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With hyperspherical coordinates, the Schrödinger equation for some quantum mechanical many-body systems such as electrons in atoms like He and charged excitons in quantum wells can be solved in a similar way with the wave functions being expanded into orthonormal complete basis sets of the hyperspherical harmonics of hyperangles and generalized Laguerre polynomials of the hyperradius. Numerical results obtained explicitly by solving a simple secular equation show good agreement with those obtained through other computationally intensive methods. The eigenenergies and particle correlation in the low-lying states are investigated. Alternatively, the solutions of multi-electron systems can be obtained by using a linear combination of one-dimensional wave functions. Numerical results of small systems like He and H2 show that the one-dimensional bases along the x, y, z axes are good choices which facilitate easy numerical integration. The method developed is an effective numerical approach to the many-body problem and could be extended to larger atomic and molecular systems. [ABSTRACT FROM AUTHOR]

In this paper, certain properties of newly defined generalized hypergeometric polynomials in two variables{Rn (β ;γ ; x, y)} have been derived, namely, recurrence relations of ascending and descending type, ordinarydifferential equation and linear generating function, which are essential to derive generating relations of various types from the group-theoretic method point of view. Furthermore, Laguerre polynomials of single and two variables, Meixner, Gottlieb and Krawtchouk polynomials are deduced as special cases, which arises in mathematical physics and communication engineering. [ABSTRACT FROM AUTHOR]

A modified collocation method is proposed for the numerical solution of quantum mechanical eigenvalue problems for the Schrödinger equation. An increase in the accuracy of the numerical solution is achieved by a special irregular arrangement of collocation nodes. This leads to an improvement in the quality of the numerical model. Various systems of basic functions are used. The investigated numerical method allows one to obtain an approximate solution of quantum mechanical problems in an analytical form, which significantly expands the capabilities of the method. As an example, the Schrödinger equation is numerically solved to determine the eigenvalues of the electron energy in the hydrogen atom. For a hydrogen atom, the energy values at n=1, 2, and 3 exactly correspond to the theoretical values. The wave functions corresponding to the energy levels, obtained numerically, turned out to be in full agreement with the calculated theoretical formulas - Laguerre polynomials. Calculated the value of the electron wave function in the hydrogen atom when n=3, using the minimum number of collocations nodes, differs from the theoretically obtained value by 10−3%. [ABSTRACT FROM AUTHOR]

This paper describes about the Laguerre polynomials of eigenvalue degree, which we derive from solving from an eigenfunction of a type of Sturm-Liouville eigenvalue problem. Frobenius method had been applied to achieve this purpose. Hence, using this Laguerre polynomials of eigenvalue degree, we propose a numerical scheme to solve multi-term fractional differential equations in Caputo sense. Here, we use the Laguerre polynomial of eigenvalue degree to transform the fractional differential equations into a system of algebraic equations. By solving the system of algebraic equations, we able to solve the multi-term fractional differential equation easily and efficiently. [ABSTRACT FROM AUTHOR]

In this study, a matrix method which depends on Laguerre polynomials and with collocation points is developed for solving the approximate solutions of systems of high-order delay differential equations involving variable coefficients and variable delays. These kinds of systems characterized by the present functional delays and which explain many different phenomena and particularly, arise in studies based on biology, physics, chemistry, electrodynmics, and economy and in industrial applications. The proposed method reduces the solution of the mentioned delay system subject to the initial conditions to the solution of a matrix equation with the unknown Laguerre coefficients. Moreover, the approximate solution is obtained in terms of Laguerre polynomials. Besides, some examples along with different error techniques are performed for the illustration of the method’s applicability; the obtained findings are scrutinized and interpreted. [ABSTRACT FROM AUTHOR]

A generalized spectral-analytical method (GSAM) for analyzing data of various physical nature has been developed. It is based on the generalized Fourier representation of the original experimental and model digital arrays. In this report, it is used in the implementation of a resonant-impedance method for diagnosing and studying the characteristics of mechanical systems (in particular, linear one-dimensional acoustic systems with local anomalies). In the framework of this approach, the anomaly (damage) of the system is recognized, as well as the evaluation of its dynamic parameters including its localization. The proposed computational procedure involves carrying out a complete processing of the object under study in the space of the Fourier coefficients. Here we use the decomposition of the signal into generalized Laguerre polynomials. The main ideas of the method and the results of its application to the model and experimental data of well logging are considered. The use of expansions in generalized Laguerre polynomials allows one to obtain optimal representations of the original dataю The optimal approximation is explained by the nature of the signal behavior and the ability of the method to adapt to the current part of the signal, as well as by the character of the attenuation parameters in the system. The advantages of the proposed approach can be attributed to a sufficiently high noise immunity, which is the consequence of the integrating operations of the analyzed signal in the process of its decomposition within the chosen basis. The method allows remote diagnostics of the geophysical system, placing a source of an external impact and a receiver on the earth surface at the open end of the well. The results obtained can also be used as initial and auxiliary data in more contact methods such as electromagnetic and radioactive logging. [ABSTRACT FROM AUTHOR]

In this article some properties of the generalized Laguerre functions (GLF) will be presented. The application of the generalized Laguerre functions for modeling of the dead time in the time domain will be shown. The speed of convergence for the approximating series with the generalized Laguerre functions will be compared with the simple Laguerre functions series. [ABSTRACT FROM AUTHOR]

In this article the question of the optimal choice of the generalization parameter ff and the time-scale parameter p for truncated series of generalized Laguerre functions (GLF) will be studied. Short introduction of the GLF identification method will be given. The impact of the optimal choice of parameters ff and p for approximating output signals of LTI dynamical systems will be presented on the examples. [ABSTRACT FROM AUTHOR]

The complete bound state structure of energy eigenvalues and their associated eigenfunctions of quantum mechanical Morse oscillator are investigated analytically in the setting of the commutative ring of bicomplex numbers. The expressions for the eigenstates are derived in explicit closed forms using the Nikiforov - Uvarov (NU) method and amongst them the excited states are found in terms of generalized Laguerre orthogonal polynomials. The bound states in this connection are found finite in number. [ABSTRACT FROM AUTHOR]

In this research, we propose a new technique for solving fuzzy fractional differential equations of order 0 < α < 1. In this regard, fractional Laguerre polynomials (FLP) are employed to approximate the solution, then, the corresponding operational matrix is derived. Next, the fuzzy-like residual function of the original problem is obtained and finally the unknown fuzzy coefficients are achieved to get the approximate solution under uncertainty. The algorithm is validated by solving a test problem. [ABSTRACT FROM AUTHOR]

An approximate but closed-form expression for a Poisson-like steady state wealth distribution in a kinetic model of gambling was formulated from a finite number of its moments, which were generated from a βa,b(x) exchange distribution. The obtained steady-state wealth distributions have tails which are qualitatively similar to those observed in actual wealth distributions. [ABSTRACT FROM AUTHOR]

The short introduction of existing methods based on the Least squares estimation with state variable filters for the identification of the continuous time models from sampled data will be given. The identification method with the use of the Simple and Generalized Laguerre function will be proposed and compared with the traditional Least squares based approach on the examples of continuous dynamical systems. [ABSTRACT FROM AUTHOR]

Some interesting identities of Sheffer polynomials was given by Roman [11], [12]. In this paper, we study a q-analogue of Laguerre polynomials which are also q-Sheffer polynomials. Furthermore, we give some new properties and formulas of these q-Laguerre polynomials of higher order by using the theory of the umbral algebra and umbral calculus. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of some properties of the Laguerre transform.We define new properties of the Laguerre transform in a weighted L2-space. Moreover, we present some results concerning the action of this integral transform over some class of polynomials. [ABSTRACT FROM AUTHOR]

In this work, it is considered the system of Wilczynski type, and from this system it is defined a range of systems having the solutions in the form of confluent hypergeometric functions and two variables Laguerre polynomials. It is shown that two variables confluent hypergeometric functions being the main tool of investigation of Laguerre polynomials and two variables Whittaker functions. This aspects still remaining little-investigated. [ABSTRACT FROM AUTHOR]

This paper considers the problem of model order reduction by approximate balanced truncation with Laguerre polynomials approximation of the system cross gramian. The cross gramian contains information both for the reachability of the system as well as for its observability. The main property of the cross gramian for a square symmetric stable linear system is that its square is equal to the product of the reachability and observability gramians and therefore, the absolute values of its eigenvalues are equal to the Hankel singular values. This is the reason to use the cross gramian for computing balancing transformations for model reduction. Laguerre polynomial series representations are used to approximate the cross gramian of the system at infinity. The orthogonal polynomials of Laguerre possess good convergence properties and allow to reduce the computational complexity of the model reduction problem. Numerical experiments are performed confirming the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A numerical method solving moment equations with a large number of moments in the gas kinetic theory is presented. The distribution function is expanded in series with the product of Laguerre polynomials and spherical harmonics as basis functions, and a special moment closure is applied to achieve global hyperbolicity. The linear collision terms, including the BGK model, the Shakhov model and the linearized hard-sphere model are considered. Numerical results are validated by comparison with the DSMC results, and the differences between various collision models are exhibited. [ABSTRACT FROM AUTHOR]

This paper considers the problem of model order reduction by balanced truncation with Laguerre polynomials approximation of system gramians. The proposed method combines the system properties of balanced truncation with the computational effectiveness of the snapshots approach from proper orthogonal decomposition. Laguerre polynomial series representations are used to approximate the reachability and observability gramians at infinity of the system. The orthogonal polynomials of Laguerre possess good rate of convergence, which allows further to reduce the size of the computational problem. The approximation capability of the problem is measured by the Hankel singular values, which are used to determine the truncated states for the reduced order model. Several experiments are performed confirming the good approximation properties of the proposed method. [ABSTRACT FROM AUTHOR]

The Schrödinger equation with the interaction of inversely quadratic effective and Mie-type potential has been solved for any angular momentum quantum number l using the Nikiforov-Uvarov method. The bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values obtained. [ABSTRACT FROM AUTHOR]

The classical mathematical elementary motion detector (EMD) model stimulated with sinusoidal and pulsatile input signals is treated analytically. Drifting sinusoidal gratings are often used in insect vision research, enabling direct comparison with biological data. When displayed on a cathode ray tube monitor, the sinusoidal grating is modulated by the refresh rate of the monitor. Due to the resulting pulsatile nature of the visual stimuli and the corresponding biological response, a Laguerre domain identification method for estimating the dynamics of a single EMD appears to be suitable. A pool of spatially distributed EMDs is considered as the model for the measured neural output. The weights of the contributing EMDs are evaluated by a sparse optimization method to fit the experimental data. [ABSTRACT FROM AUTHOR]

We present new approximation results related to generalized Laguerre functions which extend error estimates derived earlier in [10, 7, 11]. The analysis is carried out directly in weighted Bessel potential spaces. This approach helps to avoid certain difficulties related to interpolation spaces and leads to sharper error estimates. [ABSTRACT FROM AUTHOR]

In this paper we present an efficient way to both compute and extract salient information from trace transform signatures to perform object identification tasks. We also present a feature selection analysis of the classical trace-transform functionals, which reveals that most of them retrieve redundant information causing misleading similarity measurements. In order to overcome this problem, we propose a set of functionals based on Laguerre polynomials that return orthonormal signatures between these functionals. In this way, each signature provides salient and non-correlated information that contributes to the description of an image object. The proposed functionals were tested considering a vehicle identification problem, outperforming the classical trace transform functionals in terms of computational complexity and identification rate. [ABSTRACT FROM PUBLISHER]

This paper uses a new set of feature vectors that is based on modulation spectrum of cepstral coefficients by means of Laguerre regression method. The performance of the proposed method is investigated by a gender classification of a noisy speech. Compared with other regression methods, our proposed feature set demonstrates high performance in the sense of gender classification. Low classification errors obtained in different noisy scenarios proves the superiority of the new feature vectors for the classification task. [ABSTRACT FROM PUBLISHER]

The coupling of reaction and separation in reactive distillation has lead to high nonlinearity in the reactive distillation. Model based predictive control schemes are preferred for the control of nonlinear process. Orthonormal basis filter (OBF) models are more flexible and comprehensive, to capture the dynamics of a process with few numbers of terms. Among the various types of orthonormal basis filters, laguerre filters are more appropriate for well damped processes with incorporation of a single real pole. These filters cannot be used when the system has complex poles. Generalized Orthonormal Basis Filter (GOBF) models can overcome the drawback of laguerre networks by allowing non-identical poles and complex poles in the construction of the networks. The response of GOBF model based predictive controller for set point and load changes has been studied and a satisfactory performance is ensured with the controller. [ABSTRACT FROM AUTHOR]

For each τ0>0 we denote by S(τ0) = {z∈C:|Imz|<τ0} the strip parallel to the real axes. Further by H(τ0) and Ĥ(τ0) we denote the Frèshet spaces of all analytic functions, defined in S(τ0) and having representation there in series of Hermite polynomials, respectively in series of Hermite functions. Also, by Δ(λ0), λ0>0 we denote the interior of parabola p(λ0rpar; with focus at the origin and vertex at the point -λ02, and by L(λ0)-the Frèshet space of all analytic functions, defined in Δ(λ0) and having representations in series of Laguerre polynomials {Ln(α)(z)}n = 0∞, α∈C (we mean here only the values of parameter α, for which the sets of functions, having representation in series of Laguerre polynomials with parameter α coincide). We prove that the spaces H(τ0) and Ĥ(τ0) are isomorphic, also that the space L(λ0) is isomorphic to the closed subspace He(λ0)⊂H(τ0) of all even functions. [ABSTRACT FROM AUTHOR]

The generalized Laguerre-generalized Hermite mixed spectral method is developed. Various orthogonal projections are investigated. Some mixed approximation results are established. As an example of their important applications, a mixed spectral scheme is proposed for a two-dimensional nonlinear problem. Its convergence is proved. Numerical results demonstrate the high accuracy of this new spectral method. [ABSTRACT FROM AUTHOR]

We generalize the coupled-mode theory, used to calculate eigenmodes of spatially-broadband optical parametric amplifiers, for Hermite- and Laguerre-Gaussian pumps of any order. [ABSTRACT FROM AUTHOR]

By using the Nikiforov-Uvarov (NU) method, the Schrödinger equation has been solved for the interaction of inversely quadratic Hellmann (IQHP) and inversely quadratic potential (IQP) for any angular momentum quantum number, l. The energy eigenvalues and their corresponding eigenfunctions have been obtained in terms of Laguerre polynomials. Special cases of the sum of these potentials have been considered and their energy eigenvalues also obtained. [ABSTRACT FROM AUTHOR]

A marching-on-in-degree solver of time domain finite element-boundary integral method is proposed for the analysis of transient electromagnetic scattering from complex materials and fine detail structures. Unlike the marching-on-in-time solver of time domain finite element-boundary integral method, the proposed algorithm uses weighted Laguerre polynomials as temporal basis functions and testing functions, which leads to a recursive matrix equation solved degree by degree. Due to the property of weighted Laguerre polynomials, the proposed method does not involve late-time instability. Numerical results demonstrate that the proposed method is both stable and accurate. [ABSTRACT FROM PUBLISHER]

A new orthogonal system of generalized Laguerre polynomials is introduced. Some approximation results are established. As an example of applications, a mixed spherical harmonic-generalized Laguerre spectral scheme for the nonlinear partial differential equation in three dimensional exterior domain is proposed. Its convergence is proved. [ABSTRACT FROM AUTHOR]

The DGLAP evolution equations for non-singlet sector of parton density is solved in x-space based on Laguerre polynomial expansion. High numerical accuracy is achieved by expanding over a set of approximately 30 polynomials. The result of evolved parton densities to high energy scales are in good agreement with phenomenological GRV model. To improve the results we can employ a constituent quark model. [ABSTRACT FROM AUTHOR]