Hyperbolic Reaction Research Articles

A hyperbolic reaction–diffusion model of chronic wasting disease

A hyperbolic reaction–diffusion model of chronic wasting disease

Enhancing photocatalytic performance of regular porous silver bromide structures through 3D printing

This study introduces the innovative method of in-situ precipitation (ISP) 3D printing technology for fabricating photocatalyst structures capable of efficiently degrading water pollutants. The technology involves using silver nitrate aqueous solution to print directly on the surface of solidified NaBr aqueous solution, resulting in the formation of regular porous silver bromide structures. The structures are then densified through drying and sintering, followed by photoreduction using ultraviolet radiation to create Ag@(silver atom cluster)AgBr photocatalyst structures. Experimental results demonstrate the effectiveness of this photocatalyst structure in degrading azo dyes (Orange II) and eliminating Escherichia coli (E. coli) under visible light and ultraviolet light irradiation. The degradation of azo dyes and the sterilization of E. coli follow first-order kinetic reactions, with the photocatalyst module maintaining an 84.8% degradation rate after five recycling cycles. Complete elimination E. coli of the photocatalytic module is achieved within 120 min, and its sterilization effect follows a hyperbolic reaction. The study further confirms the durability and reusability of the photocatalyst structure, highlighting ISP 3D printing technology as a promising approach for fabricating efficient photocatalyst structures to address water pollution challenges in water treatment and environmental protection.

Dynamic characteristics of a hyperbolic reaction–diffusion predator–prey system with self‐diffusion and nonidentical inertia

Hyperbolic reaction–diffusion (HRD) systems have emerged as a better descriptor of the macroscopic spatial interaction models used for studying pattern formation in chemical and biological systems. In contrast to the parabolic reaction–diffusion (PRD) models, the spatial disturbances in an HRD system travel through space with finite velocity. This paper considers the simplest two‐species HRD model with different inertia based on the telegraph equation. The underlying reaction terms are considered as a predator–prey interaction with type III response function and prey refuge. We prescribe the analytical conditions for the existence of diffusion‐driven instabilities of the considered system. Simulation results further verify the theoretical results. A connection between the refuge parameter and inertial time is discussed in generating different spatiotemporal patterns. Extension of the two‐species PRD system to HRD system with diagonal diffusion matrix causes a diffusion‐driven wave instability, which is never possible for its parabolic counterpart. Furthermore, the patterns under pure wave instability are dependent on the initial population density.

Rhombic and hexagonal pattern formation in 2D hyperbolic reaction–transport systems in the context of dryland ecology

In this paper the formation of rhombic and hexagonal Turing patterns in bidimensional hyperbolic reaction–transport systems is addressed in the context of dryland ecology. To this aim, exploiting the guidelines of the Extended Thermodynamics Theory, a hyperbolic extension of the classical modified Klausmeier vegetation model for flat arid environments is here proposed. This framework accounts explicitly for the inertial effects associated with the involved species and allows a more suitable description of transient dynamics between a spatially-homogeneous steady-state towards a patterned configuration. To characterize the emerging Turing patterns, linear stability analysis on the uniform steady-states is firstly addressed. Then, to determine the Stuart–Landau equations ruling the time evolution of the pattern amplitudes close to onset, multiple-scale weakly nonlinear analysis is performed for the two abovementioned geometries. Finally, to validate the analytical predictions as well as to provide more insights into the spatio-temporal evolution of the resulting pattern in both transient and stationary regimes, numerical simulations have been also performed.

Instabilities in hyperbolic reaction–diffusion system with cross diffusion and species-dependent inertia

The hyperbolic reaction–diffusion (HRD) equation may overcome the physical shortcomings of the parabolic reaction–diffusion (PRD) equation, where the initially localized disturbance propagates infinitely fast through space. Instead, species often exhibit inertia, resulting in delayed effect in their spatial movement. Incorporating such response time for the onset of species flow due to a concentration gradient leads to an HRD equation with inertia. In this paper, we develop the general theory for pattern-forming instabilities in a two-species HRD system, which becomes a PRD system in the limiting case, with cross-diffusion and species-dependent inertia to explore how they play a role in the pattern forming instabilities. In particular, we determine various criteria for diffusion-induced instabilities (like Turing, wave, wave–Turing) and Hopf-induced instabilities (like pure Hopf, Hopf–wave, Hopf–Turing, and Hopf–wave–Turing) arise due to the cross-diffusion and inertial time. The theoretical results are demonstrated with an example where the Brusselator system represents the local interaction.

Eckhaus instability of stationary patterns in hyperbolic reaction–diffusion models on large finite domains

We have theoretically investigated the phenomenon of Eckhaus instability of stationary patterns arising in hyperbolic reaction–diffusion models on large finite domains, in both supercritical and subcritical regime. Adopting multiple-scale weakly-nonlinear analysis, we have deduced the cubic and cubic–quintic real Ginzburg–Landau equations ruling the evolution of pattern amplitude close to criticality. Starting from these envelope equations, we have provided the explicit expressions of the most relevant dynamical features characterizing primary and secondary quantized branches of any order: stationary amplitude, existence and stability thresholds and linear growth rate. Particular emphasis is given on the subcritical regime, where cubic and cubic–quintic Ginzburg–Landau equations predict qualitatively different dynamical pictures. As an illustrative example, we have compared the above-mentioned analytical predictions to numerical simulations carried out on the hyperbolic modified Klausmeier model, a conceptual tool used to describe the generation of stationary vegetation stripes over flat arid environments. Our analysis has also allowed to elucidate the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips. In particular, we have inspected the functional dependence of time and location at which wavelength adjustment takes place as well as the possibility to control these quantities, independently of each other.

Turing and wave instabilities in hyperbolic reaction–diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation

Hyperbolic reaction–diffusion equations have recently attracted attention both for their application to a variety of biological and chemical phenomena, and for their distinct features in terms of propagation speed and novel instabilities not present in classical two-species reaction–diffusion systems. We explore the onset of diffusive instabilities and resulting pattern formation for such systems. Starting with a rather general formulation of the problem, we obtain necessary and sufficient conditions for the Turing and wave instabilities in such systems, thereby classifying parameter spaces for which these diffusive instabilities occur. We find that the additional temporal terms do not strongly modify the Turing patterns which form or parameters which admit them, but only their regions of existence. This is in contrast to the case of additional space derivatives, where past work has shown that resulting patterned structures are sensitive to second-order cross-diffusion and first-order advection. We also show that additional temporal terms are necessary for the emergence of spatiotemporal patterns under the wave instability. We find that such wave instabilities exist for parameters which are mutually exclusive to those parameters leading to stationary Turing patterns. This implies that wave instabilities may occur in cases where the activator diffuses faster than the inhibitor, leading to routes to spatial symmetry breaking in reaction–diffusion systems which are distinct from the well studied Turing case.

Application of high specific surface area Ag/AgCl/TiO2 coupled photocatalyst fabricated by fused filament fabrication

This study used a three-dimensional (3D) printing process to develop the Ag/AgCl/TiO2 coupled photocatalyst with a specific surface area of 748 mm2/g. We examined the catalytic capability of this photocatalyst in degrading methyl blue (MB) dye and sterilizing Escherichia coli as well as the durability and reliability of its continuous use. A TiO2 module was constructed through fused filament fabrication (FFF), and the adopted 3D printing filament was composed of anatase TiO2 nanoparticles, stearic acid, wax, and a plasticizer. The green compact of the TiO2 module was subjected to solvent debinding, thermal debinding, and sintering to obtain a fundamental structure that was subsequently coupled with AgCl through a precipitation reaction. Ultraviolet radiation was used for the photoreduction to obtain the Ag/AgCl/TiO2 coupled photocatalyst coupling module. This photocatalyst can effectively degrade MB dye and disinfect E. coli. The degradation of MB dye and sterilization of E. coli were conducted under visible and ultraviolet light. The degradation of MB dye by the Ag/AgCl/TiO2 coupled photocatalyst was a first-order reaction. In addition, this coupled photocatalyst could retain its MB dye degradation rate (95%) for five cycles. E. coli was sterilized using the prepared photocatalytic module in a 120-min test, and this sterilization phenomenon could be presented as a hyperbolic reaction. The photocatalytic module manufactured in this study through FFF could efficiently degrade pollutants in water, and its durability and reliability after repeated use have been approved.

Long-time solutions of scalar hyperbolic reaction equations incorporating relaxation and the Arrhenius combustion nonlinearity

Abstract We consider an initial-value problem based on a class of scalar nonlinear hyperbolic reaction–diffusion equations of the general form $$\begin{align*} & u_{\tau\tau}+u_{\tau}=u_{{xx}}+\varepsilon (F(u)+F(u)_{\tau} ), \end{align*}$$in which ${x}$ and $\tau $ represent dimensionless distance and time, respectively, and $\varepsilon>0$ is a parameter related to the relaxation time. Furthermore, the reaction function, $F(u)$, is given by the Arrhenius combustion nonlinearity, $$\begin{align*} & F(u)=e^{-{E}/{u}}(1-u), \end{align*}$$in which $E>0$ is a parameter related to the activation energy. The initial data are given by a simple step function with $u({x},0)=1$ for ${x} \le 0$ and $u({x},0)=0$ for ${x}> 0$. The above initial-value problem models, under certain simplifying assumptions, combustion waves in premixed gaseous fuels; here, the variable $u$ represents the non-dimensional temperature. It is established that the large-time structure of the solution to the initial-value problem involves the evolution of a propagating wave front, which is of reaction–diffusion or reaction–relaxation type depending on the values of the problem parameters $E$ and $\varepsilon $.

Simulation of Biochemical Reactions with ANN-Dependent Kinetic Parameter Extraction Method

The measurement of thermodynamic properties of chemical or biological reactions were often confined to experimental means, which produced overall measurements of properties being investigated, but were usually susceptible to pitfalls of being too general. Among the thermodynamic properties that are of interest, reaction rates hold the greatest significance, as they play a critical role in reaction processes where speed is of essence, especially when fast association may enhance binding affinity of reaction molecules. Association reactions with high affinities often involve the formation of a intermediate state, which can be demonstrated by a hyperbolic reaction curve, but whose low abundance in reaction mixture often preclude the possibility of experimental measurement. Therefore, we resorted to computational methods using predefined reaction models that model the intermediate state as the reaction progresses. Here, we present a novel method called AKPE (ANN-Dependent Kinetic Parameter Extraction), our goal is to investigate the association/dissociation rate constants and the concentration dynamics of lowly-populated states (intermediate states) in the reaction landscape. To reach our goal, we simulated the chemical or biological reactions as system of differential equations, employed artificial neural networks (ANN) to model experimentally measured data, and utilized Particle Swarm Optimization (PSO) algorithm to obtain the globally optimum parameters in both the simulation and data fitting. In the Results section, we have successfully modeled a protein association reaction using AKPE, obtained the kinetic rate constants of the reaction, and constructed a full concentration versus reaction time curve of the intermediate state during the reaction. Furthermore, judging from the various validation methods that the method proposed in this paper has strong robustness and accuracy.

Diffusive instability in hyperbolic reaction-diffusion equation with different inertia.

This work considers a two-dimensional hyperbolic reaction-diffusion system with different inertia and explores criteria for various instabilities, like a wave, Turing, and Hopf, both theoretically and numerically. It is proven that wave instability may occur in a two-species hyperbolic reaction-diffusion system with identical inertia if the diffusion coefficients of the species are nonidentical but cannot occur if diffusion coefficients are identical. Wave instability may also arise in a two-dimensional hyperbolic reaction-diffusion system if the diffusivities of the species are equal, which is never possible in a parabolic reaction-diffusion system, provided the inertias are different. Interestingly, Turing instability is independent of inertia, but the stability of the corresponding local system depends on the inertia. Theoretical results are demonstrated with an example where the local interaction is represented by the Schnakenberg system.

Large-time solutions of a class of scalar, nonlinear hyperbolic reaction–diffusion equations

Large-time solutions of a class of scalar, nonlinear hyperbolic reaction–diffusion equations

On the propagation and bifurcation of singular surface shocks under a class of wave equations based on second-sound flux models and logistic growth

Working in the context of hyperbolic reaction–diffusion–acoustic theory, we present a detailed study of singular surface shock phenomena under two hyperbolic versions of the Fisher–KPP equation, both of which are based on flux laws originally used to describe the phenomenon of second-sound. Employing both analytical and numerical methods, we investigate the propagation, evolution, and qualitative behavior of density shocks, i.e., propagating surfaces across which the density exhibits a jump, under the two models considered. In the process, we identify a class of wave equations with the property that members of the class all exhibit the same shock structure, evolution, and velocity. Lastly, possible follow-on investigations are proposed and applications to other areas of continuum physics are noted.

Pattern formation in hyperbolic models with cross-diffusion: Theory and applications

A class of hyperbolic reaction–diffusion models with cross-diffusion is derived within the context of Extended Thermodynamics.Linear stability analysis on the uniform steady states is performed to derive the conditions for the occurrence of Hopf, Turing and wave instabilities. The weakly nonlinear analysis is then employed to describe the time evolution of the pattern amplitude close to the stability threshold. The effects of the inertial times on the pattern formation as well as on the transient regimes are highlighted. As an illustrative example, our analysis is applied to the prototype Schnakenberg model and the theoretical results are illustrated both analytically and numerically.

Speed of domain walls in thin nanotubes: The transition from the linear to the magnonic regime

Numerical simulations of domain wall propagation in thin nanotubes when an external magnetic field is applied along the nanotube axis have shown an unexpected behavior described as a transition from a linear to a magnonic regime. As the applied magnetic field increases, the initial regime of linear growth of the speed with the field is followed by a sudden change in slope accompanied by the emission of spin waves. In this work an analytical formula for the speed of the domain wall that explains this behavior is derived by means of an asymptotic study of the Landau Lifshitz Gilbert equation for thin nanotubes. We show that the dynamics can be reduced to a one dimensional hyperbolic reaction diffusion equation, namely, the damped double Sine Gordon equation, which shows the transition to the magnonic regime as the domain wall speed approaches the speed of spin waves. This equation has been previously found to describe domain wall propagation in weak ferromagnets with the mobility proportional to the Dzyaloshinskii-Moriya interaction constant, for Permalloy nanotubes the mobility is proportional to the nanotube radius.

Long-time solutions of scalar nonlinear hyperbolic reaction equations incorporating relaxation I. The reaction function is a bistable cubic polynomial

We consider an initial-value problem based on a class of scalar nonlinear hyperbolic reaction–diffusion equations of the general formuττ+uτ=uxx+ε(F(u)+F(u)τ), in which x and τ represent dimensionless distance and time respectively and ε>0 is a parameter related to the relaxation time. Furthermore the reaction function, F(u), is given by the bistable cubic polynomial,F(u)=u(1−u)(u−μ), in which 0<μ<1/2 is a parameter. The initial data is given by a simple step function with u(x,0)=1 for x≤0 and u(x,0)=0 for x>0. It is established, via the method of matched asymptotic expansions, that the large-time structure of the solution to the initial-value problem involves the evolution of a propagating wave front which is either of reaction–diffusion or of reaction–relaxation type. The one exception to this occurs when μ=12 in which case the large time attractor for the solution of the initial-value problem is a stationary state solution of kink type centred at the origin.

Wave features of a hyperbolic reaction–diffusion model for Chemotaxis

The Extended Thermodynamic theory is used to derive a hyperbolic reaction–diffusion model for Chemotaxis. Linear stability analysis is performed to study the nature of the equilibrium states against uniform and nonuniform perturbations. A particular emphasis is given to the occurrence of the Turing bifurcation. The existence of traveling wave solutions connecting the two steady states is investigated and the governing equations are numerically integrated to validate the analytical results. The propagation of plane harmonic waves is analyzed and the stability regions in terms of the model parameters are shown. The frequency dependence of the phase velocity and of the attenuation is also illustrated. Finally, in order to have a measure of the non linear stability, the propagation of acceleration waves is studied, the wave amplitude is derived and the critical time is evaluated.

Wave features of a hyperbolic model for the within-season dynamics of insect pathogens

Hyperbolic models are suitable for describing invasive phenomena with a well-defined boundary. In fact, for a class of hyperbolic reaction–diffusion models derived in the context of extended thermodynamics (ET), the non-existence of smooth travelling waves has been proved under suitable assumptions on the wave speed. In this paper a hyperbolic model for the within-season dynamics of insect pathogens is derived and smooth and discontinuous travelling wave solutions are investigated. Validation of the model in point is also accomplished by searching for numerical solutions of the system of PDEs.

The HearmscaÞa and the Handshake: Desire and Disruption in the Grendel Episode

This essay exposes a foundational misreading that occludes three puzzles in the Grendel episode: the anomaly of Beowulf’s hand-to-hand grip, the lack of counterattack, and the absence of heroic action in Grendel’s mutilation. I trace this misreading to Frederick Klaeber’s analysis of two analogue episodes from Grettir’s Saga. Within the saga I outline a shared plot structure that uses patterns of strategic interaction to confirm heroic agency. Conversely, I show how the Grendel episode subverts this masterplot, in favor of a sacred plot that confirms the agency of God. The shift is accomplished in a riddling design forged in Beowulf’s unwavering tenacity and Grendel’s unwavering attempt to flee. As this unvarying opposition replaces anticipated interaction, it charges Grendel’s hyperbolic reaction with enigma. The essential counter-clue comes, however, as demonic signs insinuate Grendel’s metaphysical antipathy to Beowulf’s inadvertent handfæstung (hand-fasting), an iconic gesture of covenant. Yet even as metaphysical tropes spotlight Grendel’s impotence, figures of equivalency extend to Beowulf, the Danes, and the Geats, eventually scuttling every confirmation of heroic agency. Accordingly, a farce of unintended outcomes ultimately subsumes the masterplot, as the tension between superhuman tenacity and superhuman desperation makes inevitable Grendel’s demonic self-destruction.

Exact solutions and qualitative features of nonlinear hyperbolic reaction—diffusion equations with delay

New classes of exact solutions to nonlinear hyperbolic reaction—diffusion equations with delay are described. All of the equations under consideration depend on one or two arbitrary functions of one argument, and the derived solutions contain free parameters (in certain cases, there can be any number of these parameters). The following solutions are found: periodic solutions with respect to time and space variable, solutions that describe the nonlinear interaction between a standing wave and a traveling wave, and certain other solutions. Exact solutions are also presented for more complex nonlinear equations in which delay arbitrarily depends on time. Conditions for the global instability of solutions to a number of reaction—diffusion systems with delay are derived. The generalized Stokes problem subject to the periodic boundary condition, which is described by a linear diffusion equation with delay, is solved.