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In this work, the Caputo-type Hadamard fractional derivative is utilized to introduce a coupled system of time fractional Klein–Gordon-Schrödinger equations. The classical and shifted Jacobi polynomials are simultaneously applied to make a numerical technique for this system. To this aim, two operational matrices for the Caputo-type Hadamard fractional derivatives of the shifted Jacobi polynomials are gained. In the developed strategy, by considering a hybrid approximation of the problem's solution via the expressed polynomials and applying the obtained matrices, solving the original fractional system turns into solving an associated algebraic system of equations. Two test problems are examined to investigate the high accuracy of the developed procedure. [ABSTRACT FROM AUTHOR]

The present paper investigates a new method for computationally solving the time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) with initial boundary conditions (IBCs). The method utilizes a set of generalized shifted Jacobi polynomials (GSJPs) that adhere to the specified initial and boundary conditions (IBCs). Our approach involves constructing operational matrices (OMs) for both ordinary derivatives (ODs) and fractional derivatives (FDs) of the GSJPs we employ. We subsequently employ the collocation spectral method using these OMs. This method successfully converts the TFCKdVEs into a set of algebraic equations, greatly simplifying the task. In order to assess the efficiency and precision of the proposed numerical technique, we utilized it to solve two distinct numerical instances. [ABSTRACT FROM AUTHOR]

This paper presents a novel spectral algorithm for the numerical solution of multi-dimensional fractional-order telegraph equations, a critical model used to capture the combined effects of diffusion and wave propagation. The core innovation of this work is the application of Jacobi-Romanovski polynomials as the basis functions for spectral discretization. These polynomials offer unique advantages, including the ability to handle nonstandard domains and boundary conditions, making them particularly suitable for partial differential equation (PDE) applications. A comprehensive error analysis is conducted, providing deep insights into the convergence rates and factors affecting the accuracy of the numerical solutions. Extensive numerical experiments further demonstrate the superior performance of the proposed spectral algorithm in solving a wide range of multi-dimensional fractional-order telegraph equation models. The results show a significant improvement in accuracy and computational efficiency compared to traditional numerical methods, such as finite difference or finite element techniques. This research advances the field of computational science by offering a robust, efficient, and versatile numerical framework for the precise solution of complex multi-dimensional PDEs. [ABSTRACT FROM AUTHOR]

The study of −1$-1$ orthogonal polynomials viewed as q→−1$q\rightarrow -1$ limits of the q$q$‐orthogonal polynomials is pursued. This paper presents the continuous polynomials part of the −1$-1$ analog of the q$q$‐Askey scheme. A compendium of the properties of all the continuous −1$-1$ hypergeometric polynomials and their connections is provided. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate Sobolev orthogonality and families of orthogonal polynomials on the triangle as a generalization of the results in Xu, Y. Constr. Approx. 46, 349-434 (2017). We define inner products or pseudo-inner products that contain derivatives up to third-order in the Sobolev space on a triangle and study associated orthogonal polynomials that are analogs of the Jacobi polynomials J k , n α , β , γ but with α , β , γ being - 1 , - 2 , or - 3 , and two other families derived under simultaneous permutations of x , y , 1 - x - y and α , β , γ . [ABSTRACT FROM AUTHOR]

With this paper we continue the investigations started in [6] and concerned with stability conditions for collocation-quadrature methods based on the zeros of classical Jacobi polynomials, not only Chebyshev polynomials. While in [6] we only proved the necessity of certain conditions, here we will show also their sufficiency in particular cases. [ABSTRACT FROM AUTHOR]

In the present work, we investigate certain algebraic and differential properties of the orthogonal polynomials with respect to a discrete–continuous Sobolev-type inner product defined in terms of the Jacobi measure. [ABSTRACT FROM AUTHOR]

The primary aim of this paper is to investigate the asymptotic distribution of the zeros of certain classes of hypergeometric $ {}_{q+1}F_{q} $ q + 1 F q polynomials. We employ classical analytical techniques, including Watson's lemma and the method of steepest descent, to understand the asymptotic behavior of these polynomials: $$\begin{align*} & _{q+1}F_{q}\left(-n,kn+\alpha,\ldots, kn+\alpha+\frac{q-1}{q};kn+\beta,\ldots,kn+\beta+\frac{q-1}{q};z\right)\\ &\quad (n\rightarrow \infty), \end{align*} $$ q + 1 F q (− n , kn + α , ... , kn + α + q − 1 q ; kn + β , ... , kn + β + q − 1 q ; z) (n → ∞) , where n is a nonnegative integer, q is a positive integer and the constant parameters α and β are constrained by $ \alpha { α < β. By applying the general results established in this paper, we generate numerical evidence and graphical illustrations using Mathematica to show the clustering of zeros on certain curves. [ABSTRACT FROM AUTHOR]

This article aims to present novel identities for elementary and complete symmetric polynomials and explore their applications, particularly to generalized Vandermonde and special tri-diagonal matrices. It also extends existing results on Jacobi polynomials P (α,β) n (x) and introduces an explicit formula based on the zeros of P (α,β) n−1 (x). Several illustrative examples are included. [ABSTRACT FROM AUTHOR]

In this work, the Caputo-type Hadamard fractional derivative is utilized to introduce a coupled system of time fractional Klein–Gordon-Schrödinger equations. The classical and shifted Jacobi polynomials are simultaneously applied to make a numerical technique for this system. To this aim, two operational matrices for the Caputo-type Hadamard fractional derivatives of the shifted Jacobi polynomials are gained. In the developed strategy, by considering a hybrid approximation of the problem’s solution via the expressed polynomials and applying the obtained matrices, solving the original fractional system turns into solving an associated algebraic system of equations. Two test problems are examined to investigate the high accuracy of the developed procedure.

Abstract The present paper investigates a new method for computationally solving the time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) with initial boundary conditions (IBCs). The method utilizes a set of generalized shifted Jacobi polynomials (GSJPs) that adhere to the specified initial and boundary conditions (IBCs). Our approach involves constructing operational matrices (OMs) for both ordinary derivatives (ODs) and fractional derivatives (FDs) of the GSJPs we employ. We subsequently employ the collocation spectral method using these OMs. This method successfully converts the TFCKdVEs into a set of algebraic equations, greatly simplifying the task. In order to assess the efficiency and precision of the proposed numerical technique, we utilized it to solve two distinct numerical instances.

Abstract This paper presents a novel spectral algorithm for the numerical solution of multi-dimensional fractional-order telegraph equations, a critical model used to capture the combined effects of diffusion and wave propagation. The core innovation of this work is the application of Jacobi-Romanovski polynomials as the basis functions for spectral discretization. These polynomials offer unique advantages, including the ability to handle nonstandard domains and boundary conditions, making them particularly suitable for partial differential equation (PDE) applications. A comprehensive error analysis is conducted, providing deep insights into the convergence rates and factors affecting the accuracy of the numerical solutions. Extensive numerical experiments further demonstrate the superior performance of the proposed spectral algorithm in solving a wide range of multi-dimensional fractional-order telegraph equation models. The results show a significant improvement in accuracy and computational efficiency compared to traditional numerical methods, such as finite difference or finite element techniques. This research advances the field of computational science by offering a robust, efficient, and versatile numerical framework for the precise solution of complex multi-dimensional PDEs.

In this article, we propose a new class of nonlinear fractional differential equations of diabetes disease based on the concept of Caputo fractional derivative. Two numerical techniques are introduced to analyze the solution of the general fractional diabetes model, that describes glucose homeostasis. The first method, which constructed using the nonstandard finite difference technique involves an asymptotically stable difference scheme. This method maintains important properties of the solutions of the studied system, such as the positivity and boundedness. The second method is the Jacobi–Gauss–Lobatto spectral collocation approach, known for its exponential accuracy. By employing this collocation method, the problem is transformed into a set of algebraic nonlinear equations, simplifying the overall task. Numerical simulations were conducted to compare the performance of these two techniques with other standard methods and the analytic solution in specific cases. Our findings show that the Jacobi–Gauss–Lobatto spectral collocation technique provides higher accuracy in solving the fractional diabetes model system, while the nonstandard finite difference approach requires lower computational duration.

This article analyzes and computes numerical solutions for the fractional isothermal chemical (FIC) model. This work suggested a Jacobi collocation method (JCM) to examine the FIC model. In the beginning, we constructed the operational matrices for fractional order derivatives for Jacobi polynomials. Then, by using operational matrices and the collocation technique, we converted the provided model into a set of algebraic equations. The approach that is used in this article is quicker and more effective than several other schemes. We solve the system of equations for various fractional orders of differentiation. Comparison of JCM and Newton polynomial interpolation (NPI) technique is present in this article. Furthermore, an error analysis of the proposed procedure is also given.

The study of elastic wave scattering is crucial for nondestructive testing, quality assurance, and failure prevention of quasicrystal (QC) structures. In the present paper, the dynamic behavior of an interface crack in one-dimensional (1D) hexagonal piezoelectric quasicrystal (PQC) coating and semiinfinite homogeneous elastic substrate subjected to incident elastic-time harmonic waves is studied. The scattering problem of plane waves is reduced to a set of the Cauchy singular integral equations (SIEs) of the second kind by introducing the dislocation density functions. The SIEs are solved employing excellent properties of the Jacobi Polynomials, and the dynamic stress intensity factors (DSIFs) at the crack tips are obtained. The effects of the different PQC coating, incidence waves, crack size, incidence angle, and coupling coefficients on the DSIFs are displayed graphically. The consequences indicate that the presence of the phason field intensifies the fluctuation of the DSIFs at the crack tips, and the higher the coupling coefficients of the phonon-phason field, the easier the crack propagates. Additionally, the crack propagation can be suppressed by selecting the appropriate PQC coating and incidence angle. This work will benefit the understanding of the dynamic failure of PQC and its application. [ABSTRACT FROM AUTHOR]

This article aims to present novel identities for elementary and complete symmetric polynomials and explore their applications, particularly to generalized Vandermonde and special tri-diagonal matrices. It also extends existing results on Jacobi polynomials $ P_n^{(\alpha, \beta)}(x) $ and introduces an explicit formula based on the zeros of $ P_{n-1}^{(\alpha, \beta)}(x) $. Several illustrative examples are included.

Abstract We provide here a novel approach for solving IVPs in ODEs and MTFDEs numerically by means of a class of MSJPs. Using the SCM, we build OMs for RIs and RLFI for MSJPs as part of our process. These architectures guarantee accurate and efficient numerical computations. We provide theoretical assurances for the efficacy of an algorithm by establishing its convergence and error analysis features. We offer five numerical examples to prove that our method is accurate and applicable. Through these examples, we demonstrate the greater accuracy and efficiency of our approach by comparing our results with previously published findings. Tables and graphs show that the method produces exact and approximate solutions that agree quite well with each other.

We examine two binary operations on the set of algebraic polynomials, known as multiplicative and additive finite free convolutions, specifically in the context of hypergeometric polynomials. We show that the representation of a hypergeometric polynomial as a finite free convolution of more elementary blocks, combined with the preservation of the real zeros and interlacing by the free convolutions, is an effective tool that allows us to analyze when all roots of a specific hypergeometric polynomial are real. Moreover, the known limit behavior of finite free convolutions allows us to write the asymptotic zero distribution of some hypergeometric polynomials as free convolutions of Marchenko–Pastur, reciprocal Marchenko–Pastur, and free beta laws, which has an independent interest within free probability. [ABSTRACT FROM AUTHOR]

We call a translation function class a set of zonal translation networks if it is produced by the translations of a convolutional structure, which is a composite function reunited with an even function defined on $ [-1, 1] $ and a geodesic distance. In the present paper, we consider the density of a class of zonal translation networks produced by a convolution structure on the unit ball from the view of Fourier-Laplace series and give a sufficient condition to ensure the zonal translation class is density in $ L^p $ spaces defined on the unit ball. In particular, we construct with De La Vallée Poussin operators a sequence of zonal translation networks and show the approximation order. [ABSTRACT FROM AUTHOR]

For T n (x) = cos n arccos x , x ∈ [ - 1 , 1 ] , the n-th Chebyshev polynomial of the first kind, we study the quantity τ n , k : = | T n (k) (ω n , k) | T n (k) (1) , 1 ≤ k ≤ n - 2 , where T n (k) is the k-th derivative of T n and ω n , k is the largest zero of T n (k + 1) . Since the absolute values of the local extrema of T n (k) increase monotonically towards the end-points of [ - 1 , 1 ] , the value τ n , k shows how small is the largest critical value of T n (k) relative to its global maximum T n (k) (1) . This is a continuation of our (joint with Alexei Shadrin) paper "On the largest critical value of T n (k) ", SIAM J. Math. Anal.50(3), 2018, 2389–2408, where upper bounds and asymptotic formuae for τ n , k have been obtained on the basis of the Schaeffer–Duffin pointwise upper bound for polynomials with absolute value not exceeding 1 in [ - 1 , 1 ] . We exploit a 1996 result of Knut Petras about the weights of the Gaussian quadrature formulae associated with the ultraspherical weight function w λ (x) = (1 - x 2) λ - 1 / 2 to find an explicit (modulo ω n , k ) formula for τ n , k 2 . This enables us to prove a lower bound and to refine the previously obtained upper bounds for τ n , k . The explicit formula admits also a new derivation of the asymptotic formula approximating τ n , k for n → ∞ . The new approach is simpler, without using deep results about the ordinates of the Bessel function, and allows to better analyze the sharpness of the estimates. [ABSTRACT FROM AUTHOR]

This article presents a new approach of shifted Jacobi spectral Galerkin methods to solve weakly singular Fredholm integral equations with non-smooth solutions. We have incorporated the singular part of the kernel into a single Jacobi weight function, by dividing the integration into two parts and using a simple variable transformation. Taking advantage of orthogonal projection operator and weighted inner product with respect to that same Jacobi weight function, we are able to obtain improved convergence rate for iterated shifted Jacobi spectral Galerkin method (SJSGM) and iterated shifted Jacobi spectral multi-Galerkin method (SJSMGM) in both weighted and infinity norms. Further, we obtain improved superconvergence rate for iterated SJSGM and iterated SJSMGM, by improving the regularity of exact solution, using smoothing transformation. Increasing the value of the smoothing parameter we can improve the regularity of the exact solution upto the desired degree. Numerical results with a comparative study of pre and post smoothing transformation are given to illustrate the theoretical results and efficiency of our proposed methods. [ABSTRACT FROM AUTHOR]

Let m and N be positive integers and k be a negative real number. Let χ be a multiplier system of weight k on Γ 0 (N). In this paper, we give an exact formula for the Fourier coefficients of the holomorphic part of a harmonic Maass–Jacobi form of weight k, index m, and multiplier system χ on Γ 0 (N). As an application, we investigate the number of ratios of the Fourier coefficients of the holomorphic parts of two harmonic Maass–Jacobi forms. Moreover, we obtain the equivalent conditions for the principal part of a harmonic Maass–Jacobi form to be the principal part of a weak Jacobi form with poles only at infinity. [ABSTRACT FROM AUTHOR]

In this paper, optimal estimates on the decaying rates of Jacobi expansion coefficients are obtained by fractional calculus for functions with algebraic and logarithmic singularities. This is inspired by the fact that integer-order derivatives fail to deal with singularity of fractional-type, while fractional calculus can. To this end, we first introduce new fractional Sobolev spaces defined as the range of the L p -space under the Riemann-Liouville fractional integral. The connection between these new spaces and classical fractional-order Sobolev spaces is then elucidated. Under this framework, the optimal decaying rate of Jacobi expansion coefficients is obtained, based on which the projection errors under different norms are given. This work is expected to introduce fractional calculus into traditional fields in approximation theory and to explore the possibility in solving classical problems by this ‘new’ tool. [ABSTRACT FROM AUTHOR]

The human eye is a fascinating optical system, with the crystalline lens playing a significant role in focusing light onto the retina of the eye. The ray tracing through the crystalline lens problem is a challenging problem in optics. In this paper, the case of a non-homogeneous optical medium is investigated, and the ray equation is numerically solved to get the ray paths. The governing equation is an ODE with a fractional derivative given in the Caputo sense. A novel numerical scheme is based on the Jacobi polynomial collocation technique to tackle this problem. A fast and accurate Broyden's Quasi-Newton algorithm is applied to solve the nonlinear system of equations obtained from the collocation process. Numerical results are stated in detail to show the efficiency of our technique and are compared with other analytical and numerical methods using tables and illustrated figures, which will be useful to corroborate the clinical and physical data. Ray tracing through the crystalline lens is not only fascinating from a scientific perspective but also has practical implications across various domains, and the proposed scheme is considered a promising and practically reliable method to address such types of applications. [ABSTRACT FROM AUTHOR]

In this paper, we give the Jacobi polynomials for first-order generalized Reed–Muller codes. We show as a corollary the nonexistence of combinatorial 3-designs in these codes. [ABSTRACT FROM AUTHOR]

Numerical methods for solving differential equations often rely on the expansion of the approximate solution using basis functions. The choice of an appropriate basis function plays a crucial role in enhancing the accuracy of the solution. In this study, our aim is to develop algorithms that can identify an optimal basis function for any given differential equation. To achieve this, we explore fractional rational Jacobi functions as a versatile basis, incorporating hyperparameters related to rational mappings, Jacobi polynomial parameters, and fractional components. Our research develops hyperparameter optimization algorithms, including parallel grid search, parallel random search, Bayesian optimization, and parallel genetic algorithms. To evaluate the impact of each hyperparameter on the accuracy of the solution, we analyze two benchmark problems on a semi‐infinite domain: Volterra's population model and Kidder's equation. We achieve improved convergence and accuracy by judiciously constraining the ranges of the hyperparameters through a combination of random search and genetic algorithms. Notably, our findings demonstrate that the genetic algorithm consistently outperforms other approaches, yielding superior hyperparameter values that significantly enhance the quality of the solution, surpassing state‐of‐the‐art results. [ABSTRACT FROM AUTHOR]

Under arbitrary boundary conditions, a Jacobi–Ritz time domain semi-analytical method is proposed to study the vibration behaviors of uniform and stepped conical shells. Based on the idea of the differential element method and the first-order shear deformation theory (FSDT), the vibration analysis models of uniform and stepped conical shell structures are established. The axial and circumferential displacement tolerance functions are expressed by the Jacobi polynomial and the Fourier series. The complex boundary conditions are simulated by using artificial spring technology. In light of the Ritz method and the Newmark-β integral method, the free and forced vibration results of the structure can be figured out in the time domain. The numerical results demonstrate that the proposed method has excellent accuracy and remarkable dependability when compared to the finite element method (FEM). Numerical examples are used to examine the forced and free vibration behaviors under various boundary conditions, semi-vertex angles, thicknesses, and load characteristics. [ABSTRACT FROM AUTHOR]

An effective methodology is proposed to evaluate the boundary stiffnesses of beams with generally restrained edges. The boundary conditions at the ends of the beams are implemented by the penalty method. The unified Jacobi polynomials are introduced to represent the assumed mode shape functions of the beam. Hamilton’s principle with the assumed mode method is employed to derive the equation of motion of the beam with generally restrained boundaries. Subsequently, a novel approach is proposed to evaluate the boundary stiffnesses of the beam using the measured natural frequencies of the beam by vibration experiments. The accuracy of this method is verified by comparing the results with the previously published literature, the finite element method (FEM), and the vibration experiments. The effects of the types and values of the boundary spring stiffnesses on the vibration properties of the beam are also discussed. The calculation method in this research can provide an effective prediction technique for the boundary stiffnesses of engineering structures with generally restrained edges. [ABSTRACT FROM AUTHOR]

The Benjamin-Bona-Mahony-Burgers equation (BBMBE) plays a fundemental role in many application scenarios. In this paper, we study a spectral method for the BBMBE with homogeneous boundary conditions. We propose a spectral scheme using the transformed generalized Jacobi polynomial in combination of the explicit fourth-order Runge-Kutta method in time. The boundedness, the generalized stability and the convergence of the proposed scheme are proved. The extensive numerical examples show the efficiency of the new proposed scheme and coincide well with the theoretical analysis. The advantages of our new approach are as follows: (i) the use of the transformed generalized Jacobi polynomial simplifies the theoretical analysis and brings a sparse discrete system; (ii) the numerical solution is spectral accuracy in space. [ABSTRACT FROM AUTHOR]

In this paper, we present a fast divergence-free spectral algorithm for the curl-curl problem. Divergence-free bases in two and three dimensions are constructed by using the generalized Jacobi polynomials. An accurate spectral method with exact preservation of the divergence-free constraint pointwisely is then proposed, and its corresponding error estimate is established. We then present a highly efficient solution algorithm based on a combination of the matrix-free preconditioned Krylov subspace iterative method and a fully diagonalizable auxiliary problem, which is derived from the spectral discretizations of generalized eigenvalue problems of Laplace and biharmonic operators. We rigorously prove that the dimensions of the invariant subspace of the preconditioned linear system resulting from the divergence-free spectral method with respect to the dominant eigenvalue 1 are (N - 3)² and 2(N - 3)³ for two- and three-dimensional problems with (N - 1)² and 2(N - 1)³ unknowns, respectively. Thus, the proposed method usually takes only several iterations to converge, and, astonishingly, as the problem size (polynomial order) increases, the number of iterations will decrease, even for highly indefinite system and oscillatory solutions. As a result, the computational cost of the solution algorithm is only a small multiple of N³ and N4 floating number operations for two- and three-dimensional problems, respectively. Plenty of numerical examples for solving the curlcurl problem with both constant and variable coefficients in two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical approximation of initial‐boundary‐value problems of a three‐parameter family of Bona–Smith systems, derived as a model for the propagation of surface waves under a physical Boussinesq regime. The work proposed here is focused on the corresponding problem with Dirichlet boundary conditions and its approximation in space with spectral methods based on Jacobi polynomials, which are defined from the orthogonality with respect to some weighted L2$L^{2}$ inner product. Well‐posedness of the problem on the corresponding weighted Sobolev spaces is first analyzed and existence and uniqueness of solution, locally in time, are proved. Then, the spectral Galerkin semidiscrete scheme and some detailed comments on its implementation are introduced. The existence of numerical solution and error estimates on those weighted Sobolev spaces are established. Finally, the choice of the time integrator to complete the full discretization takes care of different stability issues that may be relevant when approximating the semidiscrete system. Some numerical experiments illustrate the results. [ABSTRACT FROM AUTHOR]

There are several objective tests for analyzing unreplicated two-level factorial designs. However, there is no single test that can detect all patterns of possible active effects. Tests are sensitive to the number and/or the magnitude of active effects. Therefore, it is reasonable to combine recommended tests into a single test to provide researchers with a testing approach that leverages many existing methods to detect different patterns of active effects. The problem is how to combine multiple dependent tests into a single test. In this article, we review four methods for combining dependent tests and present four combined tests. In addition, we review four recommended object tests for detecting active effects. We also propose a new test procedure that can detect active effects when the number of active effects is large. We finally evaluate these nine tests (five original tests and four combined tests) in terms of controlling the type I error rate and the power performance via a simulation study. Simulation results show that the combined test that is based on the Jacobi polynomial expansion can be recommended as a test procedure to detect active effects. [ABSTRACT FROM AUTHOR]

A new interpretation and applications of the 'Diophantine' and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of $ 0,1,\ldots,N $ 0 , 1 , ... , N , constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the 'Diophantine' and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm 'eigenvectors' of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a numerical method based on the shifted fractional order Jacobi and trapezoid methods to solve a type of distributed partial differential equations. The fractional derivatives are considered in the Caputo-Prabhakar type. By shifted fractional-order Jacobi polynomials our proposed method can provide highly accurate approximate solutions by reducing the problem under study to a set of algebraic equations which is technically simpler to handle. In order to demonstrate the error estimates, several lemmas are provided. Finally, numerical results are provided to demonstrate the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

The design of a hypersonic vehicle controller has been an active research field in the last decade, especially when the vehicle is studied as a time-varying system. A time-varying compound control method is proposed for a hypersonic vehicle controlled by the direct lateral force and the aerodynamic force. The compound control method consists of a time-varying linear quadratic regulator (LQR) control law for the aerodynamic rudder and a sliding mode control law for the lateral thrusters. When the air rudder cannot continuously produce control force and torque, the direct lateral force is added to the system. To solve the problem that LQR cannot directly obtain the analytical solution of the time-varying system, a novel approach to approximate analytical solutions using Jacobi polynomials is proposed in this paper. Finally, the stability of the time-varying compound control system is proven by the Lyapunov–Krasovskii functional (LKF). The simulation results show that the proposed compound control method is effective and can improve the fast response ability of the system. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the approximation of functions using a higher-order Hermite-Fej´er interpolation process on the unit circle. The system of nodes is composed of vertically projected zeros of Jacobi polynomials onto the unit circle with boundary points at ±1. Values of the polynomial and its first four derivatives are fixed by the interpolation conditions at the nodes. Convergence of the process is obtained for analytic functions on a suitable domain, and the rate of convergence is estimated. [ABSTRACT FROM AUTHOR]

The presented work introduces a new class of nonlinear optimal control problems in two dimensions whose constraints are nonlinear Ginzburg-Landau equations with fractal-fractional (FF) derivatives. To acquire their approximate solutions, a computational strategy is expressed using the FF derivative in the Atangana-Riemann-Liouville (A-R-L) concept with the Mittage-Leffler kernel. The mentioned scheme utilizes the shifted Jacobi polynomials (SJPs) and their operational matrices of fractional and FF derivatives. A method based on the derivative operational matrices of SJR and collocation scheme is suggested and employed to reduce the problem into solving a system of algebraic equations. We approximate state and control functions of the variables derived from SJPs with unknown coefficients into the objective function, the dynamic system, and the initial and Dirichlet boundary conditions. The effectiveness and efficiency of the suggested approach are investigated through the different types of test problems. [ABSTRACT FROM AUTHOR]

A general frame for Bernstein-type operators that preserve derivatives is given. We introduce Bernstein-type operators based in the weighted classical Jacobi inner product on the interval [0, 1] that extend the well known Bernstein–Durrmeyer operator as well as some other types of Bernstein operators that appear in the literature. Apart from standard results, we deduce properties about the preservation of derivatives and prove that classical Jacobi orthogonal polynomials on [0, 1] are the eigenfunctions of these operators. We also study the limit cases when one of the parameters of the Jacobi polynomials is a negative integer. Finally, we study several numerical examples. [ABSTRACT FROM AUTHOR]

We provide here a novel approach for solving IVPs in ODEs and MTFDEs numerically by means of a class of MSJPs. Using the SCM, we build OMs for RIs and RLFI for MSJPs as part of our process. These architectures guarantee accurate and efficient numerical computations. We provide theoretical assurances for the efficacy of an algorithm by establishing its convergence and error analysis features. We offer five numerical examples to prove that our method is accurate and applicable. Through these examples, we demonstrate the greater accuracy and efficiency of our approach by comparing our results with previously published findings. Tables and graphs show that the method produces exact and approximate solutions that agree quite well with each other. [ABSTRACT FROM AUTHOR]

Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers functions of different orders and degrees as well as a uniform asymptotic expansion are derived in this article. Simple proofs of four generating functions for Ferrers functions are given. Addition theorems for P ν - μ tanh α + β are proved by basing on generation functions for three families of hypergeometric polynomials. Relations for Gegenbauer polynomials and Ferrers associated Legendre functions (associated Legendre polynomials) are obtained as special cases. [ABSTRACT FROM AUTHOR]

The electrostatic interpretation of zeros of Jacobi polynomials, due to Stieltjes and Schur, enables us to obtain the complete asymptotic expansion as n → ∞ of the minimal logarithmic potential energy of n point charges restricted to move in the interval [ - 1 , 1 ] in the presence of an external field generated by endpoint charges. By the same methods, we determine the complete asymptotic expansion as N → ∞ of the logarithmic energy ∑ j ≠ k log (1 / | x j - x k |) of Fekete points, which, by definition, maximize the product of all mutual distances ∏ j ≠ k | x j - x k | of N points in [ - 1 , 1 ] . The results for other compact intervals differ only in the quadratic and linear term of the asymptotics. Explicit formulas and their asymptotics follow from the discriminant, leading coefficient, and special values at ± 1 of Jacobi polynomials. For all these quantities we derive complete Poincaré-type asymptotics. [ABSTRACT FROM AUTHOR]

From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric series. With these contiguous relations one can prove several recursion formulas of those series. This theoretical result allows to compute integrals over products of Jacobi polynomials in a very efficient recursive way. Moreover, the authors present an application to numerical analysis where it can be used in algorithms which compute the approximate solution of boundary value problem of partial differential equations by means of the finite elements method. With the aid of the contiguous relations, the approximate solution can be computed much faster than using numerical integration. A numerical example illustrates this effect. [ABSTRACT FROM AUTHOR]

In this article, an accurate computational approaches based on Lagrange basis and Jacobi-Gauss collocation method is suggested to solve a class of nonlinear stochastic Itô-Volterra integral equations (SIVIEs). Since the exact solutions of this kind of equations are not still available, so finding an accurate approximate solutions has attracted the interest of many scholars. In the proposed methods, using Lagrange polynomials and zeros of Jacobi polynomials, the considered system of linear and nonlinear stochastic Volterra integral equations is reduced to linear and nonlinear systems of algebraic equations. Solving the resulting algebraic systems by Newton's methods, approximate solutions of the stochastic Volterra integral equations are constructed. Theoretical study is given to validate the error and convergence analysis of these methods; the spectral rate of convergence for the proposed method is established in the L ∞ -norm. Several related numerical examples with different simulations of Brownian motion are given to prove the suitability and accuracy of our methods. The numerical experiments of the proposed methods are compared with the results of other numerical techniques. [ABSTRACT FROM AUTHOR]

In the variational formulation of a boundary value problem for the harmonic oscillator, Sobolev inner products appear in a natural way. First, we study the sequences of Sobolev orthogonal polynomials with respect to such an inner product. Second, their representations in terms of a sequence of Gegenbauer polynomials are deduced as well as an algorithm to generate them in a recursive way is stated. The outer relative asymptotics between the Sobolev orthogonal polynomials and classical Legendre polynomials is obtained. Next we analyze the solution of the boundary value problem in terms of a Fourier-Sobolev projector. Finally, we provide numerical tests concerning the reliability and accuracy of the Sobolev spectral method. [ABSTRACT FROM AUTHOR]

This paper explores the possibility of using dual representation of geometry to speed up ray tracing and ensure robustness of light propagation simulation in complex optical systems with freeform surfaces defined by high-degree polynomials (up to order 34) or Jacobi polynomials. Traditional methods for representing this geometry both as a triangular mesh and as an analytical expression are analyzed. The analysis demonstrates the disadvantages of the traditional approaches due to the insufficient accuracy of calculating coordinates of the point at which the ray intersects the triangular mesh, as well as the non-robustness of the conventional procedures for finding the point of intersection between the tangent ray and the analytical surface. Thus, it is proposed to use a dual representation of geometry as a rough approximation of the surface by a triangular mesh, which is subsequently used as an initial approximation to find the point at which the ray intersects the surface defined by an analytical expression. This approach significantly speeds up convergence of the analytical methods and improves robustness of their solutions. The use of the Intel® Embree library to quickly find the point of intersection between the ray and the rough triangular mesh, as well as a vector model to refine the coordinates of the point of intersection between the ray and the analytically represented geometry, allows us to develop and implement a ray tracing algorithm for an optical system that has surfaces with dual representation of geometry. Experiments carried out using the developed algorithm show a significant speedup of ray tracing while preserving computational accuracy and high robustness of the results. The results are demonstrated by evaluating the point spread function and glare for two lenses with freeform surfaces defined by Jacobi polynomials. In addition, for these two lenses, an image formed by an RGB-D object that simulates a real scene is calculated. [ABSTRACT FROM AUTHOR]

The field of multiplicative analysis has recently garnered significant attention, particularly in the context of solving multiplicative differential equations (MDEs). The symmetry concept in MDEs facilitates the determination of invariant solutions and the reduction of these equations by leveraging their intrinsic symmetrical properties. This study is motivated by the need for efficient methods to address MDEs, which are critical in various applications. Our novel contribution involves leveraging the fundamental properties of orthogonal polynomials, specifically Laguerre polynomials, to derive new solutions for MDEs. We introduce the definitions of Laguerre multiplicative differential equations and multiplicative Laguerre polynomials. By applying the power series method, we construct these multiplicative Laguerre polynomials and rigorously prove their basic properties. The effectiveness of our proposed solution is validated through illustrative examples, demonstrating its practical applicability and potential for advancing the field of multiplicative analysis. [ABSTRACT FROM AUTHOR]

Considering the lack of studies on the transient vibro-acoustic properties of conical shell structures, a Jacobi–Ritz boundary element method for forced vibro-acoustic behaviors of structure is proposed based on the Newmark-β integral method and the Kirchhoff time domain boundary integral equation. Based on the idea of the differential element method and the first-order shear deformation theory (FSDT), the vibro-acoustic model of conical shells is established. The axial and circumferential displacement tolerance functions are expressed using Jacobi polynomials and the Fourier series. The time domain response of the forced vibration of conical shells is calculated based on the Rayleigh–Ritz method and Newmark-β integral method. On this basis, the time domain response of radiated noise is solved based on the Kirchhoff integral equation, and the acoustic radiation characteristics of conical shells from forced vibration are analyzed. Compared with the coupled FEM/BEM method, the numerical results demonstrate the high accuracy and great reliability of this method. Furthermore, the semi-vertex angle, load characteristics, and boundary conditions related to the vibro-acoustic response of conical shells are examined. [ABSTRACT FROM AUTHOR]

We study Jacobi processes (X t) t ≥ 0 on [ - 1 , 1 ] N and [ 1 , ∞ [ N which are motivated by the Heckman–Opdam theory and associated integrable particle systems. These processes depend on three positive parameters and degenerate in the freezing limit to solutions of deterministic dynamical systems. In the compact case, these models tend for t → ∞ to the distributions of the β -Jacobi ensembles and, in the freezing case, to vectors consisting of ordered zeros of one-dimensional Jacobi polynomials. We derive almost sure analogues of Wigner's semicircle and Marchenko–Pastur limit laws for N → ∞ for the empirical distributions of the N particles on some local scale. We there allow for arbitrary initial conditions, which enter the limiting distributions via free convolutions. These results generalize corresponding stationary limit results in the compact case for β -Jacobi ensembles and, in the deterministic case, for the empirical distributions of the ordered zeros of Jacobi polynomials. The results are also related to free limit theorems for multivariate Bessel processes, β -Hermite and β -Laguerre ensembles, and the asymptotic empirical distributions of the zeros of Hermite and Laguerre polynomials for N → ∞ . [ABSTRACT FROM AUTHOR]