Chebyshev Polynomials Of Second Kind Research Articles

On tridiagonal matrices associated with Jordan blocks

Abstract This paper aims to show how some standard general results can be used to uncover the spectral theory of tridiagonal and related matrices more elegantly and simply than existing approaches. As a typical example, we apply the theory to the special tridiagonal matrices in recent papers on orthogonal polynomials arising from Jordan blocks. Consequently, we find that the polynomials and spectral theory of the special matrices are expressible in terms of the Chebyshev polynomials of second kind, whose properties yield interesting results. For special cases, we obtain results in terms of the Fibonacci numbers and Legendre polynomials.

Direct multi-dimensional Chebyshev polynomial based reconstruction for magnetic particle imaging

Magnetic Particle Imaging is a tomographic imaging technique that measures the voltage induced due to magnetization changes of magnetic nanoparticle distributions. The relationship between the received signal and the distribution of the nanoparticels is described by the system function. A common method for image reconstruction is using a measured system function to create a system matrix and set up a regularized linear system of equations. Since the measurement of the system matrix is time-consuming, different methods for acceleration have been proposed. These include modeling the system matrix or using a direct reconstruction method in time, known as X-space reconstruction. In this work, based on the simplified Langevin model of paramagnetism and certain approximations, a direct reconstruction technique for Magnetic Particle Imaging in the frequency domain with two- and three-dimensional Lissajous trajectory excitation is presented. The approach uses Chebyshev polynomials of second kind. During reconstruction, they are weighted with the frequency components of the voltage signal and additional factors and then summed up. To obtain the final nanoparticle distribution, this result is rescaled and deconvolved. It is shown that the approach works for both simulated data and real measurements. The obtained image quality is comparable to a modeled system matrix approach using the same simplified physical assumptions and no relaxation effects. The reconstruction of a 31 × 31 × 31 volume takes less than a second and is up to 25 times faster than the state-of-the-art Kaczmarz reconstruction. Besides, the derivation of the proposed method shows some new theoretical aspects of the system function and its well-known observed similarity to tensor products of Chebyshev polynomials of second kind.

Spectral tau solution of the linearized time-fractional KdV-Type equations

<abstract><p>The principal objective of the current paper is to propose a numerical algorithm for treating the linearized time-fractional KdV equation based on selecting two different sets of basis functions. The members of the first set are selected to be suitable combinations of the Chebyshev polynomials of the second kind and also to be compatible with the governing boundary conditions of the problem, while the members of the second set are selected to be the shifted second-kind Chebyshev polynomials. After expressing the approximate solutions as a double expansion of the two selected basis functions, the spectral tau method is applied to convert the equation with its underlying conditions into a linear system of algebraic equations that can be treated numerically with suitable standard procedures. The convergence analysis of the double series solution is carefully tested. Some numerical examples accompanied with comparisons with some other methods in the literature are displayed aiming to demonstrate the applicability and accuracy of the presented algorithm.</p></abstract>

Studies on coefficient estimates and Fekete-Szego problem for a class of bi-univalent functions associated with \((p,q)\)-Chebyshev polynomial

In this present work, the authors studied and investigated the concept of \((p,q)\)-Chebyshev polynomial of second kind for the subclass of analytic bi-univalent function with respect to the subordination. We give an elementary proof to estimate the coefficient bounds for the bi-univalent functions defined in the open unit disk. Also we included the result of Fekete-Szego theorem.

Hypersingular Integral Equation Formulation of the Problem of Water Wave Scattering by A Circular Arc Shaped Impermeable Barrier Submerged in Water of Finite Depth

Summary In this article, we study the problem of scattering of water waves by a thin impermeable circular arc shaped barrier submerged in ocean of finite depth under the assumption of linearised theory of water waves. The problem is formulated in terms of a hypersingular integral equation of an unknown function representing the difference of potential function across the curved barrier. The hypersingular integral equation is then solved by using two numerical methods. The first method is BEM where the domain and range of integral equation are discretised into small line segments and the unknown function satisfying the integral equation is assumed to be constant in each small subinterval. This reduces the integral equations to a system of algebraic equations which is then solved to obtain the unknown function in each sub-interval. The second method is collocation method where the unknown function is expanded in terms of Chebyshev polynomials of second kind. Choosing the collocation points suitably, the integral equation is reduced to a system of algebraic equations which is then solved to obtain the unknown function satisfying the hypersingular integral equation. The physical quantities of interest viz, the reflection coefficient, transmission coefficients, which are expressed in terms of the solution of the hypersingular integral equation, are computed by both the methods. The comparison of the reflection coefficient by the two methods shows reasonably good agreement. The reflection coefficient is depicted graphically against the wave number. The graphical results show that the size, position of the barrier and the depth of the water region has some effect on the reflected and transmitted wave.

Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomials

We say that a monic polynomial with integer coefficients is a polygomial if its each zero is obtained by squaring the edge or a diagonal of a regular n-gon with unit circumradius. We find connections of certain polygomials with Morgan-Voyce polynomials and further with Chebyshev polynomials of second kind.

Properties of the second-kind Chebyshev polynomials of complex variable

We construct a system of functions biorthogonal with Chebyshev polynomials of the second kind on closed contours in the complex plane. Properties of these functions and sufficient conditions of expansion of analytic functions into series in Chebyshev polynomials of the second kind in complex domains are investigated. The examples of such expansions are given. In addition, combinatorial identities of self-interest are obtained.

A novel numerical manner for two‐dimensional space fractional diffusion equation arising in transport phenomena

AbstractFractional diffusion equations include a consistent and efficient explanation of transport phenomena that manifest abnormal diffusion, that cannot be often represented by second‐order diffusion equations. In this article, a two‐dimensional space fractional diffusion equation (SFDE‐2D) with nonhomogeneous and homogeneous boundary conditions is considered in Caputo derivative sense. An instant and nevertheless accurate scheme is obtained by the finite‐difference discretization to get the semidiscrete in temporal derivative with convergence order . Moreover, space fractional derivative can be approximated based on the Chebyshev polynomials of second kind which are powerful methods for basing the operational matrix. The convergence and stability of the proposed scheme are discussed theoretically in detail. Finally, two numerical problems with an exact solution are given that numerical results show the effectiveness of the new techniques. These schemes can be simply extended to three spatial dimensions, which will be the subject of our subsequent research.

On the connection between tridiagonal matrices, Chebyshev polynomials, and Fibonacci numbers

Abstract In this note, we recall several connections between the determinant of some tridiagonal matrices and the orthogonal polynomials allowing the relation between Chebyshev polynomials of second kind and Fibonacci numbers. With basic transformations, we are able to recover some recent results on this matter, bringing them into one place.

Common Points Between Perturbed Chebyshev Polynomials of Second Kind

We consider some perturbation of the Chebyshev polynomials of second kind obtained by modifying one of its recurrence coefficients at an arbitrary order. The goal of this work is to point out that perturbed Chebyshev polynomials of fixed degree and different values of parameters of perturbation have some common points that are zeros of two Chebyshev polynomials of second kind of lower degrees. These common points can be simple or double. We identify the cases in which they are common zeros.

Some comments on the properties of a particular tridiagonal matrix

Recently, Q.M. Al-Hassan discussed some properties of the tridiagonal matrix with zero main diagonal and all 1’s sub- and super-diagonals. In this brief note, we show that these properties are special cases of a vast theory involving Chebyshev polynomials of second kind.

Solution of the finite slab criticality problem using an alternative phase function with the second kind of Chebyshev polynomials

The critical size of a finite homogenous slab is investigated for one-speed neutrons using the alternative phase function (AG, Anli–Gungor) in place of the scattering function of the transport equation. First of all, the neutron angular flux expanded in terms of the Chebyshev polynomials of second kind (UN approximation) together with the AG phase function is applied to the transport equation to obtain a criticality condition for the system. Then, using various values of the scattering parameters, the numerical results for the critical half-thickness of the slab are calculated and they are tabulated in the tables together with the ones obtained from the conventional spherical harmonic (PN) method for comparison. They can be said to be in good accordance with each other.

On connection coefficients of some perturbed of arbitrary order of the Chebyshev polynomials of second kind

ABSTRACTOrthogonal polynomials satisfy a recurrence relation of order two defined by two sequences of coefficients. If we modify one of these recurrence coefficients at a certain order, we obtain the so-called perturbed orthogonal sequence. In this work, we analyse perturbed Chebyshev polynomials of second kind and we deal with the problem of finding the connection coefficients that allow us to write the perturbed sequence in terms of the original one and in terms of the canonical basis. From the connection coefficients obtained, we derive some results about zeros at the origin. The analysis is valid for arbitrary order of perturbation.

Effects of vertical porous barrier on progressive waves in a two layered fluid

Two dimensional scattering of linear water wave by thin vertical permeable plate in a two-layer fluid with free surface is considered. The permeable barrier is completely submerged in the upper layer of finite depth over a layer whose depth is either infinite or finite. For discontinuity in the potential across the plate, we use Green's integral theorem to formulate the problem in terms of a hypersingular integral equation. A collocation method using a finite series of Chebyshev polynomials of second kind have been introduced to get the unknown difference potential numerically. The reflection and transmission coefficients for surface and internal modes are computed as an integral involving difference potentials. The proportion of reflected and transmitted energies of both the wavenumbers are calculated. Moreover, amount of energy dissipated due to the presence of permeable plate in the upper layer are obtained. Numerical results are calculated and depicted graphically against the wavenumber for various non-dimensionalized parameters.

Second Kind Chebyshev Polynomials for Solving Space Fractional Advection–Dispersion Equation Using Collocation Method

Fractional space derivatives are studied for modeling anomalous diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. In this paper, we discuss numerical technique for solving space fractional advection–dispersion equation (FADE) where $$1<\alpha \leq 2$$ and $$0<\beta \leq 1$$ . We implement a numerical technique for solving FADE called Chebyshev collocation method. We utilized fractional derivatives in the Caputo sense. The properties of shifted Chebyshev polynomials of second kind (SCPSK) are used to reduce FADE to a system of differential equations (ODEs), which is solved by finite difference method (FDM), then we use an iteration scheme to solve the system of equations. Specially, we are focused on error analysis and convergence analysis of the proposed method. The validation of the present scheme is tested through examples and compare with existing method and exact solution.

Stable Numerical Approach for Fractional Delay Differential Equations

In this paper, we present a new stable numerical approach based on the operational matrix of integration of Jacobi polynomials for solving fractional delay differential equations (FDDEs). The operational matrix approach converts the FDDE into a system of linear equations, and hence the numerical solution is obtained by solving the linear system. The error analysis of the proposed method is also established. Further, a comparative study of the approximate solutions is provided for the test examples of the FDDE by varying the values of the parameters in the Jacobi polynomials. As in special case, the Jacobi polynomials reduce to the well-known polynomials such as (1) Legendre polynomial, (2) Chebyshev polynomial of second kind, (3) Chebyshev polynomial of third and (4) Chebyshev polynomial of fourth kind respectively. Maximum absolute error and root mean square error are calculated for the illustrated examples and presented in form of tables for the comparison purpose. Numerical stability of the presented method with respect to all four kind of polynomials are discussed. Further, the obtained numerical results are compared with some known methods from the literature and it is observed that obtained results from the proposed method is better than these methods.

Approximate solution of fractional vibration equation using Jacobi polynomials

In this paper, we present a method based on the Jacobi polynomials for the approximate solution to fractional vibration equation (FVE) of large membranes. Proposed method converts the FVE into Sylvester form of algebraic equations, whose solution gives the approximate solution. Convergence analysis of the proposed method is given. It is also shown that our approximate method is numerically stable. Numerical results are discussed for different values of wave velocities and fractional order involved in the FVE. These numerical results are shown through figures for particular cases of Jacobi polynomials such as (1) Legendre polynomial, (2) Chebyshev polynomial of second kind, (3) Chebyshev polynomial of third kind, (4) Chebyshev polynomial of fourth kind, (5) Gegenbauer polynomial. The accuracy of the proposed method is proved by comparing results of our method and other exiting analytical methods. Comparison of results are presented in the form of tables for particular cases of FVE and Jacobi polynomials.

Numerical constructions involving Chebyshev polynomials

We propose a new algorithm for the character expansion of tensor products of finite-dimensional irreducible representations of simple Lie algebras. The algorithm produces valid results for the algebras B3, C3, and D3. We use the direct correspondence between Weyl anti-invariant functions and multivariate second-kind Chebyshev polynomials. We construct the triangular trigonometric polynomials for the algebra D3.

An exponential Chebyshev second kind approximation for solving high-order ordinary differential equations in unbounded domains, with application to Dawson's integral

Abstract A new exponential Chebyshev operational matrix of derivatives based on Chebyshev polynomials of second kind (ESC) is investigated. The new operational matrix of derivatives of the ESC functions is derived and introduced for solving high-order linear ordinary differential equations with variable coefficients in unbounded domain using the collocation method. As an application the introduced method is used to evaluate Dawson's integral by solving its differential equation. The corresponding differential equation to Dawson's integral is a boundary value problem with conditions tends to infinity. The obtained numerical results are compared with the exact solution and showed good accuracy.

A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels

A spectral method based on operational matrices of the second kind Chebyshev polynomials (SKCPs) is employed for solving fractional integro-differential equations with weakly singular kernels. Firstly, properties of shifted SKCPs, operational matrix of fractional integration and product operational matrix are introduced and then utilized to reduce the given equation to the solution of a system of linear algebraic equations. This new approach provides a significant computational advantage by converting the given original problem to an equivalent linear Volterra integral equation of the second kind with the same initial conditions. Approximate solution is achieved by expanding the functions in terms of SKCPs and employing operational matrices. Unknown coefficients are determined by solving final system of linear equations. An estimation of the error is given. Finally, illustrative examples are included to demonstrate the high precision, fast computation and good performance of the new scheme.