Ricatti Equation Research Articles

Investigation of oceanic wave solutions to a modified (2+1)-dimensional coupled nonlinear Schrödinger system

This paper explores the oceanic wave characteristics exhibited by a modified integrable generalized [Formula: see text]-dimensional nonlinear Schrödinger system of equations through variable coefficients. Two newly modified methods, specifically the improved nonlinear Ricatti equation method and the improved sub-equation method, have been proposed to investigate the aforementioned nonlinear system. Through the utilization of these methods, we successfully obtain traveling and solitary waves solutions for this nonlinear system. We emphasize several constraint conditions that serve to guarantee the existence of these solutions. More comprehensive information about the physical dynamical representation of some of the solutions presented is illustrated through graphical depictions. The Mathematica software package is employed to produce both three-dimensional and their corresponding contour plots, thereby improving the visualization and comprehension of the solutions. This paper illustrates that the two proposed approaches provide straightforward and efficient means of acquiring various types of soliton, rational, trigonometric, hyperbolic and exponential solutions. Moreover, they present a more potent mathematical tool for addressing a variety of other nonlinear partial differential equations that hold significance in the field of applied science and engineering.

Investigating (2+1)-dimensional dissipative long wave system in water waves using three innovative integration norms

This research article investigates the dissipative long wave system in (2+1) dimensions. To analyze this system, three separate integrating strategies were used: the single manifold approach, the unified method, and the generalized projective Ricatti equation method. Novel soliton solutions were effectively obtained by applying these strategies to the suggested model. The resulting solutions are various, including mixed soliton-like solutions, dark solitons, rational solutions, triangular periodic solutions, and combined Jacobi elliptic wave function solutions. Using the aforementioned approaches, an extensive variety of solution types has been discovered for the proposed model. The dynamic character of these closed-form solutions has been adequately demonstrated using both three dimensional and two dimensional graphical representations. This visualization helps to better understand the behavior of the acquired solutions.

Energy partitioning among compressional and shear waves in three-dimensional attenuating elastic media.

In strongly scattering elastic media without attenuation and dispersion, the wavefield is dominated by shear waves, and in three dimensions, the ratio of the S to P energy is given by ES/EP=2(vP/vS)3. This study investigates how this ratio is influenced by attenuation. Both the case of the ringdown mode, where the energy evolves from initial values, and the case of energy equilibrium, where the attenuation is balanced by energy injection sources, are treated. It is shown that in ringdown mode, the energy ratio ES/EP satisfies a Ricatti equation in time: hence, the energy ratio is not an exponential function of time. It is also shown that the long-time energy ratio differs from the value in non-attenuating media when the attenuation coefficients for P and S waves are different. In the case of energy equilibrium, the energy ratio only is equal to the value in non-attenuating media when (1) the time scale of P- and S-wave equilibration is much smaller than the attenuation time or (2) the energy injection rate for each wave type is balanced by the dissipation for that wave type. The latter situation happens when the wavefield is excited by thermal fluctuations in thermal equilibrium.

The chemostat reactor: A stability analysis and model predictive control

This contribution develops the model predictive control for an unstable chemostat reactor. Initially, we analyze the system’s model — a nonlinear first-order hyperbolic partial integro-differential equation (PIDE) — and carry the model linearization around an unstable operating condition. Employing the structure-preserving Cayley–Tustin transformation, we obtain a discrete-time model representation of the continuous model. Subsequently, we solve the operator Ricatti equations in the discrete-time setting to derive a full state feedback controller that stabilizes the closed-loop and design a Luenberger observer for state reconstruction given the system output measures. Finally, we formulate a dual-mode MPC ensuring constraint satisfaction and optimality, integrating the gain-based unconstrained full-state feedback optimal control obtained from the Ricatti equation. This dual-mode strategy describes an optimization problem where the predictive controller acts only if constraints become active within the control horizon. Simulation studies validate the controller performance, where the MPC only takes action if the constraints are predicted to be active within the control horizon while also guaranteeing closed-loop stabilization under only output feedback. This type of controller can be easily implemented with other control strategies and significantly decreases the computational costs of solving the optimal control problems when compared to other MPC approaches.

On soliton solutions of Fokas dynamical model via analytical approaches

The nonlinear (4+1)-dimensional Fokas equation (FE) has been demonstrated to be the integrable extension of the Kadomtsev–Petviashvili (KP) and Davey–Stewartson (DS) equations. In nonlinear wave theory, the governing model is one of the fundamental structures that explains the surface waves and interior waves in straits or channels with different depths and widths. In this study, the generalized unified approach, the generalized projective ricatti equation technique, and the new F/G-expansion technique are applied to investigate the higher dimensional nonlinear model analytically. As a result, several solutions are successfully achieved, including dark soliton, periodic type solitons, w-shaped soliton, and single-bell shaped solitons. Along with an explanation of their behavior, we also display a few of the equation’s exact solutions graphically. The results demonstrate the effectiveness and simplicity of the approaches mentioned in this article, demonstrating their applicability to a wide range of additional nonlinear evolution issues in numerous scientific and technical disciplines.

Barrier methods based on Jordan–Hilbert algebras for stochastic optimization in spin factors

Infinite-dimensional stochastic second-order cone programming involves minimizing linear functions over intersections of affine linear manifolds with infinite-dimensional second-order cones. However, even though there is a legitimate necessity to explore these methods in general spaces, there is an absence of infinite-dimensional counterparts for these methods. In this paper, we present decomposition logarithmic-barrier interior-point methods based on unital Jordan–Hilbert algebras for this class of optimization problems in the infinite-dimensional setting. The results show that the iteration complexity of the proposed algorithms is independent on the choice of Hilbert spaces from which the underlying spin factors are formed, and so it coincides with the best-known complexity obtained by such methods for the finite-dimensional setting. We apply our results to an important problem in stochastic control, namely the two-stage stochastic multi-criteria design problem. We show that the corresponding infinite-dimensional system in this case is a matrix differential Ricatti equation plus a finite-dimensional system, and hence, it can be solved efficiently to find the search direction.

Stability analysis and dynamics of solitary wave solutions of the (3+1)-dimensional generalized shallow water wave equation using the Ricatti equation mapping method

Our aim is to examine the dynamic characteristics of the (3+1)-dimensional generalized equation governing shallow water waves. When the horizontal extent of the fluid significantly surpasses the vertical dimension, the employment of shallow water equations becomes appropriate. By employing an inventive Ricatti equation mapping approach, we have obtained a range of solitary wave solutions in both explicit and generalized forms. Solitons are particularly useful in signal and energy transmission due to their ability to preserve their shape during propagation. By studying soliton solutions in nonlinear systems, we can gain valuable knowledge applicable to practical fields. To deepen our grasp of the equation’s physical implications, we have presented some solutions through 3D, 2D, and contour graphics. Employing a linear stability approach, we confirm the equation’s stability. The method used distinguishes out for its simplicity, dependability, and ability to generate novel solutions for nonlinear partial differential equations in the area of mathematical physics. The research findings reported here demonstrate the viability of the used method in studying nonlinear phenomena in the studied equation as well as other nonlinear problems in mathematical physics. By employing this tool, academics have the chance to deepen and increase their grasp of the complex mathematical concepts underlying real-world problems.

Exact solution of three dimensional schrödinger equation with power function superposition potential.

Finding an analytical solution to the Schrödinger equation with power function superposition potential is essential for the development of quantum theory. For example, the harmonic oscillator potential, Coulomb potential, and Klazer potential are all classed as power function superposition potentials. In this study, the general form of the power function superposition potential was used to decompose the second-order radial Schrödinger equation with this potential into the first-order Ricatti equation. Furthermore, two forms of the power function superposition potential are constructed with an exact analytical solution, and the exact bound-state energy level formula is obtained for these two potentials. Finally, the energy levels of some of the diatomic molecules were determined through calculation. And our results are actually consistent with those obtained by other methods.

Exact solutions of classes of second order nonlinear partial differential equations reducible to first order

This paper illustrates the successful implementation of the method of variation of parameters in combination with the method of characteristics and other techniques to obtain exact solutions for a wide range of partial differential equations. The proposed approach reduces partial differential equations (PDEs) to first-order differential equations, referred to as classical equations, including Bernoulli, Ricatti, and Abel equations. In addition, the techniques proposed have the ability to produce precise solutions for nonlinear second order PDEs. For each PDE class, the method's effectiveness is demonstrated through illustrative examples.

Sign‐indefinite linear quadratic differential game for stochastic systems via static output feedback strategy

AbstractSign‐indefinite linear quadratic differential game for stochastic systems by means of static output feedback (SOF) strategy is investigated. First, the saddle point equilibrium condition of external disturbance and control strategy is established based on stochastic cross‐coupled algebraic matrix equations (SCCAMEs). Second, the numerical algorithms for solving the SCCAMEs are discussed. In particular, the difference between the existing algorithm and the proposed one is addressed. In order to show the validity for the several algorithms, a modified practical numerical example is demonstrated. In particular, it is shown that the optimality for the cost functional cannot be attained via the existing algorithm. Furthermore, it is claimed that the new proposed algorithm by combining the numerical algorithm that consist of stochastic Ricatti and Lyapunov equations with the Newton method is useful and reliable because the setting of the initial condition is quite easy task and fast convergence is attained.

Periodic solitons of Davey Stewartson Kadomtsev Petviashvili equation in (4+1)-dimension

The aim of this paper is to study new exact solutions Davey–Stewartson–Kadomtsev–Petviashvili (DSKP) equation. We use modified tanh method along associated with new Ricatti equation. The results are demonstrated for specific values of parameters, which show periodic singular as well as nonsingular solitons for specific values of parameters. Some solutions are plotted and presented in 3D and density graphs. The Mathematica are used for computation and simulations of the results, respectively.

Legendre Polynomials’ Second Derivative Tau Method for Solving Lane-Emden and Ricatti Equations

Legendre Polynomials’ Second Derivative Tau Method for Solving Lane-Emden and Ricatti Equations

Optimal Floquet Stationkeeping under the Relative Dynamics of the Three-Body Problem

Deep space missions, and particularly cislunar endeavors, are becoming a major field of interest for the space industry, including for the astrodynamics research community. While near-Earth missions may be completely covered by perturbed Keplerian dynamics, deep space missions require a different modeling approach, where multi-body gravitational interactions play a major role. To this end, the Restricted Three-Body Problem stands out as an insightful first modeling strategy for early mission design purposes, retaining major dynamical transport structures while still being relatively simple. Dynamical Systems Theory and classical Hamiltonian Mechanics have proven themselves as remarkable tools to analyze deep-space missions within this context, with applications ranging from ballistic capture trajectory design to stationkeeping. In this work, based on this premise, a Hamiltonian derivation of the Restricted Three-Body Problem co-orbital dynamics between two spacecraft is introduced in detail. Thanks to the analytical and numerical models derived, connections between the relative and classical Keplerian and CR3BP problems are shown to exist, including first-order linear solutions and an inherited Hamiltonian normal form. The analytical linear and higher-order models derived allow the theoretical finding and unveiling of natural co-orbital phase space structures, including relative periodic and quasi-periodic orbital families, which are further exploited for general proximity operation applications. In particular, a novel reduced-order, optimal low-thrust stationkeeping controller is derived in the relative Floquet phase space, hybridizing the classical State Dependent Ricatti Equation (SDRE) with Koopman control techniques for efficient unstable manifold regulation. The proposed algorithm is demonstrated and validated within several end-to-end low-cost stationkeeping missions, and comparison against classical continuous stationkeeping algorithms presented in the literature is also addressed to reveal its enhanced performance. Finally, conclusions and open lines of research are discussed.

Continuum Energy Eigenstates via the Factorization Method

The factorization method was introduced by Schrödinger in 1940. Its use in bound-state problems is widely known, including in supersymmetric quantum mechanics; one can create a factorization chain, which simultaneously solves a sequence of auxiliary Hamiltonians that share common eigenvalues with their adjacent Hamiltonians in the chain, except for the lowest eigenvalue. In this work, we generalize the factorization method to continuum energy eigenstates. Here, one does not generically have a factorization chain—instead all energies are solved using a “single-shot factorization”, enabled by writing the superpotential in a form that includes the logarithmic derivative of a confluent hypergeometric function. The single-shot factorization approach is an alternative to the conventional method of “deriving a differential equation and looking up its solution”, but it does require some working knowledge of confluent hypergeometric functions. This can also be viewed as a method for solving the Ricatti equation needed to construct the superpotential.

Alsultani Rules to Find the Particular Solution of Ricatti's Equation

Alsultani Rules to Find the Particular Solution of Ricatti's Equation

Adaptive inverse optimal backstepping control strategy for longitudinal vibration of high-speed elevator system based on fuzzy observer

To effectively suppress the longitudinal vibration of the car under the conditions of high-speed operation and emergency braking, and improve the ride comfort of the car system, this paper proposes an adaptive fuzzy inverse optimal output feedback control strategy. Firstly, the dynamic model of the high-speed elevator system is established and the nonlinear dynamic model is approximated by the fuzzy logic system, and the auxiliary system model is established. Fuzzy state observer is designed to estimate the unmeasurable state. Furthermore, an adaptive inverse optimal output feedback controller based on fuzzy observer is designed by using adaptive backstepping technology and inverse optimal control principle. The stability analysis shows that the proposed adaptive fuzzy inverse optimal output feedback control strategy not only ensures the stability of the car attitude of high-speed elevator but also realizes the inverse optimization of the target cost function. Finally, the acceleration time–frequency response analysis of the two typical stages of high-speed elevator uniform running and emergency braking is carried out, and the numerical results are compared with the linear quadratic controller optimized by stepping quantum genetic algorithm (GA-LQR) and controller based on the state-dependent Ricatti equation (SDRE). The analysis verifies the effectiveness of the controller.

Laboratory Study of Microsatellite Control Algorithms Performance for Active Space Debris Removal Using UAV Mock-Ups on a Planar Air-Bearing Test Bed

In this paper, a planar air-bearing test bed with unmanned aerial vehicles (UAV) was used to test a microsatellite motion control system. The UAV mock-ups were controlled by four ventilator actuators that imitated the satellite thrusters and provided the required acceleration vector in the horizontal plane, and torque along the vertical direction. The mock-ups moved almost without friction along the planar air-bearing test bed due to the air cushion between the test bed surface and the flat mock-up base. The motion of the mock-ups motion imitated the motion of satellites in the orbital plane. The problem of space debris can be solved using special microsatellite missions able to dock to space debris objects and change their orbit. In this paper, two control algorithms based on the virtual potentials approach and the State Dependent Ricatti Equation (SDRE) controller, were proposed for docking to a non-cooperative space debris object. The algorithms were tested in a laboratory facility, and the results are presented and analyzed, including their main features demonstrated during the laboratory study. It was shown that the SDRE-based control was faster, although the virtual potential-based control required less characteristic velocity.

Third Order Observer Design for a Rotating Energy Harvesting System - A Brief Paper

In most nonlinear systems, not all the states are available for measurement. In such instances, an observer is utilized to estimate the unavailable states. Designing observers for nonlinear systems has received widespread attention owing to its utmost significance and intensive mathematical computations generally, an observer is used to calculate approximately the mechanical quantities from the identified electrical variables in an energy harvesting system. In the past, several estimations remained a challenge because of the rigorous computational techniques involved. In this work, we present a means of designing an observer by using the algebraic Ricatti equation.

The Prufer transform for the calculation of acoustic normal modes

The PrüferTransform for the Sturm-Liouvile (SL) equation was introduced by Prüferin 1923. An excellent reference is Numerical Solutions of Sturm-Liouville Problems by Pryce. Porter in Computational Ocean Acoustics introduced it to ocean acoustics. This author stumbled upon working on state variables at CMRE on 1978. The method derives first order equations for the phase and log-magnitude for second order equation in a phase plane. The essential feature is the phase equation is not coupled to the magnitude, so it may be solved as a nonlinear, first order equation and the surface boundary condition leads to a unique solution. New results for normal modes using the Prüfermethod are introduced. i) equations for the phase shifts and log-magnitudes at layer boundaries such as at the seabed; ii) state variable transformations based upon Ricatti equations and requiring evaluating one transcendental quantity, important for numerical methods; iii) issues of numerical stability for coupling the phase solution across the potential barriers of local sound speed maxima and poorly coupled boundaries; iv) an equation for computing the imaginary part leading to modal attenuation’s. The method is fast has survived many students using it, even those writing Matlab versions.

Exact solutions for a variable-coefficients nonisospectral nonlinear Schrödinger equation via Wronskian technique

A variable-coefficients nonisospectral nonlinear Schrödinger (vc-nNLS) equation is investigated. Integrable background in the framework of the Lax pair with a time-dependent spectral parameter satisfying a generalized Ricatti equation is given. Solutions are given in terms of the double Wronski determinant where the Hirota method is employed. Moreover, different sorts of solutions are provided. In particular, employing the freedom of choosing the variable-coefficients, illustrative examples depicting localized wave solutions are presented.