One-dimensional Reaction-diffusion Research Articles

Solution of Fishers equation using homotopy perturbation Aboodh transform method

In this study, a nonlinear one-dimensional reaction Diffusion Equation called Fisher’s equation is solved using Aboodh Transform coupled with HPM. This method is applied to the general one-dimensional Fishers equation. This technique is demonstrated with three unique examples with different initial conditions solved with the values of β=1 and a=0. This simplicity of Aboodh transform makes the rapid rate of convergence from approximate to exact solution answers makes it easy and dependable.

The numerical methods for solving of the one-dimensional anomalous reaction-diffusion equation

The numerical methods for solving of the one-dimensional anomalous reaction-diffusion equation

A deconvolution analysis strategy on the thermoreconstitution process of Co/Fe layered double hydroxides: Complementary correlations between physicochemical variations and thermokinetic analyses

A variety of physicochemical analyses were systematically conducted on the complex thermoreconstitution process of Co/Fe layered double hydroxides, and deconvolution analysis strategy was utilized in elucidating the reconstitution process. The thermal analysis of Co/Fe layered double hydroxides indicated that approximately 27.8 % weight loss, three weight loss peaks and two pronounced endothermic phenomena were detected via thermogravimetry, differential thermogravimetry, and differential scanning calorimetry, respectively. In the thermoreconstitution process of Co/Fe layered double hydroxides, scanning electron microscopy images indicated that the layered structure gradually collapsed, fractured into pieces, and partially melted. Meanwhile, significant X-ray diffraction peak broadening phenomena were detected and further confirmed by the d-spacing values in high-resolution transmission electron microscope images, which were mainly caused by the lattice expansion effect and the heterogeneous microstrain effect. Three reactions were deconvolved from the differential thermogravimetry curves, and the thermokinetic models of Reaction 2 indicated that volatile components in the form of H2O and CO2 escaped from the solid-state residuals via a one-dimensional diffusion reaction model. The major products in solid-state residuals were cubic Cobalt(III) oxide and Iron(II, III) oxide. The thermokinetic models played a complementary role in elucidating the physicochemical variations that occurred throughout the thermoreconstitution process of Co/Fe layered double hydroxides.

Mass transfer analyses of reactive boundary schemes for lattice Boltzmann method with staircase approximation

Lattice Boltzmann (LB) methods with reactive boundary conditions are widely used in pore-scale simulations of dissolution and ablation processes. The staircase approximation of curved boundary is often employed because of its simplicity in handling solid structure changes. In this work, the mass transfer of two typical LB reactive boundary schemes are analyzed for the staircase boundary. The Type I boundary scheme is based on relations of local distribution functions and a wet-node boundary mesh. The Type II boundary scheme adopts the half-way bounce-back scheme. Boundary concentrations are determined by finite difference, and a link-wise boundary mesh is used. The analyses demonstrate that for straight boundaries, both the boundary schemes have accurate mass transfer rates, which means the mass transfer calculated by exchanges of distribution functions is the same as that calculated by reaction rates. For curved boundaries with staircase approximation, including interfacial normal directions in the Type I boundary scheme can provide accurate mass transfer for inclined straight boundaries. However, if the staircase boundary geometry is used directly without normal directions, the reaction rate will be overestimated. One-dimensional and two-dimensional reaction-diffusion processes with dissolution are simulated to validate the analyses. Both the boundary schemes work well for one-dimensional simulations. For two-dimensional simulations, the Type II boundary scheme significantly overestimates the reaction rate, and stronger artificial anisotropic effects are observed. The Type I boundary scheme with normal directions has better performance, but error still exists.

Quantitative Analysis of the Complex Time Evolution of a Camphor Boat

The motion of a camphor boat on the water’s surface is a long-studied example of the direct transformation of chemical energy into a mechanical one. Recent experimental papers have reported a complex character of boat motion depending on the location of the camphor source. If the source is close to the stern, the boat moves at a constant speed. When it is shifted towards the boat center, oscillations of speed are observed. When the source is close to the boat center, pulses of speed followed by oscillations appear. Here, we focus on numerical simulations of camphor boat motion. We discuss approximations that allow us to reduce the numerical complexity of the problem and formulate a model in which the equation for boat velocity is coupled with a one-dimensional reaction–diffusion equation for camphor surface concentration. We scanned the phase space of model parameters and found the values that give qualitative agreement with the experiments. The model predicts all types of boat motion (continuous, oscillating, and pulsating) observed in experiments. Moreover, the model with selected parameter values shows that for specific locations of the camphor source, a spike in speed is followed by transient oscillations, which are an inherent part of speed relaxation.

Analytical solution of the space-time fractional reaction-diffusion equation with variable coefficients

In this paper, we solve the problem of an inhomogeneous one-dimensional fractional differential reaction-diffusion equation with variable coefficients (1.1)-(1.2) by the method of separation of variables (the Fourier method). The Caputo derivative and the Riemann-Liouville derivative are considered in the time and space directions, respectively. We prove that the obtained solution of the boundary-value problem satisfies the given boundary conditions. We discuss the convergence of the series defining the proposed solution.

2D effects enhance precision of gradient-based tissue patterning

Robust embryonic development requires pattern formation with high spatial accuracy. In epithelial tissues that are patterned by morphogen gradients, the emerging patterns achieve levels of precision that have recently been explained by a simple one-dimensional reaction-diffusion model with kinetic noise. Here, we show that patterning precision is even greater if transverse diffusion effects are at play in such tissues. The positional error, a measure for spatial patterning accuracy, decreases in wider tissues but then saturates beyond a width of about ten cells. This demonstrates that the precision of gradient-based patterning in two- or higher-dimensional systems can be even greater than predicted by 1D models, and further attests to the potential of noisy morphogen gradients for high-precision tissue patterning.

Lyapunov-based nonlinear boundary control design with predefined convergence for a class of 1D linear reaction-diffusion equations

In this paper, we treat the problem of Lyapunov-based nonlinear boundary stabilization of a class of one-dimensional reaction-diffusion systems with any predefined convergence (asymptotic or non-asymptotic). As an application, we focus on the non-asymptotic notions (finite-time and fixed-time) for which we give some particular explicit control designs followed by some numerical simulations. The key idea of our approach is to use a “spatially weighted L2-norm” as a Lyapunov functional to design a nonlinear controller and to ensure stability with any desired convergence.

On unstable contrast structures in one-dimensional reaction–diffusion–advection problems with discontinuous sources

On unstable contrast structures in one-dimensional reaction–diffusion–advection problems with discontinuous sources

Marangoni-driven nonlinear dynamics of bimolecular frontal systems: a general classification for equal diffusion coefficients.

When bimolecular fronts form in solutions, their dynamics is likely to be affected by chemically driven convection such as buoyancy- and Marangoni-driven flows. It is known that front dynamics in the presence of buoyancy-driven convection can be predicted solely on the basis of the one-dimensional reaction-diffusion concentration profiles but that those predictions fail for Marangoni-driven convection. With a two-dimensional reaction-diffusion-Marangoni convection model, we analyze here convective effects on the time scalings of the front properties, together with the influence of reaction reversibility and of the ratio of initial reactants' concentrations on the front dynamics. The effect of buoyancy forces is here neglected by assuming the reactive system to be in zero-gravity condition and/or the solution density to be spatially homogenous. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Using reaction-diffusion modeling to probe high-frequency cholinergic transmission

The vertebrate skeletal muscle fiber's endplate current is well-modeled by scaling up the spontaneous miniature endplate currents (mepcs) of voltage-clamped neuromuscular junctions (NMJ). Mepc characteristics couple tightly to binding of pre-synaptically-released acetylcholine (ACh) at post-synaptic receptors plus elimination of ACh by synaptic-cleft acetylcholinesterase (AChE). The question we address: Could the same “cholinergic machinery” muscles use for intermittent, relatively low-frequency endplate currents be redeployed to trigger the continual high-Hz output underlying electric organ discharge (EOD) in gymnotid electric fish like the glass knifefish, Eigenmannia? This question arises a) because, following a whole-genome duplication ∼¼ billion years ago, gymnotids (Amazon-Basin teleosts encompassing both Eigenmannia and their close relatives, the electric eels, Electrophorus) evolved non-contractile electric organs from the duplicate “muscle excitability-genome,” b) because Eigenmannia sustains EODs night-and-day at 200-600 Hz, and c) because supplying pre-synaptic ACh is high-cost relative to firing post-synaptic action potentials, efficient ACh-use is biologically non-trivial and motivates a need to model high-Hz cholinergic transmission. Here, via one-dimensional reaction-diffusion modeling plus a lateral escape mechanism for ACh, we simulate the events between ACh-release (singly, 200 Hz, 500 Hz) and ACh-binding to post-synaptic receptors in the presence of synaptic-cleft AChE. Using open AChRs (not channel current) as our metric, we first show that our model accords with published accounts of mepc time courses (single ACh-releases). Here, we show how the ACh-release time course, synaptic gap width, ACh-diffusion coefficient, and the density and cleft-distribution of AChE affect open-AChR(time). ACh-release at 200Hz and 500Hz is then examined. To address efficiency, the fraction of released-AChs yielding open AChRs is monitored as the quantity of ACh/release is varied. We find standard “NMJ machinery” to be operative at upper gymnotid frequencies, and thus wonder if the NMJ pre-synaptic ACh-supply machinery is, likewise, “over-engineered.”

A note on control of one-dimensional heterogeneous reaction-diffusion equations

<p style='text-indent:20px;'>We are concerned with the controllability of a general heterogeneous semilinear parabolic equation of the form <inline-formula><tex-math id="M1">\begin{document}$ y_t(x,t)=y_{xx}(x,t)+f(x,y(x,t)) $\end{document}</tex-math></inline-formula> in one space dimension using boundary static controls. More precisely, we provide conditions so that, using only static controls at the boundary, a certain target (a steady-state of the equation) is reached in infinite time. Applications of the main result are also presented, in particular, we provide several controllability results to heterogeneous equations of monostable and bistable types.</p>

Stability and dynamics of spike-type solutions to delayed Gierer-Meinhardt equations

<p style='text-indent:20px;'>For a specific set of parameters, we analyze the stability of a one-spike equilibrium solution to the one-dimensional Gierer-Meinhardt reaction-diffusion model with delay in the components of the reaction-kinetics terms. Assuming slow activator diffusivity, we consider instabilities due to Hopf bifurcation in both the spike position and the spike profile for increasing values of the time-delay parameter <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula>. Using method of matched asymptotic expansions it is shown that the model can be reduced to a system of ordinary differential equations representing the position of the slowly evolving spike solution. The reduced evolution equations for the one-spike solution undergoes a Hopf bifurcation in the spike position in two cases: when the negative feedback of the activator equation is delayed, and when delay is in both the negative feedback of the activator equation and the non-linear production term of the inhibitor equation. Instabilities in the spike profile are also considered, and it is shown that the spike solution is unstable as <inline-formula><tex-math id="M2">\begin{document}$ T $\end{document}</tex-math></inline-formula> is increased beyond a critical Hopf bifurcation value <inline-formula><tex-math id="M3">\begin{document}$ T_H $\end{document}</tex-math></inline-formula>, and this occurs for the same cases as in the spike position analysis. In all cases, the instability in the profile is triggered before the positional instability. If however the degradation of activator is delayed, we find stable positional oscillations can occur in this system.</p>

BIFURCATIONS AND HYDRA EFFECTS IN A REACTION-DIFFUSION PREDATOR-PREY MODEL WITH HOLLING Ⅱ FUNCTIONAL RESPONSE

In this paper, through bifurcation analysis and numerical simulations, we consider a reaction-diffusion predator-prey model with Holling Ⅱ functional response to analyze the existence of hydra effect and the relationship between mortality independent of predator density and different steady-state solutions of the system. The hydra effect, which is a paradoxical result in both theoretical and applied ecology, refers to the phenomenon in which an increase in population mortality enhances its own population size. We investigate the existence of the hydra effect when the positive equilibrium point is locally asymptotically stable and Turing unstable. Meanwhile, numerical simulations verify the existence of the hydra effect when the one-dimensional reaction-diffusion system has a spatially inhomogeneous steady-state solution. In addition, we introduce the existence of the Turing bifurcation, the Hopf bifurcation, and the Turing-Hopf bifurcation with the parameters $ d_{2} $ and $ m_{C} $, respectively, as well as the normal form for the Turing-Hopf bifurcation. Based on the obtained normal form, we analyze the complex spatio-temporal dynamics near the Turing-Hopf bifurcation point. Finally, the numerical simulations are carried out to corroborate the obtained theoretical results.

Temperature and Light Multiphysical Fields Modulate the Propagation of Chemical Waves on Multiple Length Scales in a System

Abstract The phenomenon of multiple length scales exists in plants during growth, which is influenced not only by light but also by other physical fields such as substance concentration and temperature, and the combined effects of various physical fields form a complex multi-scale phenomenon. However, the mechanism of the multi-physical interaction of temperature and light on the formation of multiple length scales in plants has not been investigated. In this paper, we coupled a gel with a Belousov-Zhabotinsky (BZ) chemical reaction system, constructed a one-dimensional open reaction diffusion system, and investigated the changes in the multiple length scale periodic structure of chemical waves in the one-dimensional gel system under the combined effect of multiple physical fields of light and temperature. It was found that when increasing the light intensity at T = 26 °C or decreasing the temperature at the light intensity I ≤ 50 µW·cm−2, the chemical wave multiple length scale period structure of the system changed as follows: P8 → P16 → P8 → P4 → P2 → P1, and the chemical wave period structure complexity showed a non-linear change of first increasing and then decreasing. When increasing the light intensity at T ≤ 24 °C or decreasing the temperature at the light intensity I > 50 µW·cm−2, the chemical wave period structure complexity of the system shows a linear decrease.

Discrete effects on the source term for the lattice Boltzmann modelling of one-dimensional reaction–diffusion equations

This work presents a detailed numerical analysis of one-dimensional, time-dependent (linear) reaction–diffusion type equations modelled with the lattice Boltzmann method (LBM), using the two-relaxation-time (TRT) scheme, for the D1Q3 lattice. The interest behind this study is twofold. First, because it applies to the description of many engineering problems, such as the mass transport in membranes, the heat conduction in fins, or the population growth in biological systems. Second, because this study also permits understanding the general effect of solution-dependent sources in LBM, where this problem offers a simple, yet non-trivial, canonical groundwork. Without recurring to perturbative techniques, such as the Chapman-Enskog expansion, we exactly derive the macroscopic numerical scheme that is solved by the LBM-TRT model with a solution-dependent source and show that it obeys a four-level explicit finite difference structure. In the steady-state limit, this scheme reduces to a second-order finite difference approximation of the stationary reaction–diffusion equation that, due to artefacts from the source term discretization, may operate with an effective diffusion coefficient of negative value, although still remaining stable. Such a surprising result is demonstrated through an exact stability analysis that proves the unconditional stability of the LBM-TRT model with a solution-dependent source, in line with the already proven source-less pure diffusion case (Lin et al., 2021). This proof enlarges the confidence over the LBM-TRT model robustness also for the (linear) reaction–diffusion problem class. Finally, a truncation error analysis is performed to disclose the structure of the leading order errors. From this knowledge, two strategies are proposed to improve the scheme accuracy from second- to fourth-order. One exclusively based on the tuning of the LBM-TRT scheme free-parameters, namely the two relaxation rates and the lattice weight coefficient, and the other based on the redefinition of the structure of the relaxation rates, where the leading order truncation error is absorbed into one of the relaxation rates, liberating the other to improve additional features of the scheme. Numerical tests presented in the last part of the work support the ensemble of theoretical findings.

Propagation and blocking in a two-patch reaction-diffusion model

This paper is concerned with propagation phenomena for the solutions of the Cauchy problem associated with a two-patch one-dimensional reaction-diffusion model. It is assumed that each patch has a relatively well-defined structure which is considered as homogeneous. A coupling interface condition between the two patches is involved. We first study the spreading properties of solutions in the case when the per capita growth rate in each patch is maximal at low densities, a configuration which we call the KPP-KPP case, and which turns out to have some analogies with the homogeneous KPP equation in the whole line. Then, in the KPP-bistable case, we provide various conditions under which the solutions show different dynamics in the bistable patch, that is, blocking, virtual blocking (propagation with speed zero), or spreading with positive speed. Moreover, when propagation occurs with positive speed, a global stability result is proved. Finally, the analysis in the KPP-bistable frame is extended to the bistable-bistable case.

Coupled modeling of peatlands carbon cycle and carbon dioxide emission from their peat deposits

COmplex Model of BOg LAndscapes (COMBOLA) is a set of dynamic models of carbon and nitrogen turnover, net ecosystem exchange, water balance, heat and water transport, generation and transfer of CO2 and CH4 in a peat deposit on a number of time scales. Combined modeling of carbon turnover and carbon dioxide emission from peat deposits on an annual time scale is carried out as part of COMBOLA for several bog landscapes in Western-Siberian south taiga. One-dimensional vertical reaction-diffusion model for carbon dioxide emission from peat deposit is combined with compartment carbon cycle model allowing estimate both bog type vegetation changes and greenhouse gases balance under probable climate change or perturbations initiated by human activities. Water table level (WTL) separating aerobic peat layer from the anaerobic one is calculated by means of the whole peat layer depth and its water balance relations. Minimal statistical information is used for calibrating temperature dependencies together with coefficients of heat and water transfer equations. The stationary states of carbon cycle are accompanied by steady average profiles of carbon dioxide concentration and can loose stability under human or climatic perturbations. The model dynamics is simulated with several climatic projections of the IPSL atmospheric circulation model for 21st century with anthropogenic scenarios RCP-2.6 and RCP-8.5.

Soild-state decomposition mechanisms with conversion extent for ammonium percholorate catalyzed with nanothermite particles

Colloidal ferric oxide/aluminum nanothermite mixture was incorporated into ammonium percholorate (AP). Nanothermite particles experienced an increase in AP decomposition enthalpy by 120% using DSC. Decomposition kinetic study was performed using integral isoconversional models including Kissinger and KAS models. Nanothermite particles offered decrease in AP activation energy by 55 and 44% using KAS and Kissinger models, respectively. AP nanocomposite is running through three decomposition steps according to extent of conversion: first-order decomposition with Ea = 41.08 kJ.mol−1 (α = 0: 0.25), two-dimensional decomposition with Ea = 71.90 kJ.mol−1 (α = 0.25: 0.6), and one-dimensional diffusion reaction with Ea = 97.10 kJ.mol−1 (α = 0.6:0.9).

Catalytic Co-pyrolysis of empty fruit bunch and high-density polyethylene mixtures over rice husk ash: Thermogravimetric, kinetic and thermodynamic analyses

Rice husk ash (RHA) has been used as a catalyst precursor but there are lack of studies on the application of the resulting catalyst. This study allows researchers to have an insight on using RHA-sourced catalysts in pyrolysis and be encouraged to utilize waste materials in the future. The goal of this study is to examine the effect of catalysts derived from rice husk ash (RHA) using the solvent-free method, labelled as RHA-T, on the catalytic co-pyrolysis of empty fruit bunch (EFB) and high-density polyethylene (HDPE) via thermogravimetric analyser (TGA). Comparisons were then made with co-pyrolysis and catalytic co-pyrolysis over raw RHA and Hydrogen-exchanged Zeolite Socony Mobil-5 (HZSM-5). Thermogravimetric analysis was conducted (EFB-to-HDPE mass ratio of 1:1, catalyst-to-feedstock mass ratio of 1:1) in a nitrogen atmosphere, where samples were heated from 30 °C until 700 °C (heating rate 20 °C/min). The order of runs with highest mass loss in the second phase is as follows, with the term ‘BP’ indicating the biomass-plastic feedstock: BP-RHA-T (98.17 wt%), BP-RHA (96.25 wt%), BP (86.82 wt%) and BP-HZSM-5 (70.59 wt%). Kinetic analysis using Coats-Redfern method and comparing between different diffusional reaction models showed that using BP-RHA-T follows a one-dimensional diffusion reaction, similar to the non-catalytic run. Using RHA-T resulted in higher activation energy (83.03 kJ/mol to 84.91 kJ/mol) compared to the non-catalytic run (62.39 kJ/mol to 68.97 kJ/mol). Thermodynamic analysis showed the pyrolysis runs were endothermic and non-spontaneous. Using RHA-T resulted in a higher change of enthalpy, a lower change of Gibbs free energy and a less negative change of entropy. It can be concluded that applying catalysts synthesized using low-cost materials like RHA can improve the degradation of EFB and HDPE via pyrolysis, compared to commercial HZSM-5 catalysts.