Search Results (175)

In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing $\beta$-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations. Full article

This paper addresses the problem of optimal stabilization for stochastic dynamical systems characterized by Markov switches and concentration points of jumps, which is a scenario not adequately covered by classical stability conditions. Unlike traditional approaches requiring a strictly positive minimal interval between jumps, we allow jump moments to accumulate at a finite point. Utilizing Lyapunov function methods, we derive sufficient conditions for exponential stability in the mean square and asymptotic stability in probability. We provide explicit constructions of Lyapunov functions adapted to scenarios with jump concentration points and develop conditions under which these functions ensure system stability. For linear stochastic differential equations, the stabilization problem is further simplified to solving a system of Riccati-type matrix equations. This work provides essential theoretical foundations and practical methodologies for stabilizing complex stochastic systems that feature concentration points, expanding the applicability of optimal control theory. Full article

The objective of this manuscript is to investigate the (2+1)-dimensional Chiral nonlinear Schrödinger equation (CNLSE). We employ the traveling wave transformation to convert the nonlinear partial differential equation (NLPDE) into the nonlinear ordinary differential equation (NLODE). Utilizing the two new vital techniques to derive the solitary wave solutions, the generalized Arnous method and the Riccati equation method, we obtained various types of waves like bright solitons, dark solitons, and periodic wave solutions. Sensitivity analysis is also discussed using different initial conditions. Sensitivity analysis refers to the study of how the solutions of the equations respond to changes in the parameters or initial conditions. It involves assessing the impact of variations in these factors on the behavior and properties of the solutions. To better comprehend the physical consequences of these solutions, we showcase them through different visual depictions like 3D, 2D, and contour plots. The findings of this study are original and hold significant value for the future exploration of the equation, offering valuable directions for researchers to deepen knowledge on the subject. Full article

Using Schumann resonance (SR) records from the Antarctic, we evaluate the impact of the solar activity on the global ionosphere over the period from 2002 to 2024. The updated vertical profile of the middle atmosphere conductivity is applied. The pivoted upper part of profiles above the knee altitude is adjusted to represent different levels of solar activity. The electric (lower) h_C and the magnetic (upper) h_L characteristic heights, the propagation constant ν(f) of the extremely low frequency (ELF) radio waves, and the basic resonance frequency f₁ are computed for the profiles corresponding to the solar maximum, moderate, and minimum activity conditions by using the full-wave solution in the form of the Riccati differential equation. Model data are compared with experimental observations at the Ukrainian Antarctic Station of “Akademik Vernadsky” (geographic coordinates: 65.25° S and 64.25° W). The following results are discussed: (i) Solar activity modifies the upper characteristic height h_L of the ionosphere by ±1 km over the 11-year cycle; (ii) Equations were obtained linking the current level of solar activity with the basic SR frequency, with the magnetic characteristic height, and with the ELF propagation constant; (iii) Based on SR monitoring within two complete solar cycles, a practical rule is proposed: an increase in the index of solar activity I_10.7 by ~150 units raises the first SR frequency by ~0.1 Hz and elevates the magnetic characteristic height by ~2.5 km. Full article

This paper addresses the problem of optimal $H_2$-filtering for a class of continuous-time linear McKean–Vlasov stochastic differential equations under sampled measurements. The main tool used to solve the filtering problem is a forward jump matrix linear differential equation with a Riccati-type jumping operator. More specifically, the stabilizing solution of such a jump Riccati-type equation plays a key role. Full article

This study aims to establish new oscillation criteria for solutions of a specific class of functional differential equations. Our findings extend and refine the recently developed criteria for this type of equation by various authors and also encompass classical criteria for related problems. Our approach relies on the Riccati technique to derive conditions that preclude the possibility of non-oscillatory solutions. The inherent symmetry of these solutions plays a key role in formulating the new criteria presented here. By applying techniques from the theory of symmetric differential equations and leveraging symmetric functions, we are able to establish precise conditions for oscillation. To enhance practical applicability, we propose multiple distinct criteria while minimizing the constraints typically imposed. Several examples are provided to illustrate the accuracy, applicability, and versatility of the new criteria. Full article

This paper investigates a backward stochastic linear quadratic control problem with an expected-type equality constraint on the initial state. By using the Lagrange multiplier method, the problem with a uniformly convex cost functional is first transformed into an equivalent unconstrained parameterized backward stochastic linear quadratic control problem. Then, under the surjectivity of the linear constraint, the equivalence between the original problem and the dual problem is proven by Lagrange duality theory. Subsequently, with the help of the maximum principle, an explicit solution of the optimal control for the unconstrained problem is obtained. This solution is feedback-based and determined by an adjoint stochastic differential equation, a Riccati-type ordinary differential equation, a backward stochastic differential equation, and an equality, thereby yielding the optimal control for the original problem. Finally, an optimal control for an investment portfolio problem with an expected-type equality constraint on the initial state is explicitly provided. Full article

A novel methodology for solving Caputo ${\left({}^C\mathbf{D}^{(1/n)}\right)}^k$-type fractional differential equations (FDEs), where the fractional differentiation order is $k/n$, is proposed. This approach uniquely utilizes fractional power series expansions to transform the original FDE into a higher-order FDE of type ${\left({}^C\mathbf{D}^{(1/n)}\right)}^{kn}$. Significantly, this perfect FDE is then reduced to a k-th-order ordinary differential equation (ODE) of a special form, thereby allowing the problem to be addressed using established ODE techniques rather than direct fractional calculus methods. The effectiveness and applicability of this framework are demonstrated by its application to the fractional Riccati-type differential equation. Full article

This paper investigates the oscillatory behavior of a class of mixed fractional-order nonlinear differential equations incorporating both the Liouville right-sided and conformable fractional derivatives. Symmetry plays a key role in understanding the oscillatory behavior of these systems. The motivation behind this study arises from the need for a more generalized framework to analyze oscillatory behavior in fractional differential equations, bridging the gap in the existing literature. By employing the generalized Riccati technique and the integral averaging method, we establish new oscillation criteria that extend and refine previous results. Illustrative examples are provided to validate the theoretical findings and highlight the effectiveness of the proposed methods. Full article

This paper aims to study the oscillatory behavior of second-order neutral differential equations. Using the Riccati substitution technique, we introduce new oscillation criteria that essentially improve some related criteria from the literature. We provide some examples and compare the results in this paper with earlier results to illustrate the importance of our results. Full article

This paper introduces a nonlinear Lyapunov-based Finite-Time State-Dependent Differential Riccati Equation (FT-SDDRE) control scheme, considering actuator saturation constraints and ensuring that the control system operates within safe operational limits designed for satellite reconfiguration and formation-keeping in low Earth orbit (LEO) missions. This control approach addresses the challenges of reaching the relative position and velocity vectors within a defined timeframe amid various orbital perturbations. The proposed approach guarantees precise formation control by utilizing a high-fidelity relative motion model that incorporates all zonal harmonics and atmospheric drag, which are the primary environmental disturbances in LEO. Additionally, the article presents an optimization methodology to determine the most efficient State-Dependent Coefficient (SDC) form regarding fuel consumption. This optimization process minimizes energy usage through a hybrid genetic algorithm and simulated annealing (HGASA), resulting in improved performance. In addition, this paper includes a sensitivity analysis to identify the optimized SDC parameterization for different satellite reconfiguration maneuvers. These maneuvers encompass radial, along-track, and cross-track adjustments, each with varying baseline distances. The analysis provides insights into how different parameterizations affect reconfiguration performance, ensuring precise and efficient control for each type of maneuver. The finite-time controller proposed here is benchmarked against other forms of SDRE controllers, showing reduced error margins. To further assess the control system’s effectiveness, an input saturation constraint is integrated, ensuring that the control system operates within safe operational limits, ultimately leading to the successful execution of the mission. Full article

This paper deals with the oscillatory behavior of solutions of a general class of fourth-order non-linear delay differential equations. New oscillation criteria are established using Riccati transformation and a Philos-type technique. The obtained results not only improve and extend some published results in the literature, but also relax some traditional conditions on the function $\psi \left(\chi \right(\iota \left)\right)$. Three examples are provided to illustrate the main results. Full article

Piecewise analytical solutions to scalar, nonlinear, first-order, ordinary differential equations based on the second-order Taylor series expansion of their right-hand sides that result in Riccati’s equations are presented. Closed-form solutions are obtained if the dependence of the right-hand side on the independent variable is not considered; otherwise, the solution is given by convergent series. Discrete solutions also based on the second-order Taylor series expansion of the right-hand side and the discretization of the independent variable that result in algebraic quadratic equations are also reported. Both the piecewise analytical and discrete methods are applied to two singularly perturbed initial-value problems and the results are compared with the exact solution and those of linearization procedures, and implicit and explicit Taylor’s methods. It is shown that the accuracy of piecewise analytical techniques depends on the number of terms kept in the series expansion of the solution, whereas that of the discrete methods depends on the location where the coefficients are evaluated. For Riccati equations with constant coefficients, the piecewise analytical method presented here provides the exact solution; it also provides the exact solution for linear, first-order ordinary differential equations with constant coefficients. Full article

This research focuses on studying the asymptotic and oscillatory behavior of a special class of even-order nonlinear neutral differential equations, including damping terms. The research aims to achieve qualitative progress in understanding the relationship between the solutions of these equations and their associated functions. Leveraging the symmetry between positive and negative solutions simplifies the derivation of criteria that ensure the oscillation of all solutions. Using precise techniques such as the Riccati method and comparison methods, innovative criteria are developed that guarantee the oscillation of all the solutions of the studied equations. The study provides new conditions and effective analytical tools that contribute to deepening the theoretical understanding and expanding the practical applications of these systems. Based on solid scientific foundations and previous studies, the research concludes with the presentation of examples that illustrate the practical impact of the results, highlighting the theoretical value of research in the field of neutral differential equations. Full article

The goal of this study is to derive new conditions that improve the testing of the oscillatory and asymptotic features of fourth-order differential equations with an advanced neutral term. By using Riccati techniques and comparison with lower-order equations, we establish new criteria that verify the absence of positive solutions and, consequently, the oscillation of all solutions to the investigated equation. Using our results to analyze a few specific instances of the examined equation, we can ultimately clarify the significance of the new inequalities. Our results are an extension of previous results that considered equations with a neutral delay term and also an improvement of previous results that considered only equations with an advanced neutral term. Full article