Search Results (34)

This paper shows that certain ${}_{3}F_4$ hypergeometric functions can be expanded in sums of pair products of ${}_{1}F_2$ functions. In special cases, the ${}_{3}F_4$ hypergeometric functions reduce to ${}_{2}F_3$ functions. Further special cases allow one to reduce the ${}_{2}F_3$ functions to ${}_{1}F_2$ functions, and the sums to products of ${}_{0}F_1$ (Bessel) and ${}_{1}F_2$ functions. The class of hypergeometric functions with summation theorems are thereby expanded beyond those expressible as pair-products of ${}_{2}F_1$ functions, ${}_{3}F_2$ functions, and generalized Whittaker functions, into the realm of ${}_{p}F_q$ functions where $p<q$ for both the summand and terms in the series. Full article

Summation of infinite series has played a significant role in a broad range of problems in the physical sciences and is of interest in a purely mathematical context. In a prior paper, we found that the Fourier–Legendre series of a Bessel function of the first kind $J_N\left(kx\right)$ and modified Bessel functions of the first kind $I_N\left(kx\right)$ lead to an infinite set of series involving ${}_{1}F_2$ hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes $1/\left(2^i{3}^j{5}^k{7}^l{11}^m{13}^n{17}^o{19}^p\right)$ multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving ${}_{1}F_2$ hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving ${}_{1}F_2$ hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions. That the parameters in these new cases can be varied at will significantly expands the landscape of applications for which they could provide a solution. Full article

This paper offers a thorough examination of a unified class of Humbert’s polynomials in two variables, extending beyond well-known polynomial families such as Gegenbauer, Humbert, Legendre, Chebyshev, Pincherle, Horadam, Kinnsy, Horadam–Pethe, Djordjević, Gould, Milovanović, Djordjević, Pathan, and Khan polynomials. This study’s motivation stems from exploring polynomials that lack traditional nomenclature. This work presents various expansions for Humbert–Hermite polynomials, including those involving Hermite–Gegenbauer (or ultraspherical) polynomials and Hermite–Chebyshev polynomials. The proofs enhanced our understanding of the properties and interrelationships within this extended class of polynomials, offering valuable insights into their mathematical structure. This research consolidates existing knowledge while expanding the scope of Humbert’s polynomials, laying the groundwork for further investigation and applications in diverse mathematical fields. Full article

Because of its flexibility and multiple meanings, the concept of information entropy in its continuous or discrete form has proven to be very relevant in numerous scientific branches. For example, it is used as a measure of disorder in thermodynamics, as a measure of uncertainty in statistical mechanics as well as in classical and quantum information science, as a measure of diversity in ecological structures and as a criterion for the classification of races and species in population dynamics. Orthogonal polynomials are a useful tool in solving and interpreting differential equations. Lately, this subject has been intensively studied in many areas. For example, in statistics, by using orthogonal polynomials to fit the desired model to the data, we are able to eliminate collinearity and to seek the same information as simple polynomials. In this paper, we consider the Tsallis, Kaniadakis and Varma entropies of Chebyshev polynomials of the first kind and obtain asymptotic expansions. In the particular case of quadratic entropies, there are given concrete computations. Full article

The mechanical response of a quasicrystal thin film is strongly affected by an adhesive layer along the interface. In this paper, a theoretical model is proposed to study a thin two-dimensional hexagonal quasicrystal film attached to a half-plane substrate with an adhesive layer, which undergoes a thermally induced deformation. A perfect non-slipping contact condition is assumed at the interface by adopting the membrane assumption. An analytical solution to the problem is obtained by constructing governing integral–differential equations for both single and multiple films in terms of interfacial shear stresses that are reduced to a linear algebraic system via the series expansion of Chebyshev polynomials. The solution is compared to that without adhesive layers, and the effects of the aspect ratio of films, material mismatch, and the adhesive layer, as well as the interaction between films, are discussed in detail. It is found that the adhesive layer can soften the localized stress concentration. This study is instructive to the accurate safety assessment and functional design of a quasicrystal film system. Full article

This study explores efficient methods for computing eigenvalues and function values associated with Chebyshev-type prolate spheroidal wave functions (CPSWFs). Applying the expansion of the factor $e^{\mathrm{i}cxy}$ and the inherent properties of Chebyshev polynomials, we present an exact and stable numerical approximation for the exact eigenvalues of the integral operator to CPSWFs. Additionally, we illustrate the efficiency of employing fast Fourier transform and barycentric interpolation techniques for computing CPSWF values and related quantities, which are essential for various numerical applications based on these functions. The analysis is supported by numerical examples, providing validation for the accuracy and reliability of our proposed approach. Full article

Analyzing, developing and exploiting results obtained by Laplace in 1785 on the Fourier-series expansion of the function ${(1-2\alpha cos\theta +{\alpha }^2)}^{-s}$, we obtain a family of new expansions and generating functions for the Chebyshev polynomials. A relation between the generating functions of the Chebyshev polynomials $T_n$ and the Gegenbauer polynomials $C_n^{\left(2\right)}$ is given. Full article

The total focusing method (TFM) has been increasingly applied to weld inspection given its high image quality and defect sensitivity. Oblique incidence is widely used to steer the beam effectively, considering the defect orientation and structural complexity of welded structures. However, the conventional TFM based on the delay-and-sum (DAS) principle is time-consuming, especially for oblique incidence. In this paper, a fast full-matrix imaging algorithm in the Fourier domain is proposed to accelerate the TFM under the condition of oblique incidence. The algorithm adopts the Chebyshev polynomials of the second kind to directly expand the Fourier extrapolator with lateral sound velocity variation. The direct expansion maintains image accuracy and resolution in wide-angle situations, covering both small and large angles, making it highly suitable for weld inspection. Simulations prove that the third-order Chebyshev expansion is required to achieve image accuracy equivalent to the TFM with wide-angle incidence. Experiments verify the algorithm’s performance for weld flaws using the proposed method with the transverse wave and the full-skip mode. The depth deviation is within 0.53 mm, and the sizing error is below 15%. The imaging efficiency is improved by a factor of up to 8 compared to conventional TFM. We conclude that the proposed method is applicable to high-speed weld inspection with various oblique incidence angles. Full article

Structures with inhomogeneous materials, non-uniform cross-sections, non-uniform supports, and subject to non-uniform loads are increasingly common in aerospace applications. This paper presents a simple and unified numerical dynamics model for all beams with arbitrarily axially varying cross-sections, materials, foundations, loads, and general boundary conditions. These spatially varying properties are all approximated by high-order Chebyshev expansions, and discretized by Gauss–Lobatto sampling. The discrete governing equation of non-uniform axially functionally graded beams resting on variable Winkler–Pasternak foundations subjected to non-uniformly distributed loads is derived based on the Euler–Bernoulli beam theory. A projection matrix method is employed to simultaneously assemble spectral elements and impose general boundary conditions. Numerical experiments are performed to validate the proposed method, considering different inhomogeneous materials, boundary conditions, foundations, cross-sections, and loads. The results are compared with those reported in the literature and obtained by the finite element method, and excellent agreement is observed. The convergence, accuracy, and efficiency of the proposed method are demonstrated. Full article

The linear stability of a convective flow in a vertical fluid layer caused by nonlinear heat sources in the presence of cross-flow through the walls of the channel is investigated in this paper. This study is relevant to the analysis of factors that affect the effectiveness of biomass thermal conversion. The nonlinear problem for the base flow temperature is investigated in detail using the Krasnosel’skiĭ–Guo cone expansion/contraction theorem. It is shown that a different number of solutions can exist depending on the values of the parameters. Estimates for the norm of the solutions are obtained. The linear stability problem is solved numerically by a collocation method based on Chebyshev polynomials. It is shown that the increase in the cross-flow intensity stabilizes the flow, but there is also a small region of the radial Reynolds numbers where the flow is destabilized. Full article

In this paper, we present a novel and unified model for studying the vibration of cylindrical shells based on the three-dimensional (3D) elastic theory and the Carrera Unified Formulation. Our approach represents a significant advancement in the field, as it enables us to accurately predict the vibrational behavior of cylindrical shells under arbitrary boundary conditions. To accomplish this, we expand the axial, circumferential, and radial displacements of the shell using Chebyshev polynomials and Taylor series, thereby reducing the dimensionality of the expansion and ensuring the precision and rigor of our results. In addition, we introduce three groups of artificial boundary surface springs to simulate the general end boundary conditions of the cylindrical shell and coupling springs to strongly couple the two surfaces of the cylindrical shell φ = 0 and φ = 2π to ensure continuity of displacements on these faces. Using the energy function of the entire cylindrical shell model, we obtain the characteristic equation of the system by finding the partial derivatives of the unknown coefficients of displacement in the energy function. By solving this equation, we can directly obtain the vibration characteristics of the cylindrical shell. We demonstrate the convergence, accuracy, and reliability of our approach by comparing our computational results with existing results in the literature and finite element results. Finally, we present simulation results of the frequency features of cylindrical shells with various geometrical and boundary parameters in the form of tables and figures. Overall, we believe that our novel approach has the potential to greatly enhance our understanding of cylindrical shells and pave the way for further advancements in the field of structural engineering. Our comprehensive model and simulation results contribute to the ongoing efforts to develop efficient and reliable techniques for analyzing the vibrational behavior of cylindrical shells. Full article

This article completes our studies on the formal construction of asymptotic approximations for statistics based on a random number of observations. Second order Chebyshev–Edgeworth expansions of asymptotically normally or chi-squared distributed statistics from samples with negative binomial or Pareto-like distributed random sample sizes are obtained. The results can have applications for a wide spectrum of asymptotically normally or chi-square distributed statistics. Random, non-random, and mixed scaling factors for each of the studied statistics produce three different limit distributions. In addition to the expected normal or chi-squared distributions, Student’s t-, Laplace, Fisher, gamma, and weighted sums of generalized gamma distributions also occur. Full article

The viscoelastic relaxation spectrum provides deep insights into the complex behavior of polymers. The spectrum is not directly measurable and must be recovered from oscillatory shear or relaxation stress data. The paper deals with the problem of recovery of the relaxation spectrum of linear viscoelastic materials from discrete-time noise-corrupted measurements of relaxation modulus obtained in the stress relaxation test. A class of robust algorithms of approximation of the continuous spectrum of relaxation frequencies by finite series of orthonormal functions is proposed. A quadratic identification index, which refers to the measured relaxation modulus, is adopted. Since the problem of relaxation spectrum identification is an ill-posed inverse problem, Tikhonov regularization combined with generalized cross-validation is used to guarantee the stability of the scheme. It is proved that the accuracy of the spectrum approximation depends both on measurement noises and the regularization parameter and on the proper selection of the basis functions. The series expansions using the Laguerre, Legendre, Hermite and Chebyshev functions were studied in this paper as examples. The numerical realization of the scheme by the singular value decomposition technique is discussed and the resulting computer algorithm is outlined. Numerical calculations on model data and relaxation spectrum of polydisperse polymer are presented. Analytical analysis and numerical studies proved that by choosing an appropriate model through selection of orthonormal basis functions from the proposed class of models and using a developed algorithm of least-square regularized identification, it is possible to determine the relaxation spectrum model for a wide class of viscoelastic materials. The model is smoothed and robust on measurement noises; small model approximation errors are obtained. The identification scheme can be easily implemented in available computing environments. Full article

We present an algorithm for numerical solution of the equations of magnetohydrodynamics describing the convective dynamo in a plane horizontal layer rotating about an arbitrary axis under geophysically sound boundary conditions. While in many respects we pursue the general approach that was followed by other authors, our main focus is on the accuracy of simulations, especially in the small scales. We employ the Galerkin method. We use products of linear combinations (each involving two to five terms) of Chebyshev polynomials in the vertical Cartesian space variable and Fourier harmonics in the horizontal variables for space discretisation of the toroidal and poloidal potentials of the flow (satisfying the no-slip conditions on the horizontal boundaries) and magnetic field (for which the boundary conditions mimick the presence of a dielectric over the fluid layer and an electrically conducting bottom boundary), and of the deviation of temperature from the steady-state linear profile. For the chosen coefficients in the linear combinations, the products satisfy the respective boundary conditions and constitute non-orthogonal bases in the weighted Lebesgue space. Determining coefficients in the expansion of a given function in such a basis (for instance, for computing the time derivatives of these coefficients) requires solving linear systems of equations for band matrices. Several algorithms for determining the coefficients, which are exploiting algebraically precise relations, have been developed, and their efficiency and accuracy have been numerically investigated for exponentially decaying solutions (encountered when simulating convective regimes which are spatially resolved sufficiently well). For the boundary conditions satisfied by the toroidal component of the flow, our algorithm outperforms the shuttle method, but the latter proves superior when solving the problem for the conditions characterising the poloidal component. While the conditions for the magnetic field on the horizontal boundaries are quite specific, our algorithms for the no-slip boundary conditions are general-purpose and can be applied for treating other boundary-value problems in which the zero value must be admitted on the boundary. Full article

This paper is devoted to proposing numerical algorithms based on the use of the tau and collocation procedures, two widely used spectral approaches for the numerical treatment of the initial high-order linear and non-linear equations of the singular type, especially those of the high-order Emden–Fowler type. The class of modified Chebyshev polynomials of the third-kind is constructed. This class of polynomials generalizes the class of the third-kind Chebyshev polynomials. A new formula that expresses the first-order derivative of the modified Chebyshev polynomials in terms of their original modified polynomials is established. The establishment of this essential formula is based on reducing a certain terminating hypergeometric function of the type ${}_5{F}_4\left(1\right)$. The development of our suggested numerical algorithms begins with the extraction of a new operational derivative matrix from this derivative formula. Expansion’s convergence study is performed in detail. Some illustrative examples of linear and non-linear Emden–Flower-type equations of different orders are displayed. Our proposed algorithms are compared with some other methods in the literature. This confirms the accuracy and high efficiency of our presented algorithms. Full article