Nonlinear Analysis: Modelling and Control
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Generalized solutions for singular double-phase elliptic equations under mixed boundary conditions
Full text linkIn this article, we investigate at least one or two generalized solutions for double-phase singular elliptic equations with Hardy potential. We show the existence of at least one or two distinct generalized solutions under mixed boundary conditions via variational methods when the nonlinearity f satisfying suitable hypotheses
A study of nonlinear fractional-order biochemical reaction model and numerical simulations
Full text linkThis article depicts an approximate solution of systems of nonlinear fractional biochemical reactions for the Michaelis–Menten enzyme kinetic model arising from the enzymatic reaction process. This present work is concerned with fundamental enzyme kinetics, utilised to assess the efficacy of powerful mathematical approaches such as the homotopy perturbation method (HPM), homotopy analysis method (HAM), and homotopy analysis transform method (HATM) to get the approximate solutions of the biochemical reaction model with time-fractional derivatives. The Caputo-type fractional derivatives are explored. The proposed method is implemented to formulate a fractional differential biochemical reaction model to obtain approximate results subject to various settings of the fractional parameters with statistical validation at different stages. The comparison results reveal the complexity of the enzyme process and obtain approximate solutions to the nonlinear fractional differential biochemical reaction model
Soliton-like solutions supported by refined hydrodynamic-type model of an elastic medium with soft inclusions
Full text linkA nonlinear elastic medium containing sharp inhomogeneities is considered. The properties of a modified model of such a medium are investigated. The modification consists in including in the asymptotic equation of state those terms that were discarded in the previously considered models. The main purpose of the ongoing research is to analyze the existence, stability, and dynamic properties of soliton-like solutions within the modified model, as well as to compare these solutions with analogous solutions obtained in the previously considered models
Properties of Shannon and Rényi entropies of the Poisson distribution as the functions of intensity parameter
Full text linkWe consider two types of entropy, namely, Shannon and Rényi entropies of the Poisson distribution, and establish their properties as the functions of intensity parameter. More precisely, we prove that both entropies increase with intensity. While for Shannon entropy the proof is comparatively simple, for Rényi entropy, which depends on additional parameter α > 0, we can characterize it as nontrivial. The proof is based on application of Karamata’s inequality to the terms of Poisson distribution
Adaptive synchronization of quaternion-valued neural networks with reaction–diffusion and fractional order
Full text linkThis paper is dedicated to the study of adaptive finite-time synchronization (FTS) for generalized delayed fractional-order reaction–diffusion quaternion-valued neural networks (GDFORDQVNN). Utilizing the suitable Lyapunov functional, Green’s formula, and inequalities skills, testable algebraic criteria for ensuring the FTS of GDFORDQVNN are established on the basis of two adaptive controllers. Moreover, the numerical examples validate that the obtained results are feasible. Furthermore, they are also verified in image encryption as the application
Lie group analysis and its invariants for the class of multidimensional nonlinear wave equations
Full text linkWe systematically classify Lie symmetries of a class of (2 + 1)-dimensional nonlinear wave equations. Our methodology proposes a symmetry classification for Lie generators applicable to four distinct cases inherent within this equation. For each identified category, we comprehensively analyze symmetry reduction and delineate the invariant solutions. Furthermore, we extend our Lie symmetry analysis to encompass reduced 1 + 1 partial differential equations (PDEs). Through our investigations, we establish local conservation laws corresponding to each conserved vector, employing the formal Lagrangian approach. Significantly, this classification constitutes a novel contribution to the scientific discourse, as it remains absent from extant literature to date
Hysteresis and bistability in synaptic transmission modeled as a chain of biochemical reactions with a positive feedback
Full text linkIn this paper, we employ computational analysis to investigate the long-term potentiation (LTP) and memory formation in synapses between neurons. We use a mathematical model describing the synaptic transmission as a signal transduction pathway with a positive feedback loop formed by diffusion of nitric oxide (NO) to the presynaptic site. We found that the model of synaptic transmission exhibits a hysteresis-like behavior, where the strength of synaptic transmission depends not just on instantaneous interstimulus intervals, but also on the history of activity. The switching between resting and memory states can be induced by physiologically relevant and moderate (less than 50%) changes in the duration of interstimulus intervals
Fixed point theorems for xi-alpha-eta-Gamma F-fuzzy contraction with an application to neutral fractional integro-differential equation with nonlocal conditions
No full textIn this study, we define a new fuzzy contraction principle, namely, the concept of ξ-α-η-Γ F-mappings, and prove the existence and uniqueness of the fixed point for such class of mappings. To further demonstrate the validity of our results, we furnish an application to neutral fractional integro-differential equations with nonlocal conditions. The presented results unify, generalize, and enhance a number of prior findings in the literature
Synchronization of delayed stochastic reaction–diffusion Hopfield neural networks via sliding mode control
Full text linkSynchronization of stochastic reaction–diffusion Hopfield neural networks with s-delays via sliding mode control is investigated in this article. To begin with, we choose suitable functional space for state variables, then the system is transformed into a functional differential equation in an infinite-dimensional Hilbert space by using appropriate functional analysis technique. Based on above preliminary preparation, sliding mode control (SMC) is constructed to drive the error trajectory into the designed switching surface. Specifically, the switching surface is constructed as linear combination of state variables, which is related to control gains. Then novel SMC law is designed which involving delay, reaction diffusion term, and reaching law. Furthermore, the criterion of mean-square exponential synchronization for stochastic delayed reaction–diffusion Hopfield neural networks with s-delays is given in the form of matrix form. This criterion is less restrictive and easy to check in computer. Meanwhile, a different novel Lyapunov–Krasovskii functional (LKF) mixed with Itô’s formula, Young inequality, Hanalay inequality is employed in this proof procedure. At last, a numerical example is presented to validate the availability of theoretical result. The simulation is based on the finite difference method, and numerical result coincides with the theoretical result proposed
Abstract random differential equations with state-dependent delay using measures of noncompactness
Full text linkThis paper is devoted to the existence of random mild solutions for a general class of second-order abstract random differential equations with state-dependent delay. The technique used is a generalization of the classical Darbo fixed point theorem for Fréchet spaces associated with the concept of measures of noncompactness. An application related to partial random differential equations with state-dependent delay is presented