Jacobi Orthogonal Polynomials Research Articles

Spectral solutions for fractional Klein–Gordon models of distributed order

The Klein–Gordon equation is a fundamental theoretical physics concept, governing the behavior of relativistic quantum particles with spin-zero. Its numerical solution is crucial in fields like quantum field theory, particle physics, and cosmology. The study explores numerical methodologies for solving this equation, highlighting their significance and challenges. This study uses the collocation method to approximate fractional Klein–Gordon models of distributed order based on Shifted Jacobi orthogonal polynomials and Shifted fractional order Jacobi orthogonal functions. While, the distributed term (integral term) was treat using Legendre–Gauss–Lobatto quadrature. It assesses residuals through finite expansion and yields accurate numerical results. The method is more factual and fair when initial and boundary conditions are enforced. Numerical simulations are presented to demonstrate the method’s accuracy, particularly in fractional Klein–Gordon models of distributed order. Furthermore, we offer a few numerical test scenarios to show that the method is able to maintain the non-smooth solution of the underlying issue.

Construction of Fractional Pseudospectral Differentiation Matrices with Applications

Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed explicitly through Jacobi–Jacobi transformations between two indexes. In the current paper, an algorithm is presented to construct a fractional differentiation matrix with a matrix representation for Riemann–Liouville, Caputo and Riesz derivatives, which makes the computation stable and efficient. Applications of the fractional differentiation matrix with the spectral collocation method to various problems, including fractional eigenvalue problems and fractional ordinary and partial differential equations, are presented to show the effectiveness of the presented method.

Applications of multiple orthogonal polynomials with hypergeometric moment generating functions

We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval, the positive real line, or the unit circle and the multiple orthogonal polynomials are generalizations of the Jacobi, Laguerre or Bessel orthogonal polynomials. We give explicit formulas for the type I and type II multiple orthogonal polynomials and study some of their properties. In particular, we describe the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials via the free convolution. Essential to our overall approach is the use of the Mellin transform. Finally, we discuss two applications. First, we show that the multiple orthogonal polynomials appear naturally in the study of the squared singular values of (mixed) products of truncated unitary random matrices and Ginibre matrices. Secondly, we use the multiple orthogonal polynomials to simultaneously approximate certain hypergeometric series and to provide an explicit proof of their Q-linear independence.

Connecting exceptional orthogonal polynomials of different kind

The known asymptotic relations interconnecting Jacobi, Laguerre, and Hermite classical orthogonal polynomials are generalized to the corresponding exceptional orthogonal polynomials of codimension m. It is proved that Xm-Laguerre exceptional orthogonal polynomials of type I, II, or III can be obtained as limits of Xm-Jacobi exceptional orthogonal polynomials of the same type. Similarly, Xm-Hermite exceptional orthogonal polynomials of type III can be derived from Xm-Jacobi or Xm-Laguerre ones. The quadratic transformations expressing Hermite classical orthogonal polynomials in terms of Laguerre ones is also extended to even X2m-Hermite exceptional orthogonal polynomials.

Modeling and experiments on the vibro-acoustic analysis of ring stiffened cylindrical shells with internal bulkheads: A comparative study

The vibro-acoustic response of ring stiffened cylindrical shells with internal bulkheads under forced excitation is presented. The numerical analysis model is established using the Jacobi Ritz-Boundary element method. The first-order shear deformation theory, multi-segment technique and artificial spring technology are applied to establish the theoretical model, and the Jacobi orthogonal polynomials are introduced to represent the displacement functions. The Newmark-β integration method is used to obtain the vibration response of the structure, and the time-domain Kirchhoff boundary integral formulation is applied to describe the exterior acoustic field. The vibration and sound radiation of the ring stiffened cylindrical shell under impact and narrow band random loading were measured. The comparative study reveals that the results obtained from the proposed method agree well with the experimental results. The stiffeners and bulkheads have a significant effect on the vibration characteristics of the structure. Additionally, some physical insights into the resonant peaks and sound pressure directivity of the cylindrical shells are provided.

Optimal reconstruction and recognition of images by Jacobi Fourier moments and artificial bee colony (ABC) algorithm

The orthogonal moments giving relevant results of these last years within the framework of object detection, pattern recognition and image reconstruction, this work based on orthogonal functions called Orthogonal Jacobi Polynomials (OJPs), and we introduce a new set of moments called Generalized Jacobi Fourier Moments (GJFMs), these polynomials are characterized by parameters . However, it was very important to optimize these parameters in order to obtain a good result, in this context; this study used a new approach to optimized Jacobi Fourier parameters using the artificial bee colony algorithm (ABC) in order to improves the quality of reconstruction of images of large sizes. On the one hand, to validate this technique which offers a high image reconstruction quality. On other hand, the comparison carried out with other algorithms clearly indicates the advantage of the proposed method.

ASYMPTOTIC BEHAVIOUR OF THE JACOBI SOBOLEV-TYPE ORTHOGONAL POLYNOMIALS. A NON DIAGONAL CASE

Asymptotic behaviour of the Jacobi Sobolev-type orthogonal polynomials. A non-diagonal case. Rev. Acad. Colomb. Cienc. 34 (133): 529-539, 2010. ISSN 0370-3908. MATEMÁTICAS 1 Universidad Nacional de Colombia, Bogotá, Colombia. Correo electrónico: haduenasr@unal.edu.co 2 Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo No. 340, Colima, Colima, México. Correo electrónico: garzaleg@gmail.com AMS Classification 2000: 33C47. Consider the following Sobolev type inner product (p, q) = ∫ p'(x)q(x)(1 - x)θ(1 + x)θdx + i[p'(1)q(1) - p(1)q'(1)], (1) where p and q are polynomials with real coefficients, 0 < θ > -1, ξ(x) = (p(x), q(x))', and A = [M0, M1; M1, M2] is a positive semidefinite matrix, with M0M1 ≥ 0, and A ∈ ℝ. The family of polynomials orthogonal with respect to (1), P_n^{(θ,θ)}, are called Jacobi Sobolev-type orthogonal polynomials. An expression that relates this family of polynomials with P_n^{(0,0)}, the usual Jacobi orthogonal polynomials, was obtained in [8]. Here, we obtain the outer relative asymptotic for P_n^{(0,0)}, as well as the corresponding Mehlér-Heine formula.

Plane and axisymmetric problems of an interfacial crack in a power-law graded material

Plane and axisymmetric problems of interfacial cracks in a power-law graded infinite medium are examined. It is shown that the models are governed by integral equations of Mellin’s convolution type whose kernels are expressed through the hypergeometric function. Exact solutions are derived by the method of orthogonal Jacobi polynomials in a series form and by the Wiener–Hopf method by quadratures. Mode-I stress intensity factors and the associated weight functions for power-law graded materials in the plane and axisymmetric cases are introduced and evaluated. It is shown that, although the displacement jump and the normal traction component have power singularities at the crack tip different from 1/2, the strain energy variation is proportional to the crack length change as in the case of homogeneous materials. A Griffith-type criterion of crack propagation in power-law graded materials is proposed and results of numerical tests are reported.

On some hypergeometric Sobolev orthogonal polynomials with several continuous parameters

In this paper we study the following hypergeometric polynomials: $$ \mathcal{P}_n(x) = \mathcal{P}_n(x;\alpha,\beta,\delta_1, \dots,\delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+2} F_{\rho+1} (-n,n+\alpha+\beta+1,\delta_1+1, \dots,\delta_\rho+1;\alpha+1,\kappa_1+\delta_1+1, \dots,\kappa_\rho+\delta_\rho+1;x), $$ and $$ \mathcal{L}_n(x) = \mathcal{L}_n(x;\alpha,\delta_1,\dots, \delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+1} F_{\rho+1} (-n,\delta_1+1,\dots,\delta_\rho+1; \alpha+1,\kappa_1+\delta_1+1,\dots,\kappa_\rho+\delta_\rho+1;x), \qquad n\in\mathbb{Z}_+, $$ where $\alpha,\beta,\delta_1,\dots,\delta_\rho\in(-1,+\infty)$, and $\kappa_1,\dots,\kappa_\rho\in\mathbb{Z}_+$, are some parameters. The natural number $\rho$ of the continuous parameters $\delta_1,\dots,\delta_\rho$ can be chosen arbitrarily large. It is seen that the special case $\kappa_1=\dots=\kappa_\rho=0$ leads to Jacobi and Laguerre orthogonal polynomials. Of course, such polynomials and more general ones appeared in the literature earlier. Our aim here is to show that polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$ are Sobolev orthogonal polynomials on the real line with some explicit matrices of measures. The importance of the orthogonality property was our main reason to concentrate our attention on polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$. Here we shall use some our tools developed earlier. In particular, it was shown recently that Sobolev orthogonal polynomials are related by a differential equation with orthogonal systems $\mathcal{A}$ of functions acting in the direct sums of usual $L^2_\mu$ spaces of square-summable (classes of the equivalence of) functions with respect to a positive measure $\mu$. The case of a unique $L^2_\mu$ is of a special interest, since it allows to use OPRL to obtain explicit systems of Sobolev orthogonal polynomials. The main problem here is \textit{to choose a suitable linear differential operator in order to get explicit representations for Sobolev orthogonal polynomials}. The proof of the orthogonality relations is then a verification of such a choice and it goes in another direction: we start from the already known polynomials to their properties. We also study briefly such properties of the above polynomials: integral representations, differential equations and location of zeros. A system of such polynomials with a kind of the bispectrality property is constructed.

Dynamic Analysis of Functionally Graded Porous (FGP) Elliptic Cylindrical Shell Based on Jacobi Polynomials Ritz Method

The lightweight of structure is widely applied in industrial applications, and the conflict between both dynamic stability and structural lightweight is still prominent. In this paper, functionally graded porous (FGP) elliptic cylindrical shells and panels with general boundary conditions are analyzed to explore the effect of the FGP on dynamic performance. First, the FGP elliptic cylindrical shell and panel models are established. Therein, three kinds of porosity distribution are considered, including nonsymmetric, symmetric, and uniform distributions. The energy expressions of the FGP elliptic cylindrical shell and panel are established by the first-order shear deformation theory (FSDT). To simulate various boundary conditions, the artificial spring boundary technique is employed in this study. Then, the Jacobi orthogonal polynomials and Fourier series are adopted to express the admissible displacements. Finally, the accuracy of this model is verified by comparing it with open literature and ABAQUS software. Results show that the variations of the boundary conditions, linear springs, thickness ratio, and porosity have close relation with the dynamic performance of the structure by affecting the stiffness of the structure.

Multiple orthogonal polynomials associated with branched continued fractions for ratios of hypergeometric series

The main objects of the investigation presented in this paper are branched-continued-fraction representations of ratios of contiguous hypergeometric series and type II multiple orthogonal polynomials on the step-line with respect to linear functionals or measures whose moments are ratios of products of Pochhammer symbols. This is an interesting case study of the recently found connection between multiple orthogonal polynomials and branched continued fractions that gives a clear example of how this connection leads to considerable advances on both topics.We start by obtaining new results about generating polynomials of lattice paths and total positivity of matrices and giving new contributions to the general theory of the connection between multiple orthogonal polynomials and branched continued fractions with emphasis on its application to the analysis of multiple orthogonal polynomials. Then, we construct new branched continued fractions for ratios of contiguous hypergeometric series. We give conditions for positivity of the coefficients of these branched continued fractions and we show that the ratios of products of Pochhammer symbols are generating polynomials of lattice paths for a special case of the branched continued fractions under study. Next, we introduce a family of type II multiple orthogonal polynomials on the step-line associated with those branched continued fractions. We present a formula as terminating hypergeometric series for these polynomials, we study their differential properties, and we find an explicit recurrence relation satisfied by them. Finally, we focus the analysis of the multiple orthogonal polynomials to the cases where the corresponding branched-continued-fraction coefficients are all positive. In those cases, the orthogonality conditions can be written using measures on the positive real line involving Meijer G-functions and we obtain results about the location of the zeros and the asymptotic behaviour of the polynomials.Specialisations of the multiple orthogonal polynomials studied here include the classical Laguerre, Jacobi, and Bessel orthogonal polynomials, multiple orthogonal polynomials with respect to Nikishin systems of two measures involving modified Bessel functions, confluent hypergeometric functions, and Gauss' hypergeometric function, and multiple orthogonal polynomials with respect to Meijer G-functions used to investigate the singular values of products of Ginibre random matrices as well as a r-orthogonal polynomial sequence with constant recurrence coefficients for any positive integer r and particular instances of the Jacobi-Piñeiro polynomials.

Vertical vibration of rigid strip footings on poroelastic soil layer of finite thickness

This paper presents a numerical solution that describes the response to forced vertical vibrations of rigid strip footings founded on the surface of a poroelastic soil layer of finite thickness. To mathematically describe this mixed boundary value problem, we relax the boundary conditions and cast it as a pair of dual integral equations, which are transformed to a system of linear equations by means of Jacobi orthogonal polynomials. The system is solved numerically to provide the frequency-dependent footing compliance, and the associated nearby soil displacement. Results obtained with the model at hand underline the important contribution of standing waves to the response of the soil-foundation system, and draw the limits of existing solutions that consider the foundation soil as infinite half-space, where standing waves cannot form.

Supersymmetry and shape invariance of exceptional orthogonal polynomials

We discuss the exceptional Laguerre and the exceptional Jacobi orthogonal polynomials in the framework of the supersymmetric quantum mechanics (SUSYQM). We express the differential equations for the Jacobi and the Laguerre exceptional orthogonal polynomials (EOP) as the eigenvalue equations and make an analogy with the time independent Schrödinger equation to define “Hamiltonians” enables us to study the EOPs in the framework of the SUSYQM and to realize the underlying shape invariance associated with such systems. We show that the underlying shape invariance symmetry is responsible for the solubility of the differential equations associated with these polynomials.

A semi-analytical method for dynamic analysis of a rectangular plate with general boundary conditions based on FSDT

Abstract In this paper, the unified Jacobi–Ritz method (JRM) is utilized to analyze the dynamic response of rectangular plates with general boundary conditions. First, the structural energy functional is established in the framework of the first-order shear deformation theory, and the rectangular plate is divided into several equal parts according to the domain decomposition method. Then, the artificial springs are introduced to ensure the continuity of segments and diversified boundary conditions. The Jacobi orthogonal polynomials are expanded to represent the displacement field in one direction. Finally, the free and forced vibration characteristics of the rectangular plate can be obtained by utilizing the Rayleigh–Ritz method, where the Newmark-β integration method is adopted to realize the time-domain solutions for transient vibration response. The results for different structural scale parameters and various boundary conditions are presented, and the validity and accuracy of the presented method are verified by comparing the results from published literature and FEM. The results of the study can provide technical support for vibration control of the plate structure.

Free and forced vibration analysis of uniform and stepped combined conical-cylindrical-spherical shells: A unified formulation

The free and forced vibration characteristics of uniform and stepped coupled shell structures with arbitrary boundary conditions is investigated in this paper. The coupled structure is comprised of typical conical shell, spherical shell and stepped cylindrical shell. The first-order shear deformation theory (FSDT) is applied to establish the analytical model, together with the domain decomposition method (DDM). Then the Jacobi orthogonal polynomials and standard Fourier series are introduced to express the displacement functions along axial direction and circumferential orientation, respectively. The free and forced vibration characteristics of uniform and stepped conical-cylindrical-spherical shells can be obtained by utilizing Ritz method, besides, the Newmark-β integration method is utilized to accomplish time-domain (transient) analysis of the structure. The reliability of the presented method is validated by comparing with the Finite Element Method (FEM) results, the free and forced vibration characteristics of the structure with different boundary conditions, structural scale parameters and damping parameters are analyzed by presenting several numerical examples, numerous new results obtained by present method may serve as reference values for other researches.

Series expansions among weighted Lebesgue function spaces and applications to positive definite functions on compact two-point homogeneous spaces

Geodesically isotropic positive definite functions on compact two-point homogeneous spaces of dimension d have series representation as members of weighted Lebesgue spaces L1w([−1,1]), where the weight w(x)=wα,β(x)=(1−x)α(1+x)β is the one related to the Jacobi orthogonal polynomials P(α,β)(x) in [−1,1], and the exponents α and β are related to the dimension d. We derive some recurrence relations among the coefficients of the series representations under different exponents, and we apply them to prove inheritance of positive definiteness between dimensions. Additionally, we give bounds on the curvature at the origin of such positive definite functions with compact support, extending the existing solutions from d-dimensional spheres to compact two-point homogeneous spaces.

Vibration analysis of FG beams under arbitrary load with general boundary conditions: Theoretical and experimental comparative research

In this paper, the experimental and Jacobi–Ritz method have been adopted to analyze the free and forced vibration comparative analysis of FG beams with general boundary conditions. The rigid-fixed steel beam at both ends is taken as an experimental model, and the free, steady and transient vibration characteristics of the structure under hammer and fixed exciter are recorded. The domain decomposition method and Jacobi orthogonal polynomials are introduced to ensure the convergence and accuracy of the algorithm, and the artificial spring is used to simulate different boundary restraint and continuity condition of the beam. The vibration characteristics of the beam can be obtained by using Rayleigh–Ritz method, in which the Newmark-β integration method is adopted to realize the time domain solutions for transient vibration response. The results show that the results of proposed method are in sensible agreement with those from published literature, experiment and FEM. In addition, the results for various boundary conditions, structural parameter and power-law exponent are also presented.

A semi-analytical method for the dynamic characteristics of stiffened plate with general boundary conditions

In this paper, the free, steady and transient vibration response of stiffened plate with general boundary conditions are studied through theoretical and experimental comparisons. Firstly, the structural energy functional is established in the framework of the first-order shear deformation theory (FSDT), the stiffened plate is divided into several equal parts according to the domain decomposition method (DDM). Next, the artificial springs are introduced to ensure the continuity of segments and diversified boundary conditions. The unified Jacobi orthogonal polynomials and trigonometric series are expanded to represents the displacement field along different directions, and the free, steady and transient vibration characteristics of stiffened plate can be obtained by using Rayleigh–Ritz method. Results for various boundary conditions, structure parameter are presented, the validity and accuracy of the proposed method is verified by comparing with the existing literatures and experiment, the arbitrary impulse excitation load and damped vibration are considered. It is clearly shown that the accurate solutions can be obtained by using proposed method and the research can supply technical support for the vibration control of stiffened plate structure.

Quadrature formulas for Cauchy-type integrals with the Cauchy kernel to a integer power and a Jacobi weight function with complex exponents

The paper presents quadrature formulas for hypersingular integrals of various orders. It is assumed that the density of these integrals is represented as a product of a function that satisfies the Hölder condition and a weight function of Jacobi orthogonal polynomials. In this case, the exponents of the weight function can be complex numbers, the real part of which is greater than -1. Numerical analysis of the dependence of the root-mean-square deviation of the quadrature formula of order 8 on the value of the hypersingular integral calculated using standard software packages is carried out for various complex values of the weight function exponents. For hypersingular integrals up to the fourth order inclusive, a numerical analysis of the convergence of quadrature formulas is carried out for certain complex values of the exponents of the weight function.

A Unified Solution Method for Free Vibration of Arbitrarily Shaped Plates without or with Cracks

This paper presents a new approach for analyzing the free vibration of thin plates with arbitrary piecewise smooth curvilinear contour under various boundary conditions. It can also be applied to plates with cracks. This approach is based on the transformation of energy integral expressions and the domain decomposition technique. Furthermore, boundary conditions are modeled by using linear springs to restrain the plate edges. For obtaining the expressions of energy integral, the arbitrarily shaped domain of integration is divided into several trapezoid domains with curved sides by a set of parallel lines passing through the intersection points of contour curve segments. Then, Jacobi orthogonal polynomials are introduced as the admissible functions, so that the repeated integrals in the energy expressions are reduced to definite integrals analytically. At this point, the calculation method of the energy functional is determined by the equation forms of the curve segments of the plate contour. When the equations of the curve segments allow the integrands to have analytic primitive functions, the energy functional has an analytical solution. Otherwise, the Gauss–Legendre method is used to obtain the numerical solution. Accordingly, the arbitrarily shaped plate with cracks is decomposed into several arbitrarily shaped subdomains based on the cracks. Each subdomain is handled according to the above procedure. The continuity conditions at the interconnecting interfaces of the subdomains are realized by linear springs. The plate without or with crack is modeled by setting the spring stiffness to infinity or 0. The accuracy of the proposed method is verified by comparing the obtained results with the published results. Furthermore, the vibration characteristics of plates with various shapes, such as astroid-shaped plates and cracked elliptical plates, are investigated. These new results can serve as benchmarks for further studies on the vibration of plates.