Unique Solution Research Articles

Computational and analytical analysis of integral-differential equations for modeling avoidance learning behavior

AbstractThis work emphasizes the computational and analytical analysis of integral-differential equations, with a particular application in modeling avoidance learning processes. Firstly, we suggest an approach to determine a unique solution to the given model by employing methods from functional analysis and fixed-point theory. We obtain numerical solutions using the approach of Picard iteration and evaluate their stability in the context of minor perturbations. In addition, we explore the practical application of these techniques by providing two examples that highlight the thorough analysis of behavioral responses using numerical approximations. In the end, we examine the efficacy of our suggested ordinary differential equations (ODEs) for studying the avoidance learning behavior of animals. Furthermore, we investigate the convergence and error analysis of the proposed ODEs using multiple numerical techniques. This integration of theoretical and practical analysis enhances the domain of applied mathematics by providing important insights for behavioral science research.

Third Order Dynamical Systems for the Sum of Two Generalized Monotone Operators

AbstractIn this paper, we propose and analyze a third-order dynamical system for finding zeros of the sum of two generalized operators in a Hilbert space $$\mathcal {H}$$ H . We establish the existence and uniqueness of the trajectories generated by the system under appropriate continuity conditions, and prove exponential convergence to the unique zero when the sum of the operators is strongly monotone. Additionally, we derive an explicit discretization of the dynamical system, which results in a forward–backward algorithm with double inertial effects and larger range of stepsize. We establish the linear convergence of the iterates to the unique solution using this algorithm. Furthermore, we provide convergence analysis for the class of strongly pseudo-monotone variational inequalities. We illustrate the effectiveness of our approach by applying it to structured optimization and pseudo-convex optimization problems.

Investigating the ability of PINNs to solve Burgers’ PDE near finite-time blowup

Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated Partial Differential Equations (PDEs) numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs is that there exist simple PDEs which can evolve into singular solutions in finite time starting from smooth initial conditions. In recent times some striking experiments have suggested that PINNs might be good at even detecting such finite-time blow-ups. In this work, we embark on a program to investigate this stability of PINNs from a rigorous theoretical viewpoint. Firstly, we derive error bounds for PINNs for Burgers’ PDE, in arbitrary dimensions, under conditions that allow for a finite-time blow-up. Our bounds give a theoretical justification for the functional regularization terms that have been reported to be useful for training PINNs near finite-time blow-up. Then we demonstrate via experiments that our bounds are significantly correlated to the ℓ2 -distance of the neurally found surrogate from the true blow-up solution, when computed on sequences of PDEs that are getting increasingly close to a blow-up.

Novel complexiton solutions to the new extended (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for incompressible fluid

This letter focuses on exploring some novel exact solutions to the new extended (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation (BLMPE) that has a major role in incompressible fluid. Adopting the Cole-Hopf transformation, the bilinear form of the considered equation is successfully constructed. Then the singular complexiton solutions (SCSs) are developed by applying the ansatz function method with a novel functional form. Additionally, we also find the non-singular complexiton solutions (NCSs) via imposing the restrictive conditions. To unveil the behaviors of the attained solutions better, the corresponding outlines are plotted via Maple.

Structural design and surface networking analysis of a truss antenna based on hexagonal frustum deployable mechanism units

With the development of space technology, deployable mechanism has been widely applied in large aperture space antennas. In this paper, a hexagonal frustum deployable antenna unit and an innovative modular assembling concept are proposed for large parabolic truss antenna. Screw topology diagram and screw constraint matrix are established to solve degree of freedom (DOF) of the antenna mechanism. By separating column vectors of screw constraint matrix, whole actuations of antenna mechanism are obtained according to unique solution condition. A new hexagon antenna network method based on hexagon frustum units and rotation transformation matrix is proposed, and correctness and efficiency of the modeling principle are proved by simulation. Finally, a prototype of the proposed hexagonal frustum truss deployable antenna mechanism is constructed, and a deployment experiment is conducted, which demonstrate the mobility and deployment performance of the mechanism. The mechanism proposed in this paper provides a theoretical reference for the parabolic truss deployable antenna mechanism, which is potentially valuable in the application of large aperture deployable antennas.

Improved weighted nuclear norm with total variation for removing multiplicative noise

This paper introduces an improved weighted nuclear norm with a total variation model tailored for removing multiplicative noise. The model incorporates a weight matrix to regularize the residual matrix, effectively leveraging image redundancy to differentiate various statistical properties of the noise. Since there is no guarantee of a unique solution, the model is reformulated as a linear equality constraint problem and decomposed into two subproblems. These are addressed by using the alternating direction method of multipliers and the split Bregman method, respectively. In addition, each alternative update step has a closed-form and convergent solution. After obtaining the denoised image in the log-domain, the recovered image is given by using the exponential function and bias correction. Experimental evaluations demonstrate the efficacy of our algorithms in enhancing image restoration quality.

Tensile properties of helical carbon fiber tows

The carbon fiber used in the load-bearing structural components often exhibits a relatively low toughness and a limited elongation at the point of failure within its carbon fiber tows. To address this challenge, we introduced a bionic helical structure as an innovative alternative to the conventional straight carbon fiber tows, resulting in a novel material characterized by enhanced strength and toughness. Pioneering a theoretical model, we delved into the effects of the helical angle on the tensile mechanical properties of helical carbon fiber tows for the first time. Subsequently, we conducted experimental trials to validate our theoretical findings and identified an optimal range for the helical angle. The optimized helical angle enhances additional lateral interactions, improving the bonding between the fibers and the resin. The characteristics of the helical structure increase the overall fracture elongation of the fiber tows. Tensile tests revealed a remarkable 10.8 % increase in maximum tensile strength and a significant 46 % enhancement in toughness when the tows were fully impregnated with epoxy resin using this optimal helical angle. Incorporating the helical structures at this optimized angle represents a breakthrough in improving the mechanical properties of carbon fiber tows, offering a unique solution to strike a balance between strength and toughness in carbon fiber composites.

Gravitational deflection of massive body around naked singularity

Spacetime singularity is not observed directly as it is covered by event horizon. Naked singularities may be formed due to gravitational collapse without an event horizon. Therefore, cosmic censorship conjecture entails advanced ascertainment. Recently, Joshi et al. have discovered a naked singularity solution (known as JMN naked singularity), which emerges from the dynamical gravitational collapse of a massive matter cloud. Since several studies are needed to know more extra characteristics that may help to test cosmic censorship conjecture in general relativity, therefore, we thought it is pertinent to study the deflection of massive particles by the naked singularity. We shall examine massive particles paths and deflection angles near naked singularities in this research. To get the deflection angle, we employ the Gauss-Bonet technique and the Rindler-Ishak method. The results of the two procedures are compared in detail.

Asymptotic Behavior of Singular Solution to the k-Hessian Equation with a Matukuma-Type Source

This paper is concerned with radially positive solutions of the [see formula in PDF]-Hessian equation involving a Matukuma-type source [see formula in PDF], where [see formula in PDF] is the [see formula in PDF]-Hessian operator, [see formula in PDF], and [see formula in PDF] is a suitable bounded domain in [see formula in PDF]. It turns out that there are two different types of radially positive solutions for [see formula in PDF], i.e., M-solution (singular at [see formula in PDF]) and E-solution (regular at [see formula in PDF]), which is distinct from the case when [see formula in PDF]. For [see formula in PDF], we apply an iterative approach to improve accuracy of asymptotic expansions of M-solution step by step to the desired extend. In contrast to the case [see formula in PDF], we require a more precise range of parameters due to repeated application of Taylor expansions, which also makes asymptotic expansions need more delicate investigation.

Intrinsically Decoupled Coordination Chemistries Enable Quasi-Eutectic Electrolytes with Fast Kinetics toward Enhanced Zinc-Ion Capacitors.

Eutectic electrolytes show potential beyond conventional low-concentration electrolytes (LCEs) in zinc (Zn)-ion capacitors (ZICs) yet suffer from high viscosity and sluggish kinetics. Herein, we originally develop an intrinsically decoupling strategy to address these issues, producing a novel electrolyte termed "quasi-eutectic" electrolyte (quasi-EE). Joint experimental and theoretical analyses confirm its unique solution coordination structure doped with near-LCE domains. This enables the quasi-EE well inherit the advanced properties at deep-eutectic states while provide facilitated kinetics as well as lower energy barriers via a vehicle/hopping-hybridized charge transfer mechanism. Consequently, a homogeneous electroplating pattern with much enhanced Sandꞌs time is achieved on the Zn surface, followed by a twofold prolonged service-life with drastically reduced concentration polarization. More encouragingly, the quasi-EE also delivers increased capacitance output in ZICs, which is elevated by 12.4%-144.6% compared to that before decoupling. Furthermore, the pouch cell with a cathodic mass loading of 36.6 mg cm-2 maintains competitive cycling performances over 600 cycles, far exceeding other Zn-based counterparts. This work offers fresh insights into eutectic decoupling and beyond.

Inverse kinematics solution and control method of 6-degree-of-freedom manipulator based on deep reinforcement learning

The advent of Industry 4.0 has significantly promoted the field of intelligent manufacturing, which is facilitated by the development of new technologies are emerging. Robot technology and robot intelligence methods have rapidly developed and been widely applied. Manipulators are widely used in industry, and their control is a crucial research topic. The inverse kinematics solution of manipulators is an important part of manipulator control, which calculates the joint angles required for the end effector to reach a desired position and posture. Traditional inverse kinematics solution algorithms often face the problem of insufficient generalization, and iterative methods have challenges such as large computation and long solution time. This paper proposes a reinforcement learning-based inverse kinematics solution algorithm, called the MAPPO-IK algorithm. The algorithm trains the manipulator agent using the MAPPO algorithm and calculates the difference between the end effector state of the manipulator and the target posture in real-time by designing a reward mechanism, while considering Gaussian distance and cosine distance. Through experimental comparative analysis, the feasibility, computational efficiency, and superiority of this reinforcement learning algorithm are verified. Compared with traditional inverse kinematics solution algorithms, this method has good generalization and supports real-time computation, and the obtained result is a unique solution. Reinforcement learning algorithms have better adaptability to complex environments and can handle different sudden situations in different environments. This algorithm also has the advantages of path planning, intelligent obstacle avoidance, and other advantages in dynamically processing complex environmental scenes.

Existence and uniqueness of well-posed fractional boundary value problem.

In this paper, the existence and uniqueness of solution for a fractional differential model involving well-posed boundary conditions and implicit fractional differential equation is considered. The desired goals are achieved by using Banach contraction principle and Scheafer's fixed point theorem. To show the results applicability some examples are presented. The basic mathematical concept of well-posed fractional boundary value issues is investigated in this study. It dives into the existence and uniqueness of these difficulties, offering light on the conditions that allow for both practical and singular solutions. This study contributes to a better knowledge of fractional calculus and its applications in a variety of scientific and technical areas, giving significant insights for both scholars and practitioners.

On optical soliton solutions of the higher-order Lakshmanan-Porsezian-Daniel model having the cubic-quintic-septic law in the presence of spatio-temporal and chromatic dispersion

The higher-order Lakshmanan-Porsezian-Daniel equation (LPDE) with the cubic-quintic-septic (CQS) law having spatiotemporal and chromatic dispersion terms (STD-CD) is examined to derive new optical soliton solutions. To accomplish this aim, we operated on a simple version of the new extended auxiliary equation method (SAEM26). The optical soliton solutions of the LPDE with CQS as well as STD-CD are constructed in detail. Moreover, 3D-surface, contour, and 2D plots are presented for the bright and periodic singular soliton solutions. Additionally, the effects of diverse model parameters on the bright soliton structure are surveyed, and these effects are displayed with 2D graphics. The findings established in this work can positively contribute to research in nonlinear optics, while the SAEM26 can be effectively applied to similar nonlinear models.

Novel analytic solutions of strain wave model in micro-structured solids

Abstract In this article, the modified extended direct algebraic method is implemented to investigate the strain wave model that governs the wave propagation in micro-structured solids. The proposed method provides many new exact traveling wave solutions with certain free parameters. Exact solutions are extremely important in interpreting the inner structures of the natural phenomena. Solitary and other wave solutions are provided for this model, such as bright solitary solutions, dark solitary solutions, singular solitary solutions, singular-dark combo solitary solutions. Also, periodic solutions and Jacobi elliptic function solutions are presented. To show the physical characteristics of the raised solutions, the graphical illustration of some solutions is presented.

Ethosome: An Innovative Perspective of Drug Delivery.

Ethosome offers a unique solution to the challenges faced by conventional drug delivery systems. In comparison to traditional liposomes and other nanocarriers, ethosomes exhibit a unique ability to improve drug absorption, overcoming a major limitation in the Transdermal Drug Delivery System. Their soft and flexible nano-vesicular structure facilitates faster permeation, resulting in significantly higher transdermal flux. This scientific investigation effectively traverses the changing landscape of ethosomes as an innovative drug delivery system. By conducting a thorough comparative analysis, we uncover the distinct characteristics that set them apart from other nanocarriers, offering insights into their distinct advantages. The study also includes a detailed analysis of the variables that have a complex impact on performance, elucidating transport mechanisms and addressing advanced facets pivotal for refined drug delivery strategies. This comprehensive overview highlights ethosomes as a future of medicine, offering a promising future for the safe and effective treatment of diverse diseases, impacting numerous lives.

Optimizing Advanced Trauma Life Support (ATLS®) to Maximize Readiness.

The Army utilizes Individual Critical Task Lists (ICTLs) to track and ensure competency and deployment readiness of its medical service members. ICTLs are the various skills and procedures that the Army has deemed foundational for each area of concentration (AOC)/military occupational specialty (MOS). While many ICTLs involve the patient care that military medical providers regularly provide, some procedures are not as commonly performed. This, when coupled with lower patient volume at military treatment facilities (MTF), poses a challenge for maintaining skill competency and deployment readiness. Fort Campbell's Blanchfield Army Community Hospital (BACH) has created a holistic and unique solution to meet many of these standardized requirements and support a ready medical force. By optimizing the Advanced Trauma Life Support (ATLS®) course curriculum to facilitate ICTL completion, BACH has increased its ICTL completion rates, ATLS® course exposure, and streamlined training requirements. The purpose of this article is to describe this best practice and suggest its applicability to other MTFs. By cross-referencing the ATLS® course curriculum and appendices with ICTLs, BACH has augmented ATLS® course certification with the additional completion of 12 ICTLs. This new approach not only increases ICTL completion, but also increases ATLS® curriculum exposure to medical providers, such as Registered Nurses or Nurse Practitioners, who would not typically take ATLS®. Since starting this new approach in April 2021, 73 military medical personnel have completed the ATLS® course at BACH, with 24 different medical specialties represented. A total of 361 ICTLs have been completed with specific ICTL completion counts ranging from 13 to 48. Each ICTL tested was completed 100% of its annual requirement. ATLS® is a mandatory joint interoperability standard for military physicians and it is also an Army ICTL for many AOCs/MOSs. Only counting completion of this course as one ICTL is a missed opportunity for the time spent by Army medical providers and limits the exposure of ATLS® to select AOCs/MOSs. This optimized and novel approach has been successful at BACH, suggesting its applicability at other MTFs that serve as ATLS® testing sites.

High-performance PVDF/PVB blend membranes with tailored sponge-like structure and antifouling properties for beer filtration

Currently limited research reported the design of microfiltration (MF) membranes specially for handling complex food processing streams such as rough beer. This study aimed to address this knowledge gap by tailoring for the first time a new class of polyvinylidene fluoride/poly(vinyl butyral) (PVDF/PVB) blend MF membrane with highly controllable pore structure and improved anti-fouling properties, and demonstrate its potential in real rough beer filtration. Combining an unconventional nonsolvent thermally induced phase separation (NTIPS) method and a unique quinary dope solution (i.e., PVDF/PVB/PVP/DMAc/MgCl2) with lower critical solution temperature (LCST) feature, the roles of PVB content and coagulation temperature were investigated in terms of pore structure characteristics and hydrophilicity. Although both the control PVDF membrane M0 without PVB and M6 with optimal 6.0 % PVB achieved similar turbidity removal (98–100 %), M6 exhibited enhanced permeance ranging from 27 % to 12 % with increasing simulated feed complexity from containing dextran only, yeast only to yeast/dextran mixture. The contribution of fouling of these key beer components was analyzed, indicating that the co-existence of yeast and dextran significantly aggravated fouling. Combining with membrane autopsy, in real rough beer filtration M6 showed permeance of 26.7 L·m−2·h−1·bar−1 at 1 bar that was ∼70 % higher than M0. Although with ∼50 % less irreversible fouling than M0, water cleaning implemented on M6 only recovered 62.5 % of flux. The simultaneous investigations on alkaline cleaning and operating pressure found that: a reasonably low pressure seemed favorable, e.g., 63.6 L·m−2·h−1·bar−1 at 0.2 bar, achieving up to flux recovery of ∼91 % with only ∼10 % contributed by irreversible fouling. Overall this study offers a facile yet practical strategy to tailor membranes for beer filtration, serving as an inspiration for designing future membrane processes for treating complex food processing and industry waste streams.

Dynamics of Hepatitis B Virus Transmission with a Lévy Process and Vaccination Effects

This work proposes a novel stochastic model describing the propagation dynamics of the hepatitis B virus. The model takes into account numerous disease characteristics, and environmental disturbances were collected using Lévy jumps and the conventional Brownian motions. Initially, the deterministic model is developed, and the asymptotic behavior of the model’s solution near the equilibria is examined. The deterministic model is transformed into a stochastic model while retaining the Lévy jumps and conventional Brownian motions. Under specific assumptions, the stochastic system is shown to have a unique solution. The study further investigates the conditions that ensure the extinction and persistence of the infection. The numerical solutions to both stochastic and deterministic systems were obtained using the well-known Milstein and RK4 techniques, and the analytical findings are theoretically confirmed. The simulation suggests that the noise intensities have a direct relationship with the amplitudes of the stochastic curves around the equilibria of the deterministic system. Smaller values of the intensities imply negligible fluctuations of trajectories around the equilibria and, hence, better describe the extinction and persistence of the infection. It has also been found that both Brownian motions and the Lévy jump had a significant influence on the oscillations of these curves. A discussion of the findings of the study reveals other important aspects as well as some future research guidelines. In short, this study proposes a novel stochastic model to describe the propagation dynamics of the hepatitis B virus.

Dynamics of invariant solutions of the DNA model using Lie symmetry approach

The utilization of the Lie group method serves to encapsulate a diverse array of wave structures. This method, established as a robust and reliable mathematical technique, is instrumental in deriving precise solutions for nonlinear partial differential equations (NPDEs) across a spectrum of domains. Its applications span various scientific disciplines, including mathematical physics, nonlinear dynamics, oceanography, engineering sciences, and several others. This research focuses specifically on the crucial molecule DNA and its interaction with an external microwave field. The Lie group method is employed to establish a five-dimensional symmetry algebra as the foundational element. Subsequently, similarity reductions are led by a system of one-dimensional subalgebras. Several invariant solutions as well as a spectrum of wave solutions is obtained by solving the resulting reduced ordinary differential equations (ODEs). These solutions govern the longitudinal displacement in DNA, shedding light on the characteristics of DNA as a significant real-world challenge. The interactions of DNA with an external microwave field manifest in various forms, including rational, exponential, trigonometric, hyperbolic, polynomial, and other functions. Mathematica simulations of these solutions confirm that longitudinal displacements in DNA can be expressed as periodic waves, optical dark solitons, singular solutions, exponential forms, and rational forms. This study is novel as it marks the first application of the Lie group method to explore the interaction of DNA molecules.

Dynamical Behaviors of Stochastic SIS Epidemic Model with Ornstein–Uhlenbeck Process

Controlling infectious diseases has become an increasingly complex issue, and vaccination has become a common preventive measure to reduce infection rates. It has been thought that vaccination protects the population. However, there is no fully effective vaccine. This is based on the fact that it has long been assumed that the immune system produces corresponding antibodies after vaccination, but usually does not achieve the level of complete protection for undergoing environmental fluctuations. In this paper, we investigate a stochastic SIS epidemic model with incomplete inoculation, which is perturbed by the Ornstein–Uhlenbeck process and Brownian motion. We determine the existence of a unique global solution for the stochastic SIS epidemic model and derive control conditions for the extinction. By constructing two suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process, we establish sufficient conditions for the existence of stationary distribution, which means the disease will prevail. Furthermore, we obtain the exact expression of the probability density function near the pseudo-equilibrium point of the stochastic model while addressing the four-dimensional Fokker–Planck equation under the same conditions. Finally, we conduct several numerical simulations to validate the theoretical results.