Search Results (32)

Fluoride solid solutions exhibit exceptional optical and thermodynamic properties that make them valuable for advanced technological applications, and the PbF₂-EuF₃ system represents a particularly promising quasi-binary system for developing high-performance materials. However, the comprehensive understanding of the thermodynamic conditions governing phase equilibria and the precise boundaries of homogeneity regions in this system remains incomplete, limiting the rational design of single-phase materials with desired properties. Therefore, we conducted a comprehensive investigation of the thermodynamic conditions (temperature and composition) controlling the existence of cubic and rhombohedral phases within the homogeneity regions of the PbF₂-EuF₃ system. Solid solution samples were synthesized using both solid-phase synthesis and co-precipitation techniques from aqueous nitrate solutions. Phase equilibria were systematically investigated in two critical regions: the solvus line spanning 0–10 mol% EuF₃ and the ordered rhombohedral R-phase region spanning 35–45 mol% EuF₃. Structural characterization was performed at temperatures below the phase transition temperature in lead fluoride (365 °C) using X-ray phase analysis, optical probing, and Raman scattering. Our investigation successfully demonstrated the possibility of obtaining cubic preparations of high purity across the 0–37 mol% EuF₃ composition range. Additionally, we precisely defined the region of existence of the ordered rhombohedral R-phase within the concentration range of 37–39 to 43–44 mol% EuF₃. These findings provide essential thermodynamic data for the rational design of PbF₂-EuF₃ solid solutions and establish clear compositional boundaries for obtaining desired phase structures in this technologically important fluoride system. Full article

In this article, we investigate the dynamical analysis and soliton solutions of the microtubule equation. First, the Lie symmetry method is applied to the considered model to reduce the governing partial differential equation into an ordinary differential equation. Next, the multivariate generalized exponential rational integral function method is employed to derive exact soliton solutions. Finally, the bifurcation analysis of the corresponding dynamical system is discussed to explore the qualitative behavior of the obtained solutions. When an external force influences the system, its behavior exhibits chaotic and quasi-periodic phenomena, which are detected using chaos detection tools. We detect the chaotic and quasi-periodic phenomena using 2D phase portrait, time analysis, fractal dimension, return map, chaotic attractor, power spectrum, and multistability. Phase portraits illustrating bifurcation and chaotic patterns are generated using the RK4 algorithm in Matlab version 24.2. These results offer a powerful mathematical framework for addressing various nonlinear wave phenomena. Finally, conservation laws are explored. Full article

In this study, the complete discrimination system for the polynomial method (CDSPM) is employed to analyze the integrable Gardner Equation (IGE). Through a traveling wave transformation, the model is reduced to a nonlinear ordinary differential equation, enabling the derivation of a wide class of exact solutions, including trigonometric, hyperbolic, rational, and Jacobi elliptic functions. For example, a bright soliton solution is obtained for parameters $A=1.3$, $\beta =-0.1$, and $\gamma =0.8$. Qualitative analysis reveals diverse phase portraits, indicating the presence of saddle points, centers, and cuspidal points depending on parameter values. Chaos and quasi-periodic dynamics are investigated via Poincaré maps and time-series analysis, where chaotic patterns emerge for values like ${\nu }_1=1.45$, ${\nu }_2=2.18$, ${\Xi }_0=4$, and $\lambda =2\pi$. Sensitivity analysis confirms the model’s sensitivity to initial conditions $\chi =2.2,2.4,2.6$, reflecting real-world unpredictability. Additionally, the energy balance method (EBM) is applied to approximate periodic solutions by conserving kinetic and potential energies. These results highlight the IGE’s ability to capture complex nonlinear behaviors relevant to fluid dynamics, plasma waves, and nonlinear optics. Full article

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 paper advances a new technique for constructing the exceptional differential polynomial systems (X-DPSs) and their infinite and finite orthogonal subsets. First, using Wronskians of Jacobi polynomials (JPWs) with a common pair of the indexes, we generate the Darboux–Crum nets of the rational canonical Sturm–Liouville equations (RCSLEs). It is shown that each RCSLE in question has four infinite sequences of quasi-rational solutions (q-RSs) such that their polynomial components from each sequence form a X-Jacobi DPS composed of simple pseudo-Wronskian polynomials (p-WPs). For each p-th order rational Darboux Crum transform of the Jacobi-reference (JRef) CSLE, used as the starting point, we formulate two rational Sturm–Liouville problems (RSLPs) by imposing the Dirichlet boundary conditions on the solutions of the so-called ‘prime’ SLE (p-SLE) at the ends of the intervals (−1, +1) or (+1, ∞). Finally, we demonstrate that the polynomial components of the q-RSs representing the eigenfunctions of these two problems have the form of simple p-WPs composed of p Romanovski–Jacobi (R-Jacobi) polynomials with the same pair of indexes and a single classical Jacobi polynomial, or, accordingly, p classical Jacobi polynomials with the same pair of positive indexes and a single R-Jacobi polynomial. The common, fundamentally important feature of all the simple p-WPs involved is that they do not vanish at the finite singular endpoints—the main reason why they were selected for the current analysis in the first place. The discussion is accompanied by a sketch of the one-dimensional quantum-mechanical problems exactly solvable by the aforementioned infinite and finite EOP sequences. Full article

We give some of our results over the past few years about rogue waves concerning some partial differential equations, such as the focusing nonlinear Schrödinger equation (NLS), the Kadomtsev–Petviashvili equation (KPI), the Lakshmanan–Porsezian–Daniel equation (LPD) and the Hirota equation (H). For the NLS and KP equations, we give different types of representations of the solutions, in terms of Fredholm determinants, Wronskians and degenerate determinants of order $2N$. These solutions are called solutions of order N. In the case of the NLS equation, the solutions, explicitly constructed, appear as deformations of the Peregrine breathers $P_N$ as the last one can be obtained when all parameters are equal to zero. At order N, these solutions are the product of a ratio of two polynomials of degree $N(N+1)$ in x and t by an exponential depending on time t and depending on $2N-2$ real parameters: they are called quasi-rational solutions. For the KPI equation, we explicitly obtain solutions at order N depending on $2N-2$ real parameters. We present different examples of rogue waves for the LPD and Hirota equations. Full article

This paper develops a new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm–Liouville equations (RCSLEs) with quasi-rational solutions (q-RSs) were obtained by applying rational Rudjak–Zakhariev transformations (RRZTs) to the Jacobi equation re-written in the canonical form. The presented analysis was focused on the RRZTs leading to the canonical form of the Heun equation. It was demonstrated that the latter equation preserves its form under the second-order Darboux–Crum transformation. The associated Sturm–Liouville problems (SLPs) were formulated for the so-called ‘prime’ SLEs solved under the Dirichlet boundary conditions (DBCs). It was proven that one of the two X₁-Jacobi DPSs composed of Heun polynomials contains both the X₁-Jacobi orthogonal polynomial system (OPS) and the finite EOP sequence composed of the pseudo-Wronskian transforms of Romanovski–Jacobi (R-Jacobi) polynomials, while the second analytically solvable Heun equation does not have the discrete energy spectrum. The quantum-mechanical realizations of the developed formalism were obtained by applying the Liouville transformation to each of the SLPs formulated in such a way. Full article

In this article, the modified $\alpha$ equation is solved using the direct algebraic approach. As a result, numerous new and more generalized exact solutions for such equations have been found, taking into account the wide range of travelling structures. The rational, trigonometric, hyperbolic, and exponential functions with a couple of licentious parameters are thus included in these exact answers. Analytical solutions feature a variety of physical structures, which are visually studied to demonstrate their dynamic behavior in 2D and 3D. Considering the parameters, all feasible phase portraits are shown. Furthermore, we used numerical approaches to determine the nonlinear periodic structures of the mentioned model, and the data are graphically displayed. Additionally, we employed numerical approaches to determine the nonlinear conditions that contribute to the presented model, and the data are graphically displayed. After evaluating the influence of frequency following the application of an external periodic factor, sensitivity exploration is used to study quasi-periodic and chaotic behavior for several starting value problems. Furthermore, the function of physical characteristics is investigated using an external periodic force. Quasi-periodic and quasi-periodic-chaotic patterns are described with the inclusion of a perturbation term. The direct algebraic methodology would be used to derive the soliton solution of modified $\alpha$ equation, from which the Galilean transformation derives traveling wave solutions of the considered and a bifurcation behavior is reported. Analytical and numerical methods have been used to have the condition of the travelling wave phase transformation. The well-judged values of parameters are enhanced well with a graphically formal analysis of such specific solutions to illustrate their propagation. Then a planer dynamical system is introduced, and a bifurcation analysis is utilized to identify the bifurcation structures of the dynamical model’s nonlinear wave propagation solutions. Additionally, the periodic and quasi-periodic behavior of the discussed equation is analyzed using sensitivity analysis for a range of beginning values. To further comprehend the dynamical behaviors of the resultant solutions, a graphic analysis is conducted. Full article

Robot acceptance is rapidly increasing in many different industrial applications. The advancement of production systems and machines requires addressing the productivity complexity and flexibility of current manufacturing processes in quasi-real time. Nowadays, robot placement is still achieved via industrial practices based on the expertise of the workers and technicians, with the adoption of offline expensive software that demands time-consuming simulations, detailed time-and-motion mapping activities, and high competencies. Current challenges have been addressed mainly via path planning or robot-to-workpiece location optimization. Numerous solutions, from analytical to physical-based and data-driven formulation, have been discussed in the literature to solve these challenges. In this context, the machine learning approach has proven its superior performance. Nevertheless, the industrial environment is complex to model, generating extra training effort and making the learning procedure, in some cases, inefficient. The industrial problems concern workstation productivity; path-constrained minimal-time motions, considering the actuator’s torque limits; followed by robot vibration and the reduction in its accuracy and lifetime. This paper presents a procedure to find the robot base location for a prescribed task within the robot’s workspace, complying with multiple criteria. The proposed hybrid procedure includes analytical, physical-based, and data-driven modeling to solve the optimization problem. The contribution of the algorithm, for a given user-defined task, is the search for the best robot base location that enables the target points, maximizing the manipulability, avoiding singularities, and minimizing energy consumption. Firstly, the established method was verified using an anthropomorphic robot that considers different levels of a priori kinematics and system dynamics knowledge. The feasibility of the proposed method was evaluated through various simulations for small- and medium-sized robots. Then, a commercial offline program was compared, considering three scenarios and fourteen robots demonstrating an energy reduction in the 7.6–13.2% range. Moreover, the unknown joint dependency in real robot applications was investigated. From 11 robot positions for each active joint, a direct kinematic was appraised with an automatic DH scheme that generates the 3D workspace with an RMSE lower than 65.0 µm. Then, the inverse kinematic was computed using an ANN technique tuned with a genetic algorithm showing an RMSE in an S-shape task close to 702.0 µm. Finally, three experimental campaigns were performed with a set of tasks, repetitions, end-effector velocity, and payloads. The energy consumption reduction was observed in the 12.7–22.9% range. Consequently, the proposed procedure supports the reduction in workstation setup time and energy saving during industrial operations. Full article

From the finite gap solutions of the KdV equation expressed in terms of abelian functions we construct solutions to the Schrödinger equation with a KdV potential in terms of fourfold Fredholm determinants. For this we establish a connection between Riemann theta functions and Fredholm determinants and we obtain multi-parametric solutions to this equation. As a consequence, a double Wronskian representation of the solutions to this equation is constructed. We also give quasi-rational solutions to this Schrödinger equation with rational KdV potentials. Full article

This paper focuses on the influence of the fractional-order (FO) resonant capacitor on the zero-voltage-switching quasi-resonant converter (ZVS QRC). The FO impedance model of the capacitor is introduced to the circuit model of the ZVS QRC; hence, a piecewise smooth FO model is developed for the converter. Numerical solutions of the converter are obtained by using both the fractional Adams–Bashforth–Moulton (F-ABM) method and Oustaloup’s rational approximation method. In addition, the analytical solution of the converter is obtained by the Grünwald–Letnikov (GL) definition, which reveals the influence of the FO resonant capacitor on the zero-crossing point (ZCP) and resonant state of the converter. An experimental platform was built to verify the results of the theoretical analysis and numerical calculation. Full article

The paper reveals some remarkable form-invariance features of the ‘Jacobi-reference’ canonical Sturm–Liouville equation (CSLE) in the particular case of the density function with the simple pole at the origin. It is proven that the CSLE under consideration preserves its form under the two second-order Darboux–Crum transformations (DCTs) with the seed functions represented by specially chosen pairs of ‘basic’ quasi-rational solutions (q-RSs), i.e., such that their analytical continuations do not have zeros in the complex plane. It is proven that both transformations generally either increase or decrease by 2 the exponent difference (ExpDiff) for the mentioned pole while keeping two other parameters unchanged. The change is more complicated in the latter case if the ExpDiff for the pole of the original CSLE at the origin is smaller than 2. It was observed that the DCTs in question do not preserve bound energy levels according to the conventional supersymmetry (SUSY) rules. To understand this anomaly, we split the DCT in question into the two sequential Darboux deformations of the Liouville potentials associated with the CSLEs of our interest. We found that the first Darboux transformation turns the initial CSLE into the Heun equation written in the canonical form while the second transformation brings us back to the canonical form of the hypergeometric equation. It is shown that the first of these transformations necessarily places the mentioned ExpDiff into the limit-circle (LC) range and then the second transformation keeps the pole within the LC region, violating the conventional prescriptions of SUSY quantum mechanics. Full article

This paper presents a quasi-analytical method that allows the derivation of a rigorous series-form representation for the mutual inductance of two co-axial coil antennas located above an arbitrarily layered earth structure. Starting from Biot–Savart law, which gives the integral representation for the primary vector potential generated by the source coil, the potential reflected by the layered ground is derived, and the resulting total vector potential is then integrated along the external circumference of the receiving coil to give the mutual inductance of the two antennas. The obtained representation for the flux is then evaluated analytically through the usage of the Gegenbauer addition theorem once an accurate, rational approximation is used in place of the factor of the integrand that exhibits branch cuts. It is shown how the resulting explicit solution exhibits the same degree of accuracy as purely numerical approaches like the finite-difference time-domain (FDTD) method and conventional numerical quadrature schemes, while it is less time-demanding than the latter methods. Full article

We take into account the (2 + 1)-dimensional stochastic Kadomtsev–Petviashvili equation with beta-derivative (SKPE-BD) in this paper. To develop new hyperbolic, trigonometric, elliptic, and rational solutions, the Riccati equation and Jacobi elliptic function methods are employed. Because the KP equation is required for explaining the development of quasi-one-dimensional shallow-water waves, the solutions obtained can be used to interpret various attractive physical phenomena. To display how the multiplicative white noise and beta-derivative impact the exact solutions of the SKPE-BD, we plot a few graphs in MATLAB and display different 3D and 2D figures. We deduce how multiplicative noise stabilizes the solutions of SKPE-BD at zero. Full article

The paper examines common elements between Lévai’s and Milson’s potentials obtained by Liouville transformations of two rational canonical Sturm–Liouville equations (RCSLEs) with even density functions which are exactly solvable via Jacobi polynomials in a real or accordingly imaginary argument. We refer to the polynomial numerators of the given rational density function as ‘tangent polynomial’ (TP) and thereby term the aforementioned potentials as ‘e-TP’. Special attention is given to the overlap between the two potentials along symmetric curves which represent two different rational forms of the Ginocchio potential exactly quantized via Gegenbauer and Masjed-Jamei polynomials, respectively. Our analysis reveals that the actual interconnection between Lévai’s parameters for these two rational realizations of the Ginocchio potential is much more complicated than one could expect based on the striking resemblance between two quartic equations derived by Lévai for ‘averaged’ Jacobi indexes. Full article