Search Results (52)

The present review discusses the solution of the Heun, confluent, biconfluent, double confluent, and triconfluent equations in terms of polynomial expansions, and applies the results to generate exactly solvable Schrödinger potentials. Although there are more general approaches to solve these differential equations in terms of the expansions of certain special functions, the importance of polynomial solutions is unquestionable, as most of the known potentials are solvable in terms of the hypergeometric and confluent hypergeometric functions; i.e., Natanzon-class potentials possess bound-state solutions in terms of classical orthogonal polynomials, to which the (confluent) hypergeometric functions can be reduced. Since some of the Heun-type equations contain the hypergeometric and/or confluent hypergeometric differential equations as special limits, the potentials generated from them may also contain Natanzon-class potentials as special cases. A power series expansion is assumed around one of the singular points of each differential equation, and recurrence relations are obtained for the expansion coefficients. With the exception of the triconfluent Heun equations, these are three-term recurrence relations, the termination of which is achieved by prescribing certain conditions. In the case of the biconfluent and double confluent Heun equations, the expansion coefficients can be obtained in the standard way, i.e., after finding the roots of an (N + 1)th-order polynomial in one of the parameters, which, in turn, follows from requiring the vanishing of an (N + 1) × (N + 1) determinant. However, in the case of the Heun and confluent Heun equations, the recurrence relation can be solved directly, and the solutions are obtained in terms of rationally extended X₁-type Jacobi and Laguerre polynomials, respectively. Examples for solvable potentials are presented for the Heun, confluent, biconfluent, and double confluent Heun equations, and alternative methods for obtaining the same potentials are also discussed. These are the schemes based on the rational extension of Bochner-type differential equations (for the Heun and confluent Heun equation) and solutions based on quasi-exact solvability (QES) and on continued fractions (for the biconfluent and double confluent equation). Possible further lines of investigations are also outlined concerning physical problems that require the solution of second-order differential equations, i.e., the Schrödinger equation with position-dependent mass and relativistic wave equations. Full article

The generation and propagation of water waves in a numerical wave flume with Ursell numbers (Ur) ranging from 0.67 to 43.81 were investigated using the wave generation theory of Goring and Raichlen and a two-dimensional numerical viscous wave flume model. The unsteady Navier–Stokes equations, along with nonlinear free surface boundary conditions and upstream boundary conditions at the wavemaker, were solved to build the numerical wave flume. The generated waves included small-amplitude, finite-amplitude, cnoidal, and solitary waves. For computational efficiency, the Jacobi elliptic function representing the surface elevation of a cnoidal wave was expressed as a Fourier series expansion. The accuracy of the generated waveforms and associated flow fields was validated through comparison with theoretical solutions. For $Ur<26.32$, small-amplitude waves generated using Goring and Raichlen’s wave generation theory matched those obtained from linear wave theory, while finite-amplitude waves matched those obtained using Madsen’s wave generation theory. For $Ur>26.32$, nonlinear wave generated using Goring and Raichlen’s theory remained permanent, whereas that generated using Madsen’s theory did not. The evolution of a cnoidal wave train with $Ur=43.81$ was examined, and it was found that, after an extended propagation period, the leading waves in the wave train evolved into a series of solitary waves, with the tallest wave positioned at the front. Full article

This study explores the application of Romanovski–Jacobi polynomials (RJPs) in spectral Galerkin methods (SGMs) for solving differential equations (DEs). It uses a suitable class of modified RJPs as basis functions that meet the homogeneous initial conditions (ICs) given. We derive spectral Galerkin schemes based on modified RJP expansions to solve three models of high-order ordinary differential equations (ODEs) and partial differential equations (PDEs) of first and second orders with ICs. We provide theoretical assurances of the treatment’s efficacy by validating its convergent and error investigations. The method achieves enhanced accuracy, spectral convergence, and computational efficiency. Numerical experiments demonstrate the robustness of this approach in addressing complex physical and engineering problems, highlighting its potential as a powerful tool to obtain accurate numerical solutions for various types of DEs. The findings are compared to those of preceding studies, verifying that our treatment is more effective and precise than that of its competitors. Full article

In this work, we present both analytical and numerical solutions to a seven-parameter confluent Heun-type differential equation. This second-order linear differential equation features three singularities: two regular singularities and one irregular singularity at infinity. First, employing the tridiagonal representation method (TRA), we derive an analytical solution expressed in terms of Jacobi polynomials. The expansion coefficients of the series are determined as solutions to a three-term recurrence relation, which is satisfied by a modified form of continuous Hahn orthogonal polynomials. Second, we develop a numerical scheme based on the basis functions used in the TRA procedure, enabling the numerical solution of the seven-parameter confluent Heun-type differential equation. Through numerical experiments, we demonstrate the robustness of this approach near singularities and establish its superiority over the finite difference method. Full article

This review discusses two triatomic Hamiltonians and their applications to some non-adiabatic spectroscopic and collision problems. Carter and Handy in 1984 presented the first Hamiltonian in bond lengths–bond angle coordinates, that is here applied for studying the NO₂ spectroscopy: vibronic states, internal dynamics, and interaction with the radiation due to the $\tilde{\mathrm{X}}$²A′(A₁)−$\tilde{\mathrm{A}}$²A′(B₂) conical intersection. The second Hamiltonian was reported by Tennyson and Sutcliffe in 1983 in Jacobi coordinates and is here employed in the study of the Kr + OH(A²Σ⁺) electronic quenching due to conical intersection and Renner–Teller interactions among the 1²A′, 2²A′, and 1²A″ electronic species. Within the non-relativistic approximation and the expansion method in diabatic electronic representations, the formalism is exact and allows a unified study of various non-adiabatic interactions between electronic states. The rotation, inversion, and nuclear permutation symmetries are considered for defining rovibronic representations, which are symmetry adapted for ABC and AB₂ molecules, and the matrix elements of the Hamiltonians are then computed. 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

Differential beamforming has attracted much research since it can utilize an array with a small aperture size to form frequency-invariant beampatterns and achieve high directional gains. It has recently been applied to the loudspeaker line array to produce a broadside frequency-invariant radiation pattern. However, designing steerable frequency-invariant beampatterns for the loudspeaker line array has yet to be explored. This paper proposes a method to design a steerable differential beamformer with a loudspeaker line array. We first determine the target differential beampatterns according to the desired direction, the main lobe width, and the beampattern order. Then, we transform the target beampattern into the modal domain for representation. The Jacobi-Anger expansion is subsequently used to design the beamformer so that the resulting beampattern matches the target differential beampattern. Furthermore, based on the criterion of minimizing the mean square error between the synthesized beampattern and the ideal one, a multi-constraint optimization problem, which compromises between the robustness and the mean square error, is formulated to calculate the optimal desired weighting vector. Simulations and experimental results show that the proposed method can achieve steerable frequency-invariant beamforming from 300 Hz–4 kHz. Full article

As an improvement on the traditional model-based Yamaguchi four-component decomposition method, in recent years, to fully utilize the polarization information in the coherency matrix, four-component target decomposition methods Y4R and S4R have been proposed, which are based on the rotation of the coherency matrix and the expansion of the volume model, respectively. At the same time, there is also an improved G4U method proposed based on Y4R and S4R. Although these methods have achieved certain decomposition results, there are still problems with overestimation of volume scattering and insufficient utilization of polarization information. In this paper, a unitary transformation extension to the four-component target decomposition method of PolSAR based on the properties of the Jacobi method is proposed. By analyzing the terms in the basic scattering models, such as volume scattering, in the existing four-component decomposition methods, it is clear that the reason for the existence of the residual matrix in the existing decomposition methods is that the off-diagonal term $T_{13}$ and the real part of $T_{23}$ of the coherency matrix $\left[T\right]$ do not participate in the four-component decomposition. On this basis, a matrix transformation method is proposed to decouple terms $T_{13}$ and $\mathrm{Re}\left(T_{23}\right)$, and the residual matrix decomposed based on this method is derived. The performance of the proposed method was validated and evaluated using two datasets. The experimental results indicate that, compared with model-based methods such as Y4R, S4R and G4U, the proposed method can enhance the contribution of double-bounce scattering and odd-bounce scattering power in urban areas in both sets of data. The computational time of the proposed method is equivalent to Y4R, S4R, etc. Full article

Mode decomposition is a powerful tool for analyzing the modal content of optical multimode radiation. There are several basic principles on which this tool can be implemented, including near-field intensity analysis, machine learning, and spatial correlation filtering (SCF). The latter is meant to be applied to a spatial light modulator and allows one to obtain information on the mode amplitudes and phases of temporally stable beams by only analyzing experimental data. As a matter of fact, techniques based on SCF have already been successfully used in several studies, e.g., for investigating the Kerr beam self-cleaning effect and determining the modal content of Raman fiber lasers. Still, such techniques have a major drawback, i.e., they require acquisition times as long as several minutes, thus being unfit for the investigation of fast mode distribution dynamics. In this paper, we numerically study three types of digital holograms, which permits us to determine, at the same time, the parameters of a set of modes of multimode beams. Because all modes are simultaneously characterized, the processing speed of these real-time mode decomposition methods in experimental realizations will be limited only by the acquisition rate of imaging devices, e.g., state-of-the-art CCD camera performance may provide decomposing rates above 1 kHz. Here, we compare the accuracy of conjugate symmetric extension (CSE), double-phase holograms (DPH), and phase correlation filtering (PCF) methods in retrieving the mode amplitudes of optical beams composed of either three, six, or ten modes. In order to provide a statistical analysis of the outcomes of these three methods, we propose a novel algorithm for the effective enumeration of mode parameters, which covers all possible beam modal compositions. Our results show that the best accuracy is achieved when the amplitude-phase mode distribution associated with multiple frequency PCF techniques is encoded by Jacobi–Anger expansion. Full article

The Fokas system with M-truncated derivative (FS-MTD) was considered in this study. To get analytical solutions of FS-MTD in the forms of elliptic, rational, hyperbolic, and trigonometric functions, we employed the extend $\mathcal{F}$-expansion approach and the Jacobi elliptic function method. Since nonlinear pulse transmission in monomode optical fibers is explained by the Fokas system, the derived solutions may be utilized to analyze a broad range of important physical processes. In order to comprehend the impacts of MTD on the solutions, the dynamic behavior of the various generated solutions are shown using 2D and 3D figures. Full article

The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization. Full article

We developed a theoretical formalism for the generation of optical vortices by phased arrays of atoms. Using the Jacobi–Anger expansion, we demonstrated the resulting field topology and determined the least number of individual atoms necessary for the generation of vortices with a given topological charge. Vector vortices were considered, taking into account both the spin and orbital angular momenta of electromagnetic fields. It was found for the vortex field that, in the far field limit, the spatial variation in spin-density matrix parameters—orientation and alignment—is independent of the distance to the radiation source. Full article

The exact traveling wave solutions to coupled KdV equations with variable coefficients are obtained via the use of quadratic Jacobi’s elliptic function expansion. The presented coupled KdV equations have a more general form than those studied in the literature. Nine couples of quadratic Jacobi’s elliptic function solutions are found. Each couple of traveling wave solutions is symmetric in mathematical form. In the limit cases $m\to 1$, these periodic solutions degenerate as the corresponding soliton solutions. After the simple parameter substitution, the trigonometric function solutions are also obtained. Full article

In the current study, we investigate the stochastic Benjamin–Bona–Mahony equation with beta derivative (SBBME-BD). The considered stochastic term is the multiplicative noise in the Itô sense. By combining the $\mathcal{F}$-expansion approach with two separate equations, such as the Riccati and elliptic equations, new hyperbolic, trigonometric, rational, and Jacobi elliptic solutions for SBBME-BD can be generated. The solutions to the Benjamin–Bona–Mahony equation are useful in understanding various scientific phenomena, including Rossby waves in spinning fluids and drift waves in plasma. Our results are presented using MATLAB, with numerous 3D and 2D figures illustrating the impacts of white noise and the beta derivative on the obtained solutions of SBBME-BD. Full article

In the sense of a conformable fractional operator, we consider a generalized fractional–stochastic nonlinear wave equation (GFSNWE). This equation may be used to depict several nonlinear physical phenomena occurring in a liquid containing gas bubbles. The analytical solutions of the GFSNWE are obtained by using the F-expansion and the Jacobi elliptic function methods with the Riccati equation. Due to the presence of noise and the conformable derivative, some solutions that were achieved are shown together with their physical interpretations. Full article