Solutions Of Reaction Research Articles

Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion

Using the nonclassical symmetry of nonlinear reaction–diffusion equations, some exact multi-dimensional time-dependent solutions are constructed for a fourth-order Allen–Cahn–Hilliard equation. This models a phase field that gives a phenomenological description of a two-phase system near critical temperature. Solutions are given for the changing phase of cylindrical or spherical inclusion, allowing for a “mushy” zone with a mixed state that is controlled by imposing a pure state at the boundary. The diffusion coefficients for transport of one phase through the mixture depend on the phase field value, since the physical structure of the mixture depends on the relative proportions of the two phases. A source term promotes stability of both of the pure phases but this tendency may be controlled or even reversed through the boundary conditions.

Pyramidal traveling fronts in a nonlocal delayed diffusion equation

Recently, there have been great progresses on the study of nonplanar traveling wave solutions of reaction–diffusion equations. In this paper, we study the pyramidal traveling fronts of nonlocal delayed diffusion equation in RN with N≥2 by using the comparison principle and establishing super- and subsolution. Since the effect of nonlocal delay, we show that nonlocal delayed diffusion equation in N-dimensional space has a pyramidal traveling front u(t,x)=V(x′,xN+st) toward XN-axis for each s>c>0. In particular, two-dimensional traveling curved front and three-dimensional pyramidal fronts for nonlocal delayed diffusion equation in R2 and R3 are also established, respectively. Moreover, we also extend our results to generally nonlocal delayed reaction–diffusion equation.

Riccati–Ermakov systems and explicit solutions for variable coefficient reaction–diffusion equations

We present several families of nonlinear reaction–diffusion equations with variable coefficients including generalizations of Fisher–KPP and Burgers type equations. Special exact solutions such as traveling wave, rational, triangular wave and N-wave type solutions are shown. By means of similarity transformations the variable coefficients are conditioned to satisfy Riccati or Ermakov systems of equations. When the Riccati system is used, conditions are established so that finite-time singularities might occur. We explore solution dynamics across multi-parameters. In the supplementary material, we provide a computer algebra verification of the solutions and exemplify nontrivial dynamics of the solutions.

Fast flow asymptotics for stochastic incompressible viscous fluids in $$\mathbb {R}^2$$ and SPDEs on graphs

Fast advection asymptotics for a stochastic reaction–diffusion–advection equation are studied in this paper. To describe the asymptotics, we introduce a new class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow. We prove that, as the advection term becomes faster and faster, the solution of the stochastic reaction–diffusion–advection equation on the plane converges in a suitable sense to the solution of such an SPDE defined on the graph. This result is a consequence of averaging, due to a probabilistic representation of solutions, both for the SPDEs on the plane and for the SPDE on the graph.

Mechanistic study on galvanic replacement reaction and synthesis of Ag-Au alloy nanoboxes with good surface- enhanced Raman scattering activity to detect melamine

Tree-type multiple-head surfactant, bis (amidoethyl-carbamoylethyl) octadecylamine (C18N3), which functions as both reducing and capping agent in the reaction system, has been used to fabricate Ag nanocubes with size over hundreds nanometer. These as-prepared Ag nanocubes could be used as template to fabricate Ag-Au nanoboxes through galvanic replacement reaction by adding HAuCl4 aqueous solution. And the inner diameter of the Ag-Au nanoboxes could be adjusted by changing the concentration of HAuCl4 in the reacted solution. Other influences on the structure of the products of galvanic replacement reaction, such as temperature, were also studied. We find that the structure of the final products are totally different when adding ammonia into the reacted solution of galvanic replacement reaction, they became hexapod and spherical Ag-Au alloy nanostructures. It was suggested that the additive of ammonia water alter the mechanism of the Ag/Au galvanic replacement reaction, which is a new way to increase the types of prepared materials with various structures. We believe that our method opens a new way and will be useful for preparing other kinds of metallic nanostructures. And the SERS (Surface Enhanced Raman Scattering) measurements proved that the prepared Ag-Au alloy nanoboxes possess excellent potential applications in the field of food safety.

Asymptotic properties of blow-up solutions in reaction–diffusion equations with nonlocal boundary flux

This paper deals with a reaction–diffusion problem with coupled nonlinear inner sources and nonlocal boundary flux. Firstly, we propose the critical exponents on nonsimultaneous blow-up under some conditions on the initial data. Secondly, we combine the scaling technique and the Green’s identity method to determine four kinds of simultaneous blow-up rates. Thirdly, the lower and the upper bounds of blow-up time are derived by using Sobolev-type differential inequalities.

Nonlinear subgrade reaction solution for circular tunnel lining design based on mobilized strength of undrained clay

This paper presents a nonlinear solution of a radial subgrade reaction–displacement (pk–ur) curve for circular tunnel lining design in undrained clay. With the concept of soil shear strength nonlinearly mobilized with shear strain, an analytical solution of pk is obtained using the mobilized strength design method. Two typical deformation modes are considered, namely oval and uniform. A total of 197 orthogonally designed cases are used to calibrate the proposed nonlinear solution of pk using the finite element method with the hardening soil model. The calibration results are summarized using a correction factor, η, which is defined as the ratio of pk_FEM to pk_MSD. It is shown that η is correlated to some input parameters. If this correlation is removed by a regression equation, f, the modified solution f(pk_MSD) agrees very well with pk_FEM. Although in reality the mobilized soil strength varies with principal stress direction, it is found that a simple average of plane strain compression and extension results is sufficient to produce the above agreement. The proposed nonlinear pk–ur curve is applied to an actual tunnel lining design example. The predicted tunnel deformations agree very well with the measured data. In contrast, a linear pk model would produce an underestimation of tunnel convergence and internal forces by 2–4 times due to the overestimation of pk at a large strain level.

Parameter estimation in IMEX-trigonometrically fitted methods for the numerical solution of reaction–diffusion problems

In this paper, an adapted numerical scheme for reaction–diffusion problems generating periodic wavefronts is introduced. Adapted numerical methods for such evolutionary problems are specially tuned to follow prescribed qualitative behaviors of the solutions, making the numerical scheme more accurate and efficient as compared with traditional schemes already known in the literature. Adaptation through the so-called exponential fitting technique leads to methods whose coefficients depend on unknown parameters related to the dynamics and aimed to be numerically computed. Here we propose a strategy for a cheap and accurate estimation of such parameters, which consists essentially in minimizing the leading term of the local truncation error whose expression is provided in a rigorous accuracy analysis. In particular, the presented estimation technique has been applied to a numerical scheme based on combining an adapted finite difference discretization in space with an implicit–explicit time discretization. Numerical experiments confirming the effectiveness of the approach are also provided.

Exceptionally Efficient and Recyclable Heterogeneous Metal-Organic Framework Catalyst for Glucose Isomerization in Water.

Heterogeneous catalysts are desired for the conversion of glucose, the most abundant sugar in renewable biomass, but presently their synthesis requires highly toxic chemicals with long synthesis times. We report the conversion of glucose into fructose and 5‐hydroxymethylfurfural on a heterogeneous catalyst that is stable and selective and operates in the most environmentally benign solvent, water. We used a bifunctional solid with Lewis and Brønsted acid sites by partially replacing the organic linker of the zirconium organic framework UiO‐66 with 2‐monosulfo‐benzene‐1,4‐dicarboxylate. This catalyst showed high product selectivity (90 %) to 5‐hydroxymethylfurfural and fructose at 140 °C in water after a reaction time of 3 h. It was recyclable and showed only a minor loss in activity after the third recycle, offering a realistic solution for the bottleneck glucose isomerization reaction for scale‐up and industrial application of biomass utilization.

Existence and Stability of Traveling Waves for Degenerate Reaction–Diffusion Equation with Time Delay

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction–diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of $$c\ge c^*$$ for the degenerate reaction–diffusion equation without delay, where $$c^*>0$$ is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay $$\tau >0$$ . Furthermore, we prove the global existence and uniqueness of $$C^{\alpha ,\beta }$$ -solution to the time-delayed degenerate reaction–diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted $$L^1$$ -space. The exponential convergence rate is also derived.

Effects of density dependent cross-diffusion on the chaotic patterns in a ratio-dependent prey-predator model

Spatio-temporal chaos is an intriguing part of the spatio-temporal pattern formation, observed in many interacting population models when their heterogeneous distributions within their habitats and movement from one location to the other are taken care of within the modeling approach. When the homogeneous steady-states become unstable, the solutions of the corresponding reaction–diffusion systems never approach a stationary state rather exhibit an irregular nature with respect to both space and/or time. Cross-diffusion terms are incorporated in a system of reaction–diffusion equations to model the situation where presence, absence, abundance of one species influence the movement of another species and vice-versa. In this work, cross-diffusion is considered in a prey–predator model with ratio-dependent functional response along with the self-diffusion terms. After deriving the Turing instability conditions in terms of cross-diffusion parameters, extensive numerical simulations are carried out to study the effect of cross-diffusion on the chaotic dynamics and stationary Turing patterns generated in the system containing self-diffusion terms only. Appropriate numerical tools are used to characterize the spatio-temporal chaos. Route to spatio-temporal chaos and its disappearance are discussed in detail. The chaotic dynamics of the self-diffusion model may be suppressed leading to a stationary state or preserved depending on the cross-diffusion coefficients. The stationary patches of both the species generated in the Turing domain remain stationary but their configurations may change due to the effect of density-dependent cross-diffusion. On the other hand, stationary patches generated in the Turing-Hopf domain change to spatio-temporal chaos for higher dispersal rates of predator avoidance by prey.

Solution of the periodic Belousov-Zhabotinsky reaction using a closed-loop mechanism

Belousov-Zhabotinsky reaction generates self-organized oscillatory pattern which is common in biological systems, synergistic study of oscillatory patterns will assist understanding and modeling complex processes in biological sphere. However, the analytical solution of a self-oscillator is difficult because the system exhibits nonlinear dynamics. In this study, a frequency domain analysis of Hopf bifurcation based on a closed-loop representation is addressed. For better understanding of the closed-loop mechanism, a Laplace-Borel transform is implemented, which is proved to be effective in identifying the coefficients of the harmonics.

Tailoring the Co 3d-O 2p Covalency in LaCoO3 by Fe Substitution To Promote Oxygen Evolution Reaction

LaCoO3 is an active, stable catalyst in alkaline solution for oxygen evolution reaction (OER). With lower cost, it is a potential alternative to precious metal oxides like IrO2 and RuO2 in water electrolysis. However, room still remains for improving its activity according to recent understandings of OER on perovskite oxides. In this work, Fe substitution has been introduced in LaCoO3 to boost its OER performance. Density function theory (DFT) calculation verified that the enhanced performance originates from the enhanced Co 3d-O 2p covalency with 10 at% Fe substitution in LaCoO3. Both DFT calculations and Superconducting Quantum Design (SQUID) magnetometer (MPMS-XL) showed a Co3+ spin state transition from generally low spin state (LS: t2g6 eg0, S = 0) to a higher spin state with the effect of 10 at% Fe substitution. X-ray absorption near-edge structure (XANES) supports DFT calculations on an insulator to half-metal transition with 10 at% Fe substitution, induced by spin state transition. The half-metall...

Wavefront solutions of quasilinear reaction–diffusion systems with mixed quasi-monotonicity

ABSTRACTWe study the existence of wavefront solutions in a general class of mixed quasi-monotone reaction–diffusion systems with nonlinear diffusions in the form for . Each function is quasi-monotone increasing for some components of and decreasing for other components of . Such systems model reaction–diffusion processes with a density driven diffusion mechanism. Under certain general conditions we prove the existence of a traveling wave solution that is between a pair of coupled upper and lower solutions. As examples, we discuss a predator–prey model and a multi-species competition model, both with nonlinear diffusions. In both models, the presences of wavefront solutions flowing towards the coexistence states are established as illustrations for applications of our main result, with numerical simulations.

Initiation of reaction–diffusion waves of blood coagulation

Formation of blood clot in response to the vessel damage is triggered by the complex network of biochemical reactions of the coagulation cascade. The process of clot growth can be modeled as a traveling wave solution of the bistable reaction–diffusion system. The critical value of the initial condition which leads to convergence of the solution to the traveling wave corresponds to the pulse solution of the corresponding stationary problem. In the current study we prove the existence of the pulse solution for the stationary problem in the model of the main reactions of the blood coagulation cascade using the Leray–Schauder method.

Efficient N-doping of hollow core-mesoporous shelled carbon spheres via hydrothermal treatment in ammonia solution for the electrocatalytic oxygen reduction reaction

Nitrogen-doped hollow core-mesoporous shelled carbon spheres (NHMCSs) are successfully synthesized via a simple, scalable hydrothermal treatment of hollow mesoporous carbon spheres (HMCSs) with ammonia water as the nitrogen source. This strategy not only efficiently introduces nitrogen atoms into the carbon framework, but also causes the fracture of some spheres and increases the shell thickness of hollow spheres. The subsequent pyrolysis at 900 °C in Ar results in the transformation of pyrrolic-N to graphitic-N. The resulting NHMCS-900 has a high BET specific surface area (648 m2 g−1), large pore volume (1.06 cm3 g−1), hierarchical mesoporosity (2–8, 43.9 nm), uniform shells and large hollow cores. The electrochemical experiments indicate that NHMCS-900 as a metal-free electrocatalyst possesses the improved ORR activity with a close four-electron reaction pathway (∼3.88) and a peroxide yield lower than 10% over a wide potential range in alkaline solution. Contrary to commercial Pt/C, NHMCS-900 is unaffected by the methanol crossover effect and exhibits excellent long-term durability with 97.0% of the initial current density after 5.56 h of continuous operation. The excellent activity of NHMCS-900 is mainly attributed to the high relative content of pyridinic-N and graphitic-N. Besides, the superior activity is also ascribed to its unique structural properties, promoting fast and efficient mass transfer and providing inner and outer surfaces with N-relative active sites for ORR.

Weak order in averaging principle for stochastic wave equation with a fast oscillation

This article deals with the weak error in averaging principle for a stochastic wave equation on a bounded interval [0,L], perturbed by an oscillating term arising as the solution of a stochastic reaction–diffusion equation evolving on the fast time scale. Under suitable conditions, it is proved that the rate of weak convergence of the original solution to the solution of the corresponding averaged equation is of order 1 via an asymptotic expansion approach.

On the existence of weak solutions of nonlinear degenerate parabolic system with variable exponents

The main goal of the present paper is establishing the existence and uniqueness of weak solutions for the nonlinear degenerate reaction–diffusion system with variable exponents. A model also is proposed to characterize the invasion of cancer cells towards healthy cells with acidification environment. Moreover, the main results of this paper are obtained using regularization problem, the Faedo–Galerkin approximation method, some apriori estimates, compactness results and the Gronwall Lemma.

Large time monotonicity of solutions of reaction–diffusion equations in [formula omitted

In this paper, we consider nonnegative solutions of spatially heterogeneous reaction–diffusion equations in the whole space. Under some assumptions on the initial conditions, including in particular the case of compactly supported initial conditions, we show that, above any arbitrary positive value, the solution is increasing in time at large times. Furthermore, in the one-dimensional case, we prove that, if the equation is homogeneous outside a bounded interval and the reaction is linear around the zero state, then the solution is time-increasing in the whole line at large times. The question of the monotonicity in time is motivated by a medical imagery issue.

A numerical solution for a variable-order reaction–diffusion model by using fractional derivatives with non-local and non-singular kernel

A reaction–diffusion system can be represented by the Gray–Scott model. The reaction–diffusion dynamic is described by a pair of time and space dependent Partial Differential Equations (PDEs). In this paper, a generalization of the Gray–Scott model by using variable-order fractional differential equations is proposed. The variable-orders were set as smooth functions bounded in (0,1] and, specifically, the Liouville–Caputo and the Atangana–Baleanu–Caputo fractional derivatives were used to express the time differentiation. In order to find a numerical solution of the proposed model, the finite difference method together with the Adams method were applied. The simulations results showed the chaotic behavior of the proposed model when different variable-orders are applied.