEM.The current approach has been computationally verified by the authors it terms of high frequency dynamics including such problems as: spectra of natural frequencies, modes of natural vibrations as well as wave propagation problems, especially in the aspect of high frequency responses, all in the case of selected problems involving one- and two-mode theories of 1-D structural elements. The applicability of the proposed approach has been evaluated and compared, in terms of calculated dynamic responses, with the results obtained by the use of well-established FEM approaches: classical FEM as well as TD-SFEM.In all cases investigated by the authors the proposed spFEM approach turned out to be the most accurate approach, free from the main drawback of the other tested FEM approaches thanks to the class of differentiability of approximation polynomials, which guarantees the absence of frequency band gaps in calculated spectra of natural frequencies. A direct consequence of this feature of the proposed approach is that a larger part of the calculated spectra of natural frequency, the same as modes of natural vibrations, can effectively be used for more accurate calculations of dynamic reposes even in the case of multi-mode theories. This in contrast to the other tested FEM approaches.

On the continuous dual Hahn process

We extend the continuous dual Hahn process (Tt) of Corwin and Knizel from a finite time interval to the entire real line by taking a limit of a closely related Markov process (Tt). We also characterize processes (Tt) by conditional means and variances under bidirectional conditioning, and we prove that continuous dual Hahn polynomials are orthogonal martingale polynomials for both processes.