Jacobi Polynomials Research Articles

An operational matrix strategy for time fractional Fokker-Planck equation in an unbounded space domain

In this study, the Caputo fractional derivative is used to define time fractional Fokker-Planck equation in an unbounded domain. To solve this equation, the Jacobi polynomials together with the tanh-Jacobi functions are employed. The operational matrices of the classical and fractional derivatives of these basis functions are obtained to use them in constructing a numerical method for the expressed equation. In the proposed method, the introduced basis functions are used simultaneously to approximate the equation’s unknown solution. More precisely, the shifted Jacobi polynomials are applied to approximate the solution in the temporal direction and the tanh-Jacobi functions are utilized to approximate the solution in the spatial direction. By substitute the expressed approximation into the equation and employing the introduced operational matrix, solving the problem under consideration transforms into solving an algebraic system of equations, which can be solved easily. The accuracy and efficiency of the presented method are investigated numerically by solving some numerical examples. The reported results confirms the high accuracy of the established method.

Numerical computation of fractional Bloch equation by using Jacobi operational matrix

In this work, we present a numerical scheme based on the operational matrix of fractional Caputo-Fabrizio (CF) integration for handling fractional Bloch equation (FBE) in nuclear magnetic resonance (NMR). The understanding of Bloch equation provides us a fundamental framework for describing magnetic resonance phenomena, facilitating breakthrough in diverse fields such as medical diagnostics, quantum computing and materials characterization. The non-integer order derivative and integration are presented in the Caputo-Fabrizio sense. To construct the operational matrix, Jacobi polynomial is used as a basis. The fractional Bloch equation is transformed into a set of algebraic equations by using the operational matrix. In order to examine the fractional order problem, we obtain an approximate solution for FBE and present the numerical results in graphical and tabular forms.

Vibrations and thermoelastic quality factors of hemispherical shells with fillets

In engineering applications, hemispherical shell resonators are typically machined with fillets to reduce stress concentration and enhance structure strength. The fillets will inevitably affect the dynamic properties and the mechanical quality factor of hemispherical shell resonators, which has been seldom investigated before. In this paper, an effective analytical method is developed to explore the free vibration and thermoelastic damping (TED) characteristics of the hemispherical shell with fillets. The fillets are characterized by the variation in the thickness of the hemispherical shell during modelling. The first-order shear deformation theory (FSDT) is used to describe the theoretical formulas of the hemispherical shell with fillets. By employing the unified Jacobi polynomials and Fourier series as the assumed mode shape functions, the equation of motion of the structure is established by Hamilton's principle and the assumed mode method. The analytical model for thermoelastic quality factor (QTED) which is determined by TED is obtained by computing the dissipated energy and the maximum elastic potential energy of the hemispherical shell with fillets. The validity and accuracy of the present method are confirmed by comparing the present solutions with the published results and those obtained from the finite element method (FEM). The influences of fillets on the vibration behaviors and QTED characteristics of the hemispherical shells are analyzed in detail. The present model can be used to optimize the design of the fillets of the hemispherical shell resonators with high quality factors.

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.

On Linearization Coefficients of Shifted Jacobi Polynomials

On Linearization Coefficients of Shifted Jacobi Polynomials

Convergence analysis of Jacobi spectral tau-collocation method in solving a system of weakly singular Volterra integral equations

The main purpose of this paper is to solve a system of weakly singular Volterra integral equations using the Jacobi spectral tau-collocation method from two perspectives. Since the solutions of the main system exhibit discontinuity at the origin, classical Jacobi methods may yield less accuracy. Therefore, in the first approach, we transform the proposed system through a suitable transformation into an alternative type whose solutions are as smooth as desired. Subsequently, we derive a matrix formulation of the method and analyze its convergence properties in both L2 and L∞-norms.In the second approach, instead of employing a smoothing transformation, we select fractional Jacobi polynomials as basis functions for the approximation space. This choice is motivated by their similar behavior to the exact solutions. We then derive a matrix formulation of the method and perform an error analysis analogous to the first approach. Finally, we present several illustrative examples to demonstrate the accuracy of our method.

Some Orthogonal Combinations of Legendre Polynomials

The principle objective of this article is to introduce and investigate a type of orthogonal polynomials that are written as combinations of Legendre polynomials. This kind of polynomials can be viewed as generalized Jacobi polynomials (GJPs), since they are written in terms of Jacobi polynomials (JPs) with certain negative parameters. The analytic and inversion formulas of these polynomials are established. New derivative expressions for these polynomials are derived in detail in terms of their original polynomials. Other derivative expressions for these polynomials are found but in terms of some orthogonal and non-orthogonal polynomials. Some product formulas with some other polynomials are also obtained. Certain definite and weighted definite integrals are obtained using the newly introduced connection and product formulas.

Wigner- and Marchenko–Pastur-Type Limit Theorems for Jacobi Processes

AbstractWe study Jacobi processes $$(X_{t})_{t\ge 0}$$ ( X t ) t ≥ 0 on $$[-1,1]^N$$ [ - 1 , 1 ] N and $$[1,\infty [^N$$ [ 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 $$t\rightarrow \infty $$ t → ∞ to the distributions of the $$\beta $$ 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 $$N\rightarrow \infty $$ 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 $$\beta $$ 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, $$\beta $$ β -Hermite and $$\beta $$ β -Laguerre ensembles, and the asymptotic empirical distributions of the zeros of Hermite and Laguerre polynomials for $$N\rightarrow \infty $$ N → ∞ .

Jacobi polynomials for the first-order generalized Reed–Muller codes

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.

A Spectrally Accurate Step-by-Step Method for the Numerical Solution of Fractional Differential Equations

In this paper we consider the numerical solution of fractional differential equations. In particular, we study a step-by-step procedure, defined over a graded mesh, which is based on a truncated expansion of the vector field along the orthonormal Jacobi polynomial basis. Under mild hypotheses, the proposed procedure is capable of getting spectral accuracy. A few numerical examples are reported to confirm the theoretical findings.

A Novel Deep Convolutional Neural Network Approach using Jacobi Polynomial and Laplacian Function (JPLF) in Recognition of Plant Leaf Disease

Background/Objectives: Enhancing agricultural productivity is crucial for fostering economic growth. Plant diseases significantly threaten crops, necessitating timely detection to mitigate adverse impacts on quality, quantity, and overall productivity. This research addresses the importance of early disease detection in agriculture and proposes an innovative method utilizing Jacobian Polynomial and Laplacian Function for precise identification. Methods: Efficient monitoring of large-scale crop farms with minimal workforce is essential. To achieve this, an automatic method for plant disease detection is proposed. The method leverages Jacobian polynomials to expand input features, mitigating correlation issues among input vectors. The expanded Jacobi polynomial is the input vector for a backpropagation algorithm with a novel activation function based on the Laplacian function. Findings: The efficacy of the proposed JPLF model is demonstrated through the accurate identification of leaf diseases, achieving a high testing accuracy of 92.07%. Comparative analysis with existing models, such as CNN with MobileNet V2 (85.38%) and the IoU model (83.75%), highlights the superiority of the JPLF model in plant disease detection. Novelty: To overcome the limitations of existing approaches, the incorporation of Jacobian polynomials plays a pivotal role in expanding input features. This expansion aids in eliminating correlations among input vectors, enhancing the efficacy of disease detection. The proposed model, Jacobi Polynomial and Laplacian Function (JPLF) introduces a unique activation function based on the Laplacian function, improving accuracy. Keywords: Plant Disease Detection, Jacobi Polynomial, Laplacian Transform, Deep Learning Model, Feature Expansion

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.

New Generalized Jacobi Polynomial Galerkin Operational Matrices of Derivatives: An Algorithm for Solving Boundary Value Problems

In this study, we present a novel approach for the numerical solution of high-order ODEs and MTVOFDEs with BCs. Our method leverages a class of GSJPs that possess the crucial property of satisfying the given BCs. By establishing OMs for both the ODs and VOFDs of the GSJPs, we integrate them into the SCM, enabling efficient and accurate numerical computations. An error analysis and convergence study are conducted to validate the efficacy of the proposed algorithm. We demonstrate the applicability and accuracy of our method through eight numerical examples. Comparative analyses with prior research highlight the improved accuracy and efficiency achieved by our approach. The recommended approach exhibits excellent agreement between approximate and precise results in tables and graphs, demonstrating its high accuracy. This research contributes to the advancement of numerical methods for ODEs and MTVOFDEs with BCs, providing a reliable and efficient tool for solving complex BVPs with exceptional accuracy.

A space-time Galerkin Müntz spectral method for the time fractional Fokker–Planck equation

In this paper, we propose a space-time Galerkin spectral method for the time fractional Fokker–Planck equation. This approach is based on combining temporal Müntz Jacobi polynomials spectral method with spatial Legendre polynomials spectral method. Based on the well-posedness and regularity for the re-scaled problem of a linear model problem which reflects the main difficulty for solving the equivalent equation (i.e. the time fractional convection-diffusion equation): the singularity of the solution in time, we explain in detail why we use the Müntz polynomials to approximate in time. The well-posedness and stability of the discrete scheme as well as its continuous problem are established. Moreover, the error estimation of the space-time approach is derived. We find that the proposed method can attain spectral accuracy regardless of whether the solution of the original equation is smooth or non-smooth. Numerical experiments substantiate the theoretical results.

A fast polynomial-FE method for the vibration of the composite laminate quadrilateral plates and shells based on the segmentation strategy

The finite element method is widely applicable to various kinds of models. Nonetheless, as the size and frequency domain of the solving model increase, it encounters challenges related to high memory consumption and slow solution speed. To enhance the computational efficiency while preserving its existing advantages, this study incorporates the Jacobi polynomial with a segmentation strategy into the finite element method for addressing the vibration analysis of composite laminate quadrilateral flat and curved plates. In this method, the dynamic stiffness matrix of a finite element is multiplied on both sides by a matrix composed of Jacobi polynomial basis functions, thereby transforming the expression form of the dynamic stiffness matrix from FE form to meshless form and reducing its size. Additionally, the segmentation strategy is introduced to divide the model into multiple blocks, enhancing convergence and calculation efficiency by altering the sparsity pattern of the dynamic stiffness matrix. Convergence research, validation studies, limitations studies, and parametric studies are conducted. The method demonstrates evident advantages in computational efficiency for random surface quadrilateral models requiring dense mesh description while maintaining a certain level of accuracy.

Analytical solutions for the Klein–Gordon equation with combined exponential type and ring-shaped potentials

In this study, we have successfully obtained the analytical solutions for the Klein–Gordon equation with new proposed a non-central exponential potential Vr=D1-σ0coth(αr)2+(η1+η2cosθ)/r2sin2θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V\\left( r \\right) = D\\left[ {1 - \\sigma_{0} \\coth (\\alpha r)} \\right]^{2} + (\\eta_{1} + \\eta_{2} \\cos \ heta )/r^{2} \\sin^{2} \ heta$$\\end{document}. Our approach involves a proper approximation of the centrifugal term, with l′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${l}{\\prime}$$\\end{document} representing the generalized orbital angular momentum quantum number, and the utilization of the Nikiforov–Uvarov method. The resulting radial and angular wave functions are expressed in terms of Jacobi polynomials, and the corresponding energy equation is also derived. Our calculations of the eigenvalues for arbitrary quantum numbers demonstrated significant sensitivity to potential parameters and quantum numbers. Additionally, we evaluate the dependence of energy eigenvalues on screening parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} for arbitrary quantum numbers nr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_{r}$$\\end{document} and N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N$$\\end{document} to establish the accuracy of our findings. Furthermore, we determine the non-relativistic limits of the radial wave function and energy equation, which align with corresponding previous results in the case where η1=η2=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta_{1} = \\eta_{2} = 0$$\\end{document}.

Spectral Jacobi approximations for Boussinesq systems

AbstractThis 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 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.

Generating Function Of Extended Jacobi Polynomial Involving Stirling Number of Second Kind

In this paper we have obtained the generating function of extended Jacobi polynomial containing Stirling number of the second kind. Some particular cases are also obtained

On the Fourier coefficients of the derivative with respect to celebrated orthogonal systems

The main goal of this paper is to find the coefficients of the Jacobi polynomials and the integrals of Legendre polynomials expansion of the derivative of a function in terms of the coefficients in the expansion of the original function. More precisely, if {Q_{n}} is a sequence or orthogonal polynomials, and if p(x)=∑_{j=0}ⁿa_{j}Q_{n-j}(x) is such that p′(x)=∑_{j=0}ⁿ⁻¹d_{j}Q_{n-j-1}(x), we find an explicit relation for the coefficients d_{j}, as linear combinations of the coefficients a_{j}. This will be done for two celebrated classes of orthogonal functions, namely the Jacobi polynomials and the integrals of the Legendre polynomials.

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.