Unique Solution Research Articles

Asymptotic stability results of generalized discrete time reaction diffusion system applied to Lengyel-Epstein and Dagn Harrison models

In this research paper, we delve into the analysis of a generalized discrete reaction-diffusion system. Our study begins with the discretization of a generalized reaction-diffusion model, achieved through second-order and L1-difference approximations. We explore the local stability of its unique solution, both in the absence and presence of the diffusion term. To determine the conditions for global asymptotic stability of the steady-state solution, we employ suitable techniques including the direct Lyapunov method. To illustrate the practical application of this theoretical framework, we provide several numerical simulations that examine both the Lengyel-Epstein reaction-diffusion model and the discrete Degn-Harrison reaction-diffusion model. These simulations serve to validate the predictions of asymptotic stability.

Solitary wave type solutions of nonlinear improved mKdV equation by modified techniques

In this paper, the improved and modified version of the Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM) are manipulated effectively and generously to determine the exact solitary wave soliton solutions of the improved modified KdV (mKdV) equation. The purpose of this study is to provide novel exact solutions to the improved mKdV equation. Specifically, we utilized IMSSEM and IGREMM to study different solutions of the nonlinear improved mKdV equation, focusing on exponential, trigonometric, and trigonometric hyperbolic type solutions. Furthermore, the plotting of various solutions for direct viewing analysis is provided in two and three-dimensional graphs. The new strategies are straightforward, quick, and efficient and have many other advantages, whereas, they provide the most accurate and unique solution to many other types of nonlinear partial differential equations (NPDEs), which usually arise in engineering and applied sciences. It should be noted that these methodologies are novel mathematical instruments that have shown to be the most effective mathematical tools for solving higher-order nonlinear partial differential equations in mathematical physics. Symbolic computation was used to validate all of the solutions that were established. Thus, it is also hoped that these techniques will ultimately reduce the cumbersome workload involved during the process of solutions to complicated NLPDEs.

Competitive exclusion and coexistence in a general two-species stochastic chemostat model with Markov switching

In this paper, a stochastic framework with a general nonmonotonic response function is formulated to investigate the competition dynamics between two species in a chemostat environment. The model incorporates both white noise and telegraph noise, the latter being described by Markov process. The existence of a unique global positive solution for the stochastic chemostat model is established. Subsequently, by using the ergodic theory of Markov process and utilizing techniques of stochastic analysis, the critical value differentiating between persistence in mean and extinction for the microorganism species is explored. Moreover, the existence of a unique stationary distribution is proved by using stochastic Lyapunov analysis. Finally, numerical simulations are introduced to support the obtained results.

The analysis of traveling wave structures and chaos of the cubic–quartic perturbed Biswas–Milovic equation with Kudryashov's nonlinear form and two generalized nonlocal laws

The cubic–quartic perturbed Biswas–Milovic equation, which contains Kudryashov's nonlinear form and two generalized nonlocal laws, has been explored qualitatively and quantitatively, as demonstrated in the present work. The research methods used include the complete discrimination system for polynomial method and the trial equation method. The results show that the Hamiltonian has the conservation property, and the global phase diagrams obtained via the bifurcation method reveal the existence of periodic and soliton solutions. Furthermore, we fully classify all the single traveling wave solutions to substantiate our findings, covering singular solutions, solitons, and Jacobian elliptic function solutions. We analyze their topological stabilities and present two‐dimensional graphs of solutions. We also delve deeper into the dynamic system by incorporating the perturbation item to explore the chaotic phenomena associated with the equation. These outcomes are valuable for studying the propagation of high‐order dispersive optical solitons and have potential applications in optimizing optical communication systems to improve efficiency.

Harnessing Nanoparticles to Overcome Antimicrobial Resistance: Promises and Challenges.

The rise of antimicrobial resistance (AMR) has become a serious global health issue that kills millions of people each year globally. AMR developed in bacteria is difficult to treat and poses a challenge to clinicians. Bacteria develop resistance through a variety of processes, including biofilm growth, targeted area alterations, and therapeutic drug alteration, prolonging the period they remain within cells, where antibiotics are useless at therapeutic levels. This rise in resistance is linked to increased illness and death, highlighting the urgent need for effective solutions to combat this growing challenge. Nanoparticles (NPs) offer unique solutions for fighting AMR bacteria. Being smaller in size with a high surface area, enhancing interaction with bacteria makes the NPs strong antibacterial agents against various infections. In this review, we have discussed the epidemiology and mechanism of AMR development. Furthermore, the role of nanoparticles as antibacterial agents, and their role in drug delivery has been addressed. Additionally, the potential, challenges, toxicity, and future prospects of nanoparticles as antibacterial agents against AMR pathogens have been discussed. The research work discussed in this review links with Sustainable Development Goal 3 (SDG-3), which aims to ensure disease-free lives and promote well-being for all ages.

Gradient conformal stationarity and the CMC condition in LRS spacetimes

Abstract We study the existence of gradient conformal Killing vectors (CKVs) in the class of locally rotationally symmetric (LRS) spacetimes which generalizes spherically symmetric spacetimes, and investigate some implications for the evolutionary character of marginally outer trapped surfaces. We first study existence of gradient CKVs via the obtention of a relationship between the Ricci curvature and the gradient of the divergence of the CKV. This provides an alternative set of equations, for which the integrability condition is obtained, to analyze the existence of gradient CKVs. A uniqueness result is obtained in the case of perfect fluids, where it is demonstrated that the Robertson-Walker solution is the unique perfect fluid solution with a nonvanishing pressure, admitting a timelike gradient CKV. The constant mean curvature condition for LRS spacetimes is also obtained, characterized by three distinct conditions which are specified by a set of three scalars. Linear combinations of these scalars, whose vanishing define the constant mean curvature condition, turn out to be related to the evolutions of null expansions of 2-spheres along their null normal directions. As such, some implications for the existence of black holes and the character of the associated horizons are obtained. It is further shown that dynamical black holes of increasing area, with a non-vanishing heat flux across the horizon, will be in equilibrium, with respect to the frame of the conformal observers.

Uniqueness of the static vacuum asymptotically flat spacetimes with massive particle spheres

In this paper, we establish that a four-dimensional static vacuum asymptotically flat spacetime containing a massive particle sphere is isometric to the Schwarzschild spacetime. Our results expand upon existing uniqueness theorems for static vacuum asymptotically flat spacetimes, which focus on scenarios featuring event horizons or photon spheres. Similarly to the uniqueness theorems concerning photon spheres or event horizons, only a single massive particle sphere is sufficient to obtain a unique solution. However, in contrast to previous theorems, our result leads to the existence of an entire spacetime foliation sliced by a set of massive particle spheres spanning various energies. Published by the American Physical Society 2024

Weak ψ-Contractions on Directed Graphs with Applications to Integral Equations

This article deals with a few outcomes ensuring the fixed points of a weak (G,ψ)-contraction map of metric spaces comprised with a reflexive and transitive digraph G. To validate our findings, we furnish several examples. The findings we obtain enable us to seek out the unique solution of a nonlinear integral equation. The outcomes presented herewith sharpen, subsume, unify, improve, enrich, and compile a number of existing theorems.

Hybrid-tracked finite-time path following control of underactuated underwater vehicles with 6-DOF

This paper introduces a novel hybrid-tracked finite-time path following control approach for controlling the surge speed and attitude angles of underactuated underwater vehicles with six degrees of freedom (6-DOF). This method integrates 6-DOF line-of-sight (LOS) guidance with nonzero roll dynamics. In kinematics layer, the Rodrigues rotation matrix ensures a unique solution and a slight desired roll angle in LOS guidance calculations. A second-order finite-time differentiator bridges the kinematics and dynamics layers, processing LOS guidance commands and ensuring differential errors converge to zero in finite time. In dynamics layer, a hybrid thrust calculation strategy achieves finite-time convergence of attitude angles in the inertial frame and surge speed in body-fixed frame. Extensive numerical simulations comparing various initial conditions and control strategies demonstrate the superior performance and efficiency of the proposed hybrid-tracked finite-time controller (HTFTTC). The HTFTTC significantly outperforms existing hybrid-tracked backstepping and direct finite-time controllers, reducing the root mean square error of attitude angles or speed by at least 67.2% in steady-state.

Duality and the well-posedness of a martingale problem

For two Polish state spaces EX and EY, and an operator GX, we obtain existence and uniqueness of a GX-martingale problem provided there is a bounded continuous duality function H on EX×EY together with a dual process Y on EY which is the unique solution of a GY-martingale problem. For the corresponding solutions (Xt)t≥0 and (Yt)t≥0, duality with respect to a function H in its simplest form means that the relation Ex[H(Xt,y)]=Ey[H(x,Yt)] holds for all (x,y)∈EX×EY and t≥0. While duality is well-known to imply uniqueness of the GX-martingale problem, we give here a set of conditions under which duality also implies existence without using approximating sequences of processes of a different kind (e.g. jump processes to approximate diffusions) which is a widespread strategy for proving existence of solutions of martingale problems. Given the process (Yt)t≥0 and a duality function H, to prove existence of (Xt)t≥0 one has to show that the r.h.s. of the duality relation defines for each y a measure on EX, i.e. there are transition kernels (μt)t≥0 from EX to EX such that Ey[H(x,Yt)]=∫μt(x,dx′)H(x′,y) for all (x,y)∈EX×EY and all t≥0.As examples, we treat resampling and branching models, such as the Fleming–Viot measure-valued diffusion and its spatial counterparts (with both, discrete and continuum space), as well as branching systems, such as Feller’s branching diffusion. While our main result as well as all examples come with (locally) compact state spaces, we discuss the strategy to lift our results to genealogy-valued processes or historical processes, leading to non-compact (discrete and continuum) state spaces. Such applications will be tackled in forthcoming work based on the present article.

XDL-ESI: Electrophysiological Sources Imaging via explainable deep learning framework with validation on simultaneous EEG and iEEG

Electroencephalography (EEG) or Magnetoencephalography (MEG) source imaging aims to estimate the underlying activated brain sources to explain the observed EEG/MEG recordings. Solving the inverse problem of EEG/MEG Source Imaging (ESI) is challenging due to its ill-posed nature. To achieve a unique solution, it is essential to apply sophisticated regularization constraints to restrict the solution space. Traditionally, the design of regularization terms is based on assumptions about the spatiotemporal structure of the underlying source dynamics. In this paper, we propose a novel paradigm for ESI via Explainable Deep Learning Framework, termed as XDL-ESI, which connects the iterative optimization algorithm with deep learning architecture by unfolding the iterative updates with neural network modules. The proposed framework has the advantages of (1) establishing a data-driven approach to model the source solution structure instead of using hand-crafted regularization terms; (2) improving the robustness of source solutions by introducing a topological loss that leverages the geometric spatial information applying varying penalties on distinct localization errors; (3) improving the reconstruction efficiency and interpretability as it inherits the advantages from both the iterative optimization algorithms (interpretability) and deep learning approaches (function approximation). The proposed XDL-ESI framework provides an efficient, accurate, and interpretable paradigm to solve the ESI inverse problem with satisfactory performance in both simulated data and real clinical data. Specially, this approach is further validated using simultaneous EEG and intracranial EEG (iEEG).

Schrödinger-Hirota equation in birefringent fibers with cubic-quantic nonlinearity and multiplicative white noise in the ito sense: Nucci’s reductions and soliton solutions

This paper explores innovative solutions for the Stochastic Schrödinger-Hirota equation within the context of birefringent fibers with cubic-quintic nonlinearity, emphasizing incorporating multiplicative white noise in the Itô sense. Leveraging the Nucci reduction method, the study focuses on obtaining exact solutions, shedding light on the intricate interplay between quantum mechanics and stochastic processes. The Nucci reduction method is a powerful tool to facilitate the derivation of precise solutions, showcasing its efficacy in unravelling complex mathematical structures and providing valuable insights into the behaviour of quantum systems under the influence of diverse parameters. In addition, two effective and convenient procedures are employed to extract bright, dark, and unique soliton solutions, as well as their combination. Exploring these solutions contributes to a deeper understanding of the equation’s dynamics, particularly in real-world applications such as quantum optics and condensed matter physics. Additionally, this study incorporates graphical depictions of specific solutions to demonstrate the effect of white noise on solitons visually.

Experimental and numerical investigation on fluid flow and heat transfer behaviour of foam cladded pin fin heat sink

As technology advances, there is a growing demand for more efficient cooling solutions in electronics. Open-cell metal foams offer a unique solution, combining the strength of metals with the characteristics of foams, including being lightweight and having a high heat transfer area-to-volume ratio. Fully covering heatsinks with porous materials results in high pressure drop values, which is undesirable for electronics cooling. This study proposes a novel design that incorporates cladded aluminum foams on the pin fins within a heatsink. The hydrothermal performance of the proposed design is experimentally and numerically investigated. The results indicate that increasing the cladding thickness enhances heat transfer performance while increasing the pressure drop. For a heatsink with a pin fin diameter of 5 mm at the Reynolds number of 1121, the foam thickness of 5 mm depicts 46.5 % higher hydrothermal performance compared to a pin fin heatsink without porous while a fully porous covered heatsink depicts 38.9 % higher hydrothermal performance.

The Stiffness and Damping Characteristics of a Rubber-Based SMA Composite Shock Absorber with a Hyper-Elastic SMA-Constitutive Model Considering the Loading Rate.

Shock absorbers are essential in enhancing vehicle ride comfort by mitigating vibrations. However, traditional rubber shock absorbers are constrained by their fixed stiffness and damping properties, limiting their adaptability to varying loads and thus affecting the ride comfort, especially under extreme road conditions. Shape Memory Alloys (SMAs), known for their intelligent material properties, offer a unique solution by adjusting stiffness and damping in response to temperature changes or strain rates, making them ideal for advanced vibration control applications. This study builds upon the Auricchio constitutive model to propose an enhanced SMA hyper-elastic constitutive model that accounts for different loading rates. This new model elucidates the impact of loading rates on the stiffness and damping characteristics of SMAs. Additionally, we introduce an innovative circular rubber-based SMA composite vibration reduction structure. Through a parameterized model and finite element simulation, we comprehensively analyze the stiffness and damping properties of the composite damper under various loading rates and harmonic excitations. Our findings suggest a novel approach to improving the vehicle ride comfort, offering significant potential for engineering applications and practical value.

Global solutions to the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner system with a class of large initial data

In this paper, we consider the global well-posedness of the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner (PTT) system with general initial data in the critical Besov spaces. The question of whether or not PTT system is globally well posed for large data is still open. When the density ρ is away from zero, we denote by ϱ:=1ρ−1. More precisely, we prove that the PTT system admits a unique global solution, provided that the initial data ϱ 0, the initial horizontal velocity uh0 , the product ωu30 of the coupling parameter ω and the initial vertical velocity u 30, and the initial symmetric tensor of constraints τ 0 are sufficiently small. In particular, this result includes the global well-posedness of the PTT system for small initial data in the case where ω∈[0,1) and for large initial vertical velocity in the case where ω tends to zero. As a by-product, our results can be applied to the so-called Oldroyd-B system.

Robust Recovery of Optimally Smoothed Polymer Relaxation Spectrum from Stress Relaxation Test Measurements.

The relaxation spectrum is a fundamental viscoelastic characteristic from which other material functions used to describe the rheological properties of polymers can be determined. The spectrum is recovered from relaxation stress or oscillatory shear data. Since the problem of the relaxation spectrum identification is ill-posed, in the known methods, different mechanisms are built-in to obtain a smooth enough and noise-robust relaxation spectrum model. The regularization of the original problem and/or limit of the set of admissible solutions are the most commonly used remedies. Here, the problem of determining an optimally smoothed continuous relaxation time spectrum is directly stated and solved for the first time, assuming that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. The relaxation time spectrum model that reproduces the relaxation modulus measurements and is the best smoothed in the class of continuous square-integrable functions is sought. Based on the Hilbert projection theorem, the best-smoothed relaxation spectrum model is found to be described by a finite sum of specific exponential-hyperbolic basis functions. For noise-corrupted measurements, a quadratic with respect to the Lagrange multipliers term is introduced into the Lagrangian functional of the model's best smoothing problem. As a result, a small model error of the relaxation modulus model is obtained, which increases the model's robustness. The necessary and sufficient optimality conditions are derived whose unique solution yields a direct analytical formula of the best-smoothed relaxation spectrum model. The related model of the relaxation modulus is given. A computational identification algorithm using the singular value decomposition is presented, which can be easily implemented in any computing environment. The approximation error, model smoothness, noise robustness, and identifiability of the polymer real spectrum are studied analytically. It is demonstrated by numerical studies that the algorithm proposed can be successfully applied for the identification of one- and two-mode Gaussian-like relaxation spectra. The applicability of this approach to determining the Baumgaertel, Schausberger, and Winter spectrum is also examined, and it is shown that it is well approximated for higher frequencies and, in particular, in the neighborhood of the local maximum. However, the comparison of the asymptotic properties of the best-smoothed spectrum model and the BSW model a priori excludes a good approximation of the spectrum in the close neighborhood of zero-relaxation time.

Structure of positive solutions for a reaction-diffusion model with additional food and protection zone

The additional food and protection zone provided to predator and prey respectively are introduced into a reaction-diffusion predation model with nonlinear functional response in this paper. We mainly concentrate on the combined effects of additional food and protection zone on the existence and uniqueness of solution for the model. It is shown that, compared with the models with classical Beddington-DeAngelis functional response, the integration of additional food and protection zone into Beddington-DeAngelis functional response causes the model to produce new critical values for the existence of positive solutions. Moreover, we derive that the model permits a unique positive solution when the strength of mutual interference among predators or the quantity of additional food is large, where, the effect of quantity of additional food on the positive solution is a new finding. The results shed light on that the combination of additional food and protection zone is an effective and environmentally-friendly strategy in preserving biodiversity.

Mean-field backward stochastic differential equations with mean reflection and nonlinear resistance

The present paper is devoted to the study of the well-posedness of mean field BSDEs with mean reflection and nonlinear resistance. By the contraction mapping argument, we first prove that the mean-field BSDE with mean reflection and nonlinear resistance admits a unique deterministic flat local solution on a small time interval. Moreover, we build the global solution by introducing a two-step approach, which is a combination of stitching method and fixed point method. We further provide an application to the super-hedging problem with risk constraint.

Unsupervised physics-informed deep learning for assessing pulmonary artery hemodynamics

Deep learning advancements have significantly benefited medical applications. One such helpful application is noninvasive fractional flow reserve (FFR) evaluation along the pulmonary artery tree, which aids in planning optimal balloon pulmonary angioplasty. This study proposes a surrogate model that employs an unsupervised physics-informed neural network (UPNN) to predict FFR based on pulmonary CT angiography. To ensure the UPNN strictly follows the unique solution of the governing equations, we implemented a hard boundary conditions enforcement approach. Subsequently, a finite difference convolutional filter was developed to enhance connectivity among neighboring points. This allows the neural networks to propagate boundary conditions into the unseen vessel interior and perceive the geometric structure of the computational domain. We also introduced regularization constraints on the three components contributing to pressure drop within the artery tree. A total of 4500 synthetic pulmonary artery trees were used to train the UPNN with impedance outlet boundary conditions. We found that the limits of agreement of FFR trained by UPNN ranged from −0.05 to 0.04 with computational fluid dynamics (CFD) simulation results. The testing results showed that the correlation coefficient between FFR predicted by UPNN and CFD were 96.1% and 94.7% for synthetic and patient-specific data, respectively. Using invasive FFR as reference, the accuracy of FFR predicted by UPNN and CFD was 81.4% and 84.8%, respectively. These results demonstrate that our UPNN-based surrogate model’s performance in evaluating FFR aligns closely with CFD simulations.

Probing the Activity and Selectivity of Nickel Carboxylethylphosphonate 2D-MOF for the Electrochemical Conversion of CO2 to Syngas

The increasing concentration of CO2 in the atmosphere is alarming for modern society and the reduction of CO2 into valuable products would be a unique solution.1 Metal–organic frameworks (MOFs), constructed from organic linkers interconnected with metal (oxide) nodes, with high porosity and large surface area, have become an emerging class of electrocatalysts for reduction of CO2.2,3 Herein, we report the synthesis and characterization of a two-dimensional nickel metal-organic framework (2D Ni-MOF) with 3-phosphopropionic acid and 4,4-bypyridine as ligands and Ni2+ as the node (figure 1). The electrocatalytic activity of this material is explored for the electrochemical CO2 reduction reaction (CO2RR). An onset potential for the electrochemical reduction CO2 activity of -0.51 V vs RHE was observed which is 70 mV more positive than the HER in Ar-saturated electrolyte suggesting an efficient CO2 reduction. This catalyst showed a high current density of 33.5 mA cm-2 at 1.01 V vs RHE, exhibiting superior performance to most of the MOFs tested so far for CO2 reduction (figure 2a and b). Long term electrolysis at -0.8 V (vs RHE) yielded a current density of ∼10 mA cm-2 with high selectivity towards the formation of CO and H2 with an average production ratio of 30:70%, respectively (figure 2c and d). This new Ni-MOF catalyst retained its crystallinity and morphology after electrolysis as confirmed by powder XRD, HRTEM, FT-IR and Raman spectroscopy, indicating promising stability of the catalyst.Density functional theory (DFT) was employed to calculate adsorption-free energy of the reaction intermediates to identify the active catalytic sites responsible for CO2 reduction in our 2D Ni-MOF structure. Findings show that the catalyst more readily reduces H* to H2 (with a limiting potential of 0.48 eV) than CO2 to COOH* (limiting potential of 0.78 eV). The reaction mechanism upon which this calculation was based begins with the removal of the H2O molecule bound to the Ni (II) site, followed by CO2 adsorption and the two proton-coupled electron transfer steps yielded COOH* and then CO & H2O. The H2O is more strongly adsorbed to the Ni (II) site than CO which closes the mechanistic cycle. Reference 1. M. J. B. Kabeyi and O. A. Olanrewaju, Frontiers in Energy Research, 2022, 9.2. B.-X. Dong, S.-L. Qian, F.-Y. Bu, Y.-C. Wu, L.-G. Feng, Y.-L. Teng, W.-L. Liu and Z.-W. Li, ACS Applied Energy Materials, 2018, 1, 4662-4669.3. J.-X. Wu, S.-Z. Hou, X.-D. Zhang, M. Xu, H.-F. Yang, P.-S. Cao and Z.-Y. Gu, Chemical Science, 2019, 10, 2199-2205. Figure 1