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Herein, a pure shifted Jacobi–Galerkin (SJG) method is offered for handling linear two-dimensional space-time diffusion and telegraph equations but by considering their integrated forms. The semi-analytic approximate solution is expanded in spatial and temporal variables as polynomials bases built as a linear combination of the shifted Jacobi Polynomials (JPs). Expanding the exact solution regarding these polynomials satisfies the homogeneous initial and boundary conditions. The proposed Jacobi–Galerkin (JG) procedure yields exponential convergence rates when the solution is smooth enough. A careful investigation of the convergence analysis of the suggested approximate triple series expansion examines the offered numerical algorithm. Two test illustrative examples are given to prove the high accuracy of the proposed scheme.

An exactly solvable, two-parameter implicit quantum mechanical potential is derived and characterized. With a change of variable, this potential is shown to belong to the Natanzon class, and for at least one value of the potential parameters, an explicit, rather than implicit, potential results.

We solve for the spectrum and eigenfunctions of Dirac operator on the sphere. The igenvalues are nonzero whole numbers. The eigenfunctions are two-component spinors which may be classified by representations of the SU(2) group with half-integer angular momenta. They are special linear combinations of conventional spherical spinors.

We present the results of our QCD analysis for quark distributions and structure functions for polarized deep inelastic scattering of leptons on nucleons and nuclei using the Jacobi polynomials up to NLO approximation. Finally by having proton and also neutron structure functions, we have enough motivation to extract ³He and ³H polarized structure functions.

In this work, we study an explicit calculation of the deformed Pauli–Schrödinger oscillator for nonrelativistic charged spin 1/2 particle in momentum space with Snyder–de Sitter model, where the energy eigenvalues are given in their exact forms, and the corresponding radial wave functions are expressed in associated Jacobi polynomials. We have also studied the effect of the spatial deformation parameter on the bound states of our system.

Jacobi polynomials appear to play a very important role in describing all the spin field (s = 0, 1/2, 1, 2) perturbation of the Friedmann–Robertson–Walker (FRW) spacetime. The formulation becomes very transparent when done in Newman–Penrose (NP) formalism. All the variables are separable, and the spatial eigenfunctions turn out to be Jacobian polynomials in different forms. In particular, the angular ones are expressible as spin weighted spherical harmonics which are just the spherical harmonics formed with Jacobi polynomials. The radial eigenfunctions are also Jacobi polynomials but with unconventional parameters. Various properties of these polynomials are used to describe the scalar, vector and tensor modes of the perturbation. The Green's function of the scalar perturbations and also its Lienard–Wiechert type potentials are derived, and are shown to reduce to the familiar ones in the limit to flat FRW case. Some of the components of the perturbed metric tensor h^μν have also been calculated.

By using the Nikiforov–Uvarov method, we give the approximate analytical solutions of the Dirac equation with the shifted Deng–Fan potential including the Yukawa-like tensor interaction under the spin and pseudospin symmetry conditions. After using an improved approximation scheme, we solved the resulting schrödinger-like equation analytically. Numerical results of the energy eigenvalues are also obtained, as expected, the tensor interaction removes degeneracies between spin and pseudospin doublets.

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

In this paper, the experimental and Jacobi–Ritz method (JRM) have been adopted to analyze the forced vibration analysis of uniform and stepped circular cylindrical shells with general boundary conditions. The simply supported cylindrical shell at both ends is taken as an experimental model, and the free, steady and transient vibration characteristics of structures under hammer and fixed exciter are recorded. The results show that the results of JRM are in sensible agreement with those in experiment. In addition, the results for various boundary conditions, structural parameter are also presented. On this basis, the Newmark-$\beta$ integration method is adopted to realize the time domain solutions for transient vibration response, and the frequency domain results can be obtained by using Fourier transformations from time domain results. Finally, the line spectrum vibration response results of the structure are presented under the random excitation load, and the research can supply technical support for the vibration control of cylindrical shell structure.

There are now several ways to derive an asymptotic expansion for , as n → ∞, which holds uniformly for . One of these starts with a contour integral, involves a transformation which takes this integral into a canonical form, and makes repeated use of an integration-by-parts technique. There are two advantages to this approach: (i) it provides a recursive formula for calculating the coefficients in the expansion, and (ii) it leads to an explicit expression for the error term. In this paper, we point out that the estimate for the error term given previously is not sufficient for the expansion to be regarded as genuinely uniform for θ near the origin, when one takes into account the behavior of the coefficients near θ = 0. Our purpose here is to use an alternative method to estimate the remainder. First, we show that the coefficients in the expansion are bounded for . Next, we give an estimate for the error term which is of the same order as the first neglected term.

A general scheme for tridiagonalizing differential, difference or q-difference operators using orthogonal polynomials is described. From the tridiagonal form the spectral decomposition can be described in terms of the orthogonality measure of generally different orthogonal polynomials. Three examples are worked out: (1) related to Jacobi and Wilson polynomials for a second order differential operator, (2) related to little q-Jacobi polynomials and Askey–Wilson polynomials for a bounded second order q-difference operator, (3) related to little q-Jacobi polynomials for an unbounded second order q-difference operator. In case (1) a link with the Jacobi function transform is established, for which we give a q-analogue using example (2).

Let

In this paper, an estimate of Jacobi polynomials with complex indices is proved. It improves a corresponding result of Ref. 2.

Let p > 5 be a prime. We prove congruences modulo p^3-d for sums of the general form and with d = 0, 1. We also consider the special case t = (-1)^d/16 of the former sum, where the congruences hold modulo p^5-d.

Recently, Mertens, Ono and the third author studied mock modular analogues of Eisenstein series. Their coefficients are given by small divisor functions, and have shadows given by classical Shimura theta functions. Here, we construct a class of small divisor functions ${\sigma }_{2,\chi }^{\text{sm}}$ and prove that these generate the holomorphic part of polar harmonic Maaß forms of weight $\frac{3}{2}$. To this end, we essentially compute the holomorphic projection of mixed harmonic Maaß forms in terms of Jacobi polynomials, but without assuming the structure of such forms. Instead, we impose translation invariance and suitable growth conditions on the Fourier coefficients. Specializing to a certain choice of characters, we obtain an identity between ${\sigma }_{2,𝟙}^{\text{sm}}$ and Hurwitz class numbers, and ask for more such identities. Moreover, we prove $p$-adic congruences of our small divisor functions when $p$ is an odd prime. If $\chi$ is non-trivial we rewrite the generating function of ${\sigma }_{2,\chi }^{\text{sm}}$ as a linear combination of Appell–Lerch sums and their first two normalized derivatives. Lastly, we offer a connection of our construction to meromorphic Jacobi forms of index $-1$ and false theta functions.

In this paper, the general nonlinear multi-strain Tuberculosis model is controlled using the merits of Jacobi spectral collocation method. In order to have a variety of accurate results to simulate the reality, a fractional order model of multi-strain Tuberculosis with its control is introduced, where the derivatives are adopted from Caputo’s definition. The shifted Jacobi polynomials are used to approximate the optimality system. Subsequently, Newton’s iterative method will be used to solve the resultant nonlinear algebraic equations. A comparative study of the values of the objective functional, between both the generalized Euler method and the proposed technique is presented. We can claim that the proposed technique reveals better results when compared to the generalized Euler method.

In this research work, we study the Human Immunodeficiency Virus (HIV) infection on helper T cells governed by a mathematical model consisting of a system of three first-order nonlinear differential equations. The objective of the analysis is to present an approximate mathematical solution to the model that gives the count of the numbers of uninfected and infected helper T cells and the number of free virus particles present at a given instant of time. The system of nonlinear ODEs is converted into a system of nonlinear algebraic equations using spectral collocation method with three different basis functions such as Chebyshev, Legendre and Jacobi polynomials. Some factors such as the production of helper T cells and infection of these cells play a vital role in infected and uninfected cell counts. Detailed error analysis is done to compare our results with the existing methods. It is shown that the spectral collocation method is a very reliable, efficient and robust method of solution compared to many other solution procedures available in the literature. All these results are presented in the form of tables and figures.

In this paper, we introduce a new approach to the study of finite integral formulas associated with special functions, bringing out the incomplete Aleph functions and other well-known special functions. Our method is based on the finite integral operator applied to incomplete Aleph functions. This approach makes it easier to derive integral formulas that connect the incomplete Aleph functions using well-known special functions and algebraic expressions. As a result, we can group integral formulas and special functions into different classes according to similar features. Furthermore, we can discover new integral formulas with this framework that are useful for computational applications.

The generalized spectral-analytical method is based on the use of orthogonal decompositions of signals and analytic transformations in the space of expansion coefficients. The theory of classical orthogonal bases is a generalization of the theory of Fourier series to algebraic orthogonal polynomials of continuous and discrete arguments. The choice of the optimal set of features is carried out on the basis of integral signal estimates, and the mathematical operations on the signals studied are performed in the space of the expansion coefficients. Within the framework of the generalized spectral-analytical method, the problems of adaptive analytical description of one-dimensional and multidimensional digital information arrays are effectively solved in order to reduce the redundancy of signals’ description, as well as recognition problems on the basis of pairwise comparison of objects in Fourier coefficients features space.

This chapter describes the application of this method in the following problems:

• image recognition (by using invariants when recognizing contour objects),
• bioinformatics (recognition of repeats in sequences and complex spatial configurations of biomacromolecules),
• biomedicine (analysis of the spatial and temporal organization of electrical and magnetic activity of the brain).

The alternative approach to QCD analysis for quark distributions and structure functions for polarized deep inelastic scattering is presented. We use very recently experimental data to parameterize our model for quark and gluon distributions up to NLO approximation. Our calculation is based on Jacobi polynomials expansion of the polarized structure functions. The final results are in good agreement with the other theoritical models and experimental data.