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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]

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]

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 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]

For the numerical solution of Fredholm integral equations on [ − 1 , 1 ] whose integrands have endpoint algebraic singularities, we investigate in this paper a modified collocation method based on the zeros of the Jacobi polynomials in appropriate weighted spaces. The proposed method converges faster than the standard collocation scheme, and the Sloan iteration can be used to make the solution even more accurate. The iterated collocation method is also defined in this paper. To the best of our knowledge, this work is the first to investigate superconvergent methods for such integral equations. Some numerical tests are presented to show the effectiveness of the suggested methods. [ABSTRACT FROM AUTHOR]

In this paper, the one- and two-dimensional stochastic multi-order fractional diffusion-wave equations are introduced and a collocation procedure based on the shifted Jacobi polynomials is established to find their numerical solutions. Through this way, some operational matrices regarding classical and stochastic integrals as well as fractional and classical differentiations of these polynomials, are obtained. By representing the problem solution using an expansion of these polynomials (in which the coefficients of the expansion are unknown) and substituting it into the first problem, as well as by employing the obtained operational matrices, a system containing algebraic equations is obtained. Eventually, the coefficients of expansion and subsequently the solution of the original problem are found by solving this system. The correctness of the procedure is studied by solving four examples. [ABSTRACT FROM AUTHOR]

We develop a very simple but efficient postprocessing technique for enhancing the accuracy of the C 1 -conforming finite element method for fourth-order boundary value problems. The key idea of the postprocessing technique is to add certain generalized Jacobi polynomials of degree larger than k to the Galerkin approximation of degree k. We prove that the postprocess improves the order of convergence of the Galerkin approximation under L 2 - and H 2 -norms. Numerical experiments are provided to illustrate the theoretical results. We also present computational results for a steady-state problem with to demonstrate the effectiveness of the postprocessing. • We propose a novel postprocessing technique for enhancing the accuracy of the C 1 -conforming FEM for fourth-order BVPs. • Convergence orders of the L 2 - and H 2 -errors of the Galerkin approximation are improved from O (h k + 1) and O (h k − 1) to O (h { 2 k − 2 , k + 3 }) and O (h k + 1) , respectively. • The postprocessing is local and can be done independently on each element. [ABSTRACT FROM AUTHOR]

In this paper, the variable-order (VO) time fractional version of the 2D Wu-Zhang system is introduced. A numerical method based on the spline approximation of the VO fractional terms in this system and the Jacobi polynomials (JPs) approximation of the solution is proposed to solve this system. In the presented method, the VO fractional derivatives are approximated using the spline expansions at first. Then, by employing the expressed spline expansions as well as the θ -weighted method a recurrence relation is obtained. Next, the unknown functions are expanded in terms of the 2D JPs (with unknown coefficients) and substituted into the recurrence relation. Finally, by solving a system of linear algebraic equations, the unknown coefficients and subsequently an approximate solution of the problem is obtained. The accuracy of the established method is investigated by solving some numerical examples. [ABSTRACT FROM AUTHOR]

• The dynamic model of stepped functionally graded conical shells under general boundary conditions is established. • The introduction of Jacobi polynomial enriches the selection and diversity of displacement functions. • The free and forced vibrational behaviors of the structure are delineated by utilizing the Ritz method and Newmark- β integral approach. This study investigates the dynamic behaviors of stepped functionally graded conical shells under general boundary restraints. The properties of functionally graded material change continuously through the thickness in the framework of the general four parameter power-law distributions. A numerical analysis framework is formulated utilizing the Jacobi polynomials-Ritz method. The construction of the dynamic model incorporates the first-order shear deformation theory, domain decomposition approach and virtual spring method, with the introduction of the Jacobi orthogonal polynomials to delineate displacement functions. Through the Ritz method, both the free and forced vibrational behaviors of the structure are delineated. Moreover, the vibration response of the structure in time domain is ascertained utilizing the Newmark- β integral approach. This study offers a detailed examination of the influence exerted by factors including truncation number, and the Jacobi parameter on the convergence of the presented method. The veracity of the current approach is verified by juxtaposing its findings with those derived from the Finite Element Method and existing literatures. Additionally, the investigation explores the dynamic features of the stepped structure subject to different construction parameters, boundary restraints, and power-law distribution, as demonstrated through a comprehensive set of numerical examples. [ABSTRACT FROM AUTHOR]

In the research, a quantum dot (QD) under an external magnetic field is theoretically investigated. The confining potential applied to the charge carriers is chosen as the Frost-Musulin (FM) potential model. First, the energy eigenvalues and eigenstates have been analytically obtained by Nikiforov-Uvarov (NU) procedure. Then, an electric field is imposed on the system. The Stark shift effect (SSE) has been calculated and an analytical relation has been obtained in terms of the Jacobi polynomials. The findings show that the shift of electron energy states at large, and small electric fields are different. The shift is small at weak electric fields. The electron energy states decrease with the increment of the electric field. The electron states are increased by enhancing the system size. In summary, the SSE can be tuned by setting the electric field, the potential height, and the size of QD. [ABSTRACT FROM AUTHOR]

We consider a generalized (3+1)-dimensional nonlinear Schrödinger with cubic-quintic nonlinear and self-frequency shift and self-steepening terms. We disrupt the plane wave to study the stability of the wave in this media. We examine how various factors—such as the amplitude of the plane wave, cubic and quintic nonlinear terms, self-steepening, and self-frequency shift—affect the modulational instability (MI). Our findings reveal that the amplitude of the plane wave and the quintic nonlinear term can expand the MI bands and increase the amplitude of the MI growth rate. Conversely, the self-steepening and self-frequency shift terms exert opposite effects, narrowing the MI bands and reducing the amplitude of the MI growth rate. We investigate the existence of modulated chirped rational and polynomial Jacobi elliptic function solutions and chirped optical solitons. [ABSTRACT FROM AUTHOR]

We introduce a class of orthogonal functions associated with integral and fractional differential equations with a logarithmic kernel. These functions are generated by applying a log transformation to Jacobi polynomials. We construct interpolation and projection error estimates using weighted pseudo-derivatives tailored to the involved mapping. Then, using the nodes of the newly introduced logarithmic Jacobi functions, we develop an efficient spectral logarithmic Jacobi collocation method for the integrated form of the Caputo–Hadamard fractional nonlinear differential equations. To demonstrate the proposed approach's spectral accuracy, an error estimate is derived, which is then confirmed by numerical results. [ABSTRACT FROM AUTHOR]

• A novel class of flawed functional gradient composite sandwich cylindrical shells with a corrugated-honeycomb blended core is tailored. • An analytical dynamic prediction model of the target structure under arbitrary boundary constraints undergoing a non-uniform thermal regime is developed. • The suitable model parameters are excavated for low computing costs, sufficient data output, and stable convergence of benchmark results. • Unreported transient response behaviors of such structures subjected to rectangular, exponential, triangular, and half-sine pulse loads and heat intrusion are disclosed. • Some design recommendations dedicated to improving natural frequencies and reducing response amplitudes and decay times are pointed out. In the present study, an original sandwich cylindrical shell structure composed of a corrugated-honeycomb blended core and functional gradient composite skins with even and uneven porosity defects is tailored for lightweight engineering components, where both the cylindrical honeycomb-corrugated cross-reinforced core and composite forms deliver novel design, optimization, and refinement schemes for associated sandwich shell structures. Accordingly, an analytical model for predicting and assessing the vibration and response behaviors of such structures is proposed to repair the absence of matched theoretical tools in the current market. Also, an inhomogeneous temperature field and various external loads as the endured environmental factors are introduced to enrich the prediction capacity and reference scope of the developed model. Herein, the refined equivalent mechanical parameters of the corrugated-honeycomb blended core are analytically formulated using a macroscopic homogeneity approach based on the strain energy invariant criterion, and the mechanical properties of functional gradient composite skins obeying the power, sigmoid, and exponential distributions are further determined, with even and uneven porosity defects being considered. Besides, the inhomogeneous temperature spreads at various locations in the core and skin parts are derived from the solutions of Fourier heat conduction equations. Employing the first-order shear deformation theory, the Green heat strain hypothesis, and the spring simulation technique, the analytical expressions of elastic strain, thermal strain, boundary potential, and kinetic energies are obtained, with the work generated by external loads included. Subsequently, the free and forced vibration solutions of such structures in the form of Jacobi polynomial displacement assumptions are reaped using the Rayleigh-Ritz and Newmark-beta computation routines. Convergence analyzes and comparison studies are conducted to ensure valid model parameters and output results. Lastly, the effect of the temperature spread on the inherent characteristics and vibration responses is disclosed, and optimal schemes of core configurations and skin material allocations are provided for the utmost vibration suppression capability. [ABSTRACT FROM AUTHOR]

• An effective method is developed for dynamic modelling of hemispherical shells with fillets. • Effects of fillets on thermoelastic quality factors of hemispherical shells are deeply explored. • The lower fillet has larger influence on the natural frequencies than the upper fillet does. • For the lower fillet, there exists size interval where hemispherical shell has higher energy losses. 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 (Q TED) 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 Q TED 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. [ABSTRACT FROM AUTHOR]

• The vibro-acoustic model of uniform and stepped thickness ring stiffened cylindrical shells under arbitrary boundary conditions is established. • The introduction of Jacobi polynomial enriches the selection and diversity of displacement functions. • The arbitrary impulse excitation load and damped vibro-acoustic behaviors are considered. In this paper, a semi-analytical approach is presented to study the vibro-acoustic response of stiffened cylindrical shells. The analytical model is established by using multi-segment technique, artificial spring technology and smearing method, with the introduction of standard Fourier series and Jacobi polynomials. The Newmark integration approach is adopted to obtain the time domain vibration response, and the time domain Kirchhoff boundary integral formulation is employed to describe the exterior acoustic field. On this basis, the vibro-acoustic model of ring stiffened cylindrical shell can be established by considering the external excitation acting on the cylindrical surface. The accuracy and reliability of the current model are validated by comparing with the coupled FEM/BEM method and experiment, in which the object of the vibro-acoustic response test is a simply supported cylindrical shell. Additionally, the studies on influence of load parameters, edge restraints and structural scale parameters on the vibration and acoustic response of the ring stiffened cylindrical shell are conducted, which is helpful for the design of ring stiffened cylindrical shell to some extent. [ABSTRACT FROM AUTHOR]

• The Jacobi-polynomial and the segmentation strategy are adopted to control the dynamic stiffness matrix size and sparsity of the FE method in the vibration problems. • Various types of geometric shapes of quadrilateral flats and curved plates are considered in the free and forced vibration studies. • A study of the influence of lamination angle on the vibration characteristics of the composite laminate of the curved plates is conducted. 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. [ABSTRACT FROM AUTHOR]

This paper presents a mathematical formulation that accounts for the effects of groundwater table on the dynamic responses of a rigid strip footing on a finite-thickness soil layer subjected to harmonic vertical loading. The soil layer above the groundwater table is treated as viscoelastic single-phase medium, while the bottom layer is treated as saturated two-phase medium. The governing equations of the single-phase and saturated soil are cast based on the classical elastodynamic theory and Biot's poroelastodynamic theory, respectively. The mixed-boundary value problem is converted into a pair of dual integral equations, which are subsequently transformed to a set of linear equations by means of Jacobi orthogonal polynomials and numerically solved. The validity of the derived solution is verified by comparisons with a couple of existing solutions. The effects of the variation of groundwater table and soil parameters on the dynamic compliance of footing and ground surface displacement are examined based on the proposed formulation in frequency domain. [ABSTRACT FROM AUTHOR]

In this paper an indirect spectral method of an optimal control problem governed by a space-fractional diffusion equation with variable diffusivity coefficient is studied. First-order optimality conditions of the proposed model are derived and the regularity of the solutions is analyzed. Indirect spectral methods via weighted Jacobi polynomials are built up based on the "first optimize, then discretize" strategy, and a priori error estimates of the discrete optimal control problem in weighted norms are derived. As proposed indirect spectral discrete schemes are designed to accommodate the impact of the variable coefficients, which are complicated and not formulated under the variational framework, conventional error estimate techniques of the discrete space-fractional optimal control problems do not apply. Novel treatments of the discrete variational inequality are developed to resolve the aforementioned issues and to support the error estimates. Numerical examples are presented to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

In this work, we study the numerical approximation of the initial-boundary-value problem of nonlinear pseudo-parabolic equations with Dirichlet boundary conditions. We propose a discretization in space with spectral schemes based on Jacobi polynomials and in time with robust schemes attending to qualitative features such as stiffness and preservation of strong stability for a more correct simulation of non-regular data. Error estimates for the corresponding semidiscrete Galerkin and collocation schemes are derived. The performance of the fully discrete methods is analyzed in a computational study. [ABSTRACT FROM AUTHOR]

This paper presents new efficient numerical methods for solving Volterra integro-differential equations and a system of nonlinear delay integro-differential equations which arises in biology. The principal idea of these approaches is based on a careful blend of the Petrov-Galerkin technique and the Sumudu transform method. In the proposed methods, using Lagrange polynomials and zeros of Jacobi polynomials, the considered system of linear and nonlinear integro-differential equations, with their associated initial conditions are reduced to linear and nonlinear systems of algebraic equations in the unknown expansion coefficients. Solving the resulting algebraic systems by Gaussian elimination and Newton's methods respectively, approximate solutions of the integro-differential problems are constructed. Detailed error analysis of the proposed methods is carried out to establish and ascertain the reliability and effectiveness of the methods. The methods are then tested on several examples, and the results are compared with those obtained via existing methods in the literature. The numerical results showed that the proposed methods are accurate, efficient, and reliable for solving all kinds of integro-differential equations. [ABSTRACT FROM AUTHOR]

We construct a set H of orthogonal polynomial sequences that contains all the families in the Askey scheme and the q -Askey scheme. The polynomial sequences in H are solutions of a generalized first-order difference equation which is determined by three linearly recurrent sequences of numbers. Two of these sequences are solutions of the difference equation s k + 3 = z (s k + 2 − s k + 1) + s k , where z is a complex parameter, and the other sequence satisfies a related difference equation of order five. We obtain explicit expressions for the coefficients of the orthogonal polynomials and for the generalized moments with respect to a basis of Newton type of the space of polynomials. We also obtain explicit formulas for the coefficients of the three-term recurrence relation satisfied by the polynomial sequences in H. The set H contains all the 15 families in the Askey scheme of hypergeometric orthogonal polynomials [8, p. 183] and all the 29 families of basic hypergeometric orthogonal polynomial sequences in the q -Askey scheme [8, p. 413]. Each of these families is obtained by direct substitution of appropriate values for the parameters in our general formulas. The only cases that require some limits are the Hermite and continuous q -Hermite polynomials. We present the values of the parameters for some of the families. [ABSTRACT FROM AUTHOR]

• A unified vibration model of FRP-FGPGP cylindrical shell is proposed. • The equation of motion of FRP-FGPGP cylindrical shell is derived with Jacobi polynomials and Lagrangian energy function. • Dynamic analysis with different coating patterns, boundary conditions, coating thickness, and fiber layups is studied. In this study, a unified vibration model of fiber reinforced polymer (FRP) cylindrical shell with functionally graded porous graphene platelets (FGPGP) coating is proposed. The material properties of four types of FGPGP coating are presented first, and then the displacements of the structure are expressed by using the first-order shear deformation theory. The energy expressions of coating are introduced by the multi-segment partition technique. Then, an artificial spring method is employed to simulate arbitrary boundary conditions. With the application of the Jacobi orthogonal polynomials and the multi-segment partition techniques, the equation of motion of FRPCS with FG-PGP coating is derived to solve the free and forced vibrations. Finally, the model is systematically verified with some references and dynamic analysis with different coating patterns, boundary conditions and the thickness of coating are comprehensively investigated. [ABSTRACT FROM AUTHOR]

• A time-fractional Black-Scholes model with smooth payoff function (TFBSM-APF) is considered. • A high-order numerical scheme is constructed for solving TFBSM-APF. • Convergence and stability analyses of the proposed schemes are presented. • Numerical examples are given. • The theoretical analysis and numerical approach can be extended to solve other similar time-fractional models. The purpose of this paper is to investigate a high order numerical method for solving time-fractional Black-Scholes equation in which the fractional operator is defined by the Caputo fractional derivative. The proposed space-time spectral method employs the Jacobi polynomials for the temporal discretisation and Fourier-like basis functions for the spatial discretisation. The stability and convergence of the numerical scheme are analyzed. Two numerical examples are considered to validate the accuracy and illustrate the practicability of the proposed method. The results agree with the theoretical analysis and this approach can be applied in dealing with option pricing models with smooth payoff functions. [ABSTRACT FROM AUTHOR]

In this study, we provide a novel third-order weighted essentially non-oscillatory (WENO) method to solve Hamilton-Jacobi equations. The key idea is to incorporate exponential polynomials to construct numerical fluxes and smoothness indicators. First, the new smoothness indicators are designed by using the finite difference operator annihilating exponential polynomials such that singular regions can be distinguished from smooth regions more efficiently. Moreover, to construct numerical flux, we employ an interpolation method based on exponential polynomials which yields improved results around steep gradients. The proposed scheme retains the optimal order of accuracy (i.e., three) in smooth areas, even near the critical points. To illustrate the ability of the new scheme, some numerical results are provided along with comparisons with other WENO schemes. [ABSTRACT FROM AUTHOR]

We obtain a class of exact solutions for a Bessel-type differential equation, which is a five-parameter linear ordinary differential equation of the second order with irregular (essential) singularity at the origin. The solutions are obtained using the Tridiagonal Representation Approach (TRA) as bounded series of square-integrable functions written in terms of a quasirational Laguerre function (Laguerre polynomials in the reciprocal argument). The expansion coefficients of the series are orthogonal polynomials in the equation parameters space. [ABSTRACT FROM AUTHOR]

For the numerical solution of Volterra integral equations of the second kind whose integrands have diagonal and endpoint algebraic singularities, we investigate in this paper, a fast modified collocation method based on the zeros of the Jacobi polynomials in appropriate weighted spaces. The iterated version of the standard collocation method is also defined. The proposed methods are shown to converge faster than the collocation scheme, and the Sloan iteration can be applied to the modified collocation solution to make it even more accurate. This research seems to be the first to explore superconvergent approaches for solving integral equations of this type. Some numerical tests are presented to show the effectiveness of the suggested methods. [ABSTRACT FROM AUTHOR]

We study the infrared scale expansion of single field cosmological models using the Hamilton-Jacobi formalism, showing that its specialization at unit scale parameter recovers the slow roll expansion. In particular, we show that the latter coincides with a Laurent expansion of the Hamilton-Jacobi function in powers of the Planck mass, whose terms are controlled by certain recursively-defined polynomials. This allows us to give an explicit recursion procedure for constructing all higher order terms of the slow roll expansion. We also discuss the corresponding effective potential and the action of the universal similarity group. [ABSTRACT FROM AUTHOR]

We use a polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 to prove six identities involving the q -binomial coefficients. These identities are then extended to the new infinite hierarchies of q -series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some new polynomials associated to linear codes over F q. In particular, we introduce the notion of split complete Jacobi polynomials attached to multiple sets of coordinate places of a linear code over F q , and give the MacWilliams type identity for it. We also give the notion of generalized q -colored t -designs. As an application of the generalized q -colored t -designs, we derive a formula that obtains the split complete Jacobi polynomials of a linear code over F q. Moreover, we define the concept of colored packing (resp. covering) designs. Finally, we give some coding theoretical applications of the colored designs for Type III and Type IV codes. [ABSTRACT FROM AUTHOR]

This paper proposes a method based on two-dimensional Euler polynomials combined with Gauss-Jacobi quadrature formula. The method is used to solve two-dimensional Volterra integral equations with fractional order weakly singular kernels. Firstly, we prove the existence and uniqueness of the original equation by Gronwall inequality and mathematical induction method. Secondly, we use two-dimensional Euler polynomials to approximate the unknown function of the original equation, and the Gauss-Jacobi quadrature formula is used to approximate the integrals in the original equation. Thirdly, we prove the existence and uniqueness of the solution of approximate equation, and the error analysis of the proposed method is given. Finally, some numerical examples illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

This paper contributes to develop a highly accurate numerical method for solving two-dimensional mass transfer equations during convective air drying of apple slices. The numerical scheme is based on the interpolating the solution of the mentioned equations over the roots of the orthogonal Jacobi polynomials (i.e., the Jacobi-Gauss-Lobatto points) in a nodal form. Moreover, for speeding up the procedure of numerical technique, operational matrices of differentiation are applied to discretize the derivatives of both spatial and temporal variables. After implementing the proposed technique, two-dimensional mass transfer equations would be transferred into the associated systems of linear algebraic equations which can be solved by appropriate iterative solvers such as robust Krylov subspace iterative methods. Some constructed artificial examples are provided to show the efficiency and applicability of the Jacobi pseudo-spectral method for solving two-dimensional mass transfer equations. Finally, a real example is considered and numerical results are validated by the experimental data which confirm the accuracy of the presented numerical approach. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution for a class of nonlinear weakly singular Volterra integral equation of the first kind. The existence and uniqueness issue of this nonlinear Volterra integral equations is studied completely. An hp -version collocation method in conjunction with Jacobi polynomials is introduced so as an appropriate numerical solution to be found. We analyze it properly and find an error estimation in L 2 -norm. The efficiency of the method is illustrated by some numerical experiments. [ABSTRACT FROM AUTHOR]

We employ the fact that certain divided differences can be written as weighted means of B-splines and hence are positive. These divided differences include the complete homogeneous symmetric polynomials of even degree 2 p , the positivity of which is a classical result by D.B. Hunter. We extend Hunter's result to complete homogeneous symmetric polynomials of fractional degree, which are defined via Jacobi's bialternant formula. We show in particular that these polynomials have positive real part for real degrees μ with | μ − 2 p | < 1 / 2. We also prove results on linear combinations of the classical complete homogeneous symmetric polynomials and on linear combinations of products of such polynomials. [ABSTRACT FROM AUTHOR]

In this article a two-sided variable coefficient fractional diffusion equation (FDE) is investigated, where the variable coefficient occurs outside of the fractional integral operator. Under a suitable transformation the variable coefficient equation is transformed to a constant coefficient equation. Then, using the spectral decomposition approach with Jacobi polynomials, we proved the wellposedness of the model and the regularity of its solution. A spectral approximation scheme is proposed and the accuracy of its approximation studied. Three numerical experiments are presented to demonstrate the derived error estimates. [ABSTRACT FROM AUTHOR]

A complete characterization is provided of Hankel matrices commuting with Jacobi matrices which correspond to hypergeometric orthogonal polynomials from the Askey scheme. It follows, as the main result of the paper, that the generalized Hilbert matrix is the only prominent infinite-rank Hankel matrix which, if regarded as an operator on ℓ 2 (N 0) , is diagonalizable by application of the commutator method with Jacobi matrices from the mentioned families. [ABSTRACT FROM AUTHOR]

The paper deals with the numerical approximation of integrals of the type I (f , y) : = ∫ − 1 1 f (x) k (x , y) d x , y ∈ S ⊂ R where f is a smooth function and the kernel k (x , y) involves some kinds of "pathologies" (for instance, weak singularities, high oscillations and/or endpoint algebraic singularities). We introduce and study a product integration rule obtained by interpolating f by an extended Lagrange polynomial based on Jacobi zeros. We prove that the rule is stable and convergent with the order of the best polynomial approximation of f in suitable function spaces. Moreover, we derive a general recurrence relation for the new modified moments appearing in the coefficients of the rule, just using the knowledge of the usual modified moments. The new quadrature sequence, suitable combined with the ordinary product rule, allows to obtain a "mixed" quadrature scheme, significantly reducing the number of involved samples of f. Numerical examples are provided in order to support the theoretical results and to show the efficiency of the procedure. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalized ordered Bell polynomial P n (q) defined by the following exponential generating function ∑ n ≥ 0 P n (q) n ! t n = e γ t (β β + β ′ q − β ′ q e t β ) 1 + γ ′ β ′ . Using the method of exponential Riordan arrays and orthogonal polynomials, we give the Jacobi continued fraction expansion of ∑ n ≥ 0 P n (q) t n. Then we get the q -Stieltjes moment property, strong q -log-convexity, 3- q -log-convexity and Hankel determinants of { P n (q) } n ≥ 0. [ABSTRACT FROM AUTHOR]

In this paper, we propose a structure-preserving one-sided cyclic Jacobi method for computing the singular value decomposition of a quaternion matrix. In our method, the columns of the quaternion matrix are orthogonalized in pairs by using a sequence of orthogonal JRS-symplectic Jacobi matrices to its real counterpart. We establish the quadratic convergence of our method specially. We also give some numerical examples to illustrate the effectiveness of the proposed method. • The singular value decomposition of a quaternion matrix is considered. • A structure-preserving one-sided cyclic Jacobi method is proposed. • Our method involves a sequence of orthogonal JRS-symplectic transformations. • The complete singular values and associated singular vectors are computed. • The quadratic convergence of our method is established specially. [ABSTRACT FROM AUTHOR]

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered new families of orthogonal polynomials on the real line. The properties of the physical system (e.g. energy spectrum, phase shift, density of states, etc.) are obtained from the properties of these polynomials. One of these new families is composed of four-parameter polynomials describing a discrete spectrum of the corresponding quantum mechanical system. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their three-term recursion relations and initial values. However, their other properties like weight functions, generating functions, orthogonality, Rodrigues-type formula, etc., are yet to be derived analytically. Obtaining these properties is an open problem in orthogonal polynomials. [ABSTRACT FROM AUTHOR]