Reaction-diffusion Research Articles

Global well-posedness and exponential decay estimates for semilinear Newell–Whitehead–Segel equation

Abstract This article presents the application of the Faedo–Galerkin compactness method to establish the local well-posedness of the Newell–Whitehead–Segel equation. By analyzing a finite-dimensional approximate problem, the existence and uniqueness of a local solution were demonstrated. A priori estimates were derived, enabling the transition to the limit and the recovery of the original problem’s local solution. The study further proves the uniqueness and continuous dependence of the solution on initial data. Additionally, under certain conditions, it is shown that the energy norm of the solution decays exponentially over time, and the L 2 {L}^{2} -norm of the time derivative of the solution approaches zero asymptotically.

Local discontinuous Galerkin methods with implicit–explicit BDF time marching for Newell–Whitehead–Segel equations

The Newell–Whitehead–Segel type equations with time-dependent Dirichlet boundary conditions are solved by the local discontinuous Galerkin (LDG) method coupled with the implicit–explicit backward difference formulas (IMEX-BDF). With a suitable setting of numerical fluxes and by the aid of the multiplier technique and the a priori error assumption technique, the optimal error estimate for the corresponding fully discrete LDG-IMEX-BDF schemes is obtained by energy analysis, under the condition τ ≤ C h 1 / s , where h and τ are mesh size and time step, respectively, the positive constant C is independent of h, and s = 1 , … , 5 is the order of the IMEX-BDF method. Numerical experiments are also presented to verify the accuracy of the considered schemes.

Piecewise second kind Chebyshev functions for a class of piecewise fractional nonlinear reaction–diffusion equations with variable coefficients

In this paper, a new type of piecewise fractional derivative (PFD) is introduced. The ordinary and distributed-order fractional derivatives in the Caputo sense are used to define this type of PFD. A new version of nonlinear reaction–diffusion equations with variable coefficients is defined using this type of PFD. The orthonormal piecewise second kind Chebyshev functions (CFs), as a new family of basic functions, are generated. An explicit formula is extracted for PFD of these piecewise functions. A hybrid method based on the orthonormal piecewise second kind CFs and orthonormal second kind Chebyshev polynomials is proposed to solve the aforementioned problem. The established approach transforms solving the expressed problem into solving an algebraic system of equations. To illustrate the accuracy of the developed method, some numerical examples are considered.

Fourier Neural Operator Networks for Solving Reaction–Diffusion Equations

In this paper, we used Fourier Neural Operator (FNO) networks to solve reaction–diffusion equations. The FNO is a novel framework designed to solve partial differential equations by learning mappings between infinite-dimensional functional spaces. We applied the FNO to the Surface Quasi-Geostrophic (SQG) equation, and we tested the model with two significantly different initial conditions: Vortex Initial Conditions and Sinusoidal Initial Conditions. Furthermore, we explored the generalization ability of the model by evaluating its performance when trained on Vortex Initial Conditions and applied to Sinusoidal Initial Conditions. Additionally, we investigated the modes (frequency parameters) used during training, analyzing their impact on the experimental results, and we determined the most suitable modes for this study. Next, we conducted experiments on the number of convolutional layers. The results showed that the performance of the models did not differ significantly when using two, three, or four layers, with the performance of two or three layers even slightly surpassing that of four layers. However, as the number of layers increased to five, the performance improved significantly. Beyond 10 layers, overfitting became evident. Based on these observations, we selected the optimal number of layers to ensure the best model performance. Given the autoregressive nature of the FNO, we also applied it to solve the Gray–Scott (GS) model, analyzing the impact of different input time steps on the performance of the model during recursive solving. The results indicated that the FNO requires sufficient information to capture the long-term evolution of the equations. However, compared to traditional methods, the FNO offers a significant advantage by requiring almost no additional computation time when predicting with new initial conditions.

Turing Instability and Dynamic Bifurcation for the One‐Dimensional Gray–Scott Model

ABSTRACTWe study the dynamic bifurcation of the one‐dimensional Gray–Scott model by taking the diffusion coefficient of the reactor as a bifurcation parameter. We define a parameter space of for which the Turing instability may happen. Then, we show that it really occurs below the critical number and obtain rigorous formula for the bifurcated stable patterns. When the critical eigenvalue is simple, the bifurcation leads to a continuous (resp. jump) transition for if is negative (resp. positive). We prove that when lies near the Bogdanov–Takens point . When the critical eigenvalue is double, we have a supercritical bifurcation that produces an ‐attractor . We prove that consists of four asymptotically stable static solutions, four saddle static solutions, and orbits connecting them. We also provide numerical results that illustrate the main theorems.

Numerical Study of Multi-Term Time-Fractional Sub-Diffusion Equation Using Hybrid L1 Scheme with Quintic Hermite Splines

Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and space discretization using the quintic Hermite spline collocation method. The hybrid technique reduces the problem to an iterative scheme of an algebraic system of equations. The stability analysis of the proposed numerical scheme and the optimal error bounds for the approximate solution are also studied. A comparative study of the obtained results and an error analysis of approximation show the efficiency, accuracy, and effectiveness of the technique.

A new fourth-order compact finite difference method for solving Lane-Emden-Fowler type singular boundary value problems

We develop a novel fourth-order compact finite difference scheme to solve nonlinear singular ordinary differential equations. Such problems occur in many fields of science and engineering, such as studying the equilibrium of an isothermal gas sphere, reaction–diffusion in a spherical permeable catalyst, etc. These problems are challenging to solve because of their singularity or nonlinearity. By our proposed method, we can easily solve these complex problems without removing or modifying the singularity. To construct the new fourth-order compact difference method, Initially, we created a uniform mesh within the solution domain and developed a compact finite difference scheme. This scheme approximates the derivatives at the boundary nodal points to handle the problem’s singularity effectively. Employing a matrix analysis approach, we discussed the convergence analysis of the methods. To demonstrate its efficacy, we apply our approach to solve various real-life problems from the literature. The new method offers high-order accuracy with minimal grid points and provides better numerical results than the nonstandard finite difference method and exponential compact finite difference method.

Efficient separation and recovery of cobalt from grinding waste of cemented carbide using a sulfuric acid-sodium persulfate mixed solution

The mechanical processing of cemented carbide often generates a considerable amount of grinding waste from cemented carbide (GWCC), which serves as an important secondary resource for recovering Co and W. However, efficient separation of tungsten carbide (WC) and Co from GWCC remains challenging. This study tested an efficient process for separating Co from GWCC using a mixed solution of H2SO4-Na2S2O8. Under optimum conditions of 20 g/L H2SO4, 30 g/L Na2S2O8, 60 °C, and 1 h, the leaching efficiency of Co reached 97.0 %, compared to only 58.6 % when using H2SO4 alone. The kinetics of Co leaching in H2SO4 and H2SO4-Na2S2O8 solutions were determined to be a combination of chemical reaction control and diffusion control, suggesting that Co leaching rate in H2SO4 solution is significantly enhanced by introducing Na2S2O8. Finally, complete recovery of Co was achieved, and pure Co powder with a flower-cluster morphology was prepared from the leaching solution through oxalic acid precipitation followed by vacuum pyrolysis. The regenerative WC and Co powder can be recycled for cemented carbide production.

Two for tau: Automated model discovery reveals two-stage tau aggregation dynamics in Alzheimer’s disease

Alzheimer’s disease is a neurodegenerative disorder characterized by the presence of amyloid-β plaques and the accumulation of misfolded tau proteins and neurofibrillary tangles in the brain. A thorough understanding of the local accumulation of tau is critical to develop effective therapeutic strategies. Tau pathology has traditionally been described using reaction–diffusion models, which succeed in capturing the global spread, but fail to accurately describe the local aggregation dynamics. Current mathematical models enforce a single-peak behavior in tau aggregation, which does not align well with clinical observations. Here we identify a more accurate description of tau aggregation that reflects the complex patterns observed in patients. We propose an innovative approach that uses constitutive neural networks to autonomously discover bell-shaped aggregation functions with multiple peaks from clinical positron emission tomography (PET) data of misfolded tau protein. Our method reveals previously overlooked two-stage aggregation dynamics by uncovering a two-term ordinary differential equation that links the local accumulation rate to the tau concentration. When trained on data from amyloid-β positive and negative subjects, the neural network clearly distinguishes between both groups and uncovers a more subtle relationship between amyloid-β and tau than previously postulated. In line with the amyloid-tau dual pathway hypothesis, our results show that the presence of toxic amyloid-β influences the accumulation of tau, particularly in the earlier disease stages. We expect that our approach to autonomously discover the accumulation dynamics of pathological proteins will improve simulations of tau dynamics in Alzheimer’s disease and provide new insights into disease progression.Significance StatementIn Alzheimer’s disease, understanding the local dynamics of tau protein aggregation is crucial for developing effective treatments. Traditional models for tau protein dynamics use reaction-diffusion models that fail to accurately capture these local patterns. Our study introduces a novel approach that leverages constitutive neural networks to autonomously discover the complex, multi-peak aggregation dynamics from clinical PET data. This method reveals a previously overlooked two-stage tau accumulation process and a nuanced relationship between amyloid-β and tau. By distinguishing between amyloid-β positive and negative subjects, our model supports the amyloid-tau dual pathway hypothesis and offers novel insights into tau protein aggregation that have the potent to advance our understanding of Alzheimer’s disease progression.

Outer synchronization and outer [formula omitted] synchronization for coupled fractional-order reaction–diffusion neural networks with multiweights

This paper introduces multiple state or spatial-diffusion coupled fractional-order reaction–diffusion neural networks, and discusses the outer synchronization and outer H∞ synchronization problems for these coupled fractional-order reaction–diffusion neural networks (CFRNNs). The Lyapunov functional method, Laplace transform and inequality techniques are utilized to obtain some outer synchronization conditions for CFRNNs. Moreover, some criteria are also provided to make sure the outer H∞ synchronization of CFRNNs. Finally, the derived outer and outer H∞ synchronization conditions are validated on the basis of two numerical examples.

Necessary Stability Conditions for Reaction–Diffusion–ODE Systems

Necessary Stability Conditions for Reaction–Diffusion–ODE Systems

Interface kinetic manipulation enabling efficient and reliable Mg3Sb2 thermoelectrics

Development of efficient and reliable thermoelectric generators is vital for the sustainable utilization of energy, yet interfacial losses and failures between the thermoelectric materials and the electrodes pose a significant obstacle. Existing approaches typically rely on thermodynamic equilibrium to obtain effective interfacial barrier layers, which underestimates the critical factors of interfacial reaction and diffusion kinetics. Here, we develop a desirable barrier layer by leveraging the distinct chemical reaction activities and diffusion behaviors during sintering and operation. Titanium foil is identified as a suitable barrier layer for Mg3Sb2-based thermoelectric materials due to the creation of a highly reactive ternary MgTiSb metastable phase during sintering, which then transforms to stable binary Ti-Sb alloys during operation. Additionally, titanium foil is advantageous due to its dense structure, affordability, and ease of manufacturing. The interfacial contact resistivity reaches below 5 μΩ·cm2, resulting in a Mg3Sb2-based module efficiency of up to 11% at a temperature difference of 440 K, which exceeds that of most state-of-the-art medium-temperature thermoelectric modules. Furthermore, the robust Ti foil/Mg3(Sb,Bi)2 joints endow Mg3Sb2-based single-legs as well as modules with negligible degradation over long-term thermal cycles, thereby paving the way for efficient and sustainable waste heat recovery applications.

Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix

Abstract The aim of this work is to study the global existence in time of solutions for the tridiagonal system of reaction-diffusion by order m m . Our techniques of proof are based on compact semigroup methods and some L 1 {L}^{1} -estimates. We show that global solutions exist. Our investigation can be applied for a wide class of nonlinear terms of reaction.

Existence of a Global Attractor for the Reaction–Diffusion Equation with Memory and Lower Regularity Terms

This paper investigates the large time behavior for the reaction–diffusion equation with memory and the forcing term g∈H−1(Ω). We prove the existence of a global attractor in L2(Ω)×Lμ2(R;H01(Ω)). Due to the lower regularity of g, one can hardly use the traditional energy estimates to derive the existence of a bounded absorbing set in the higher regularity space and then the compactness of the semigroup. Here, we utilize the contractive function method to establish the asymptotic smoothness of the semigroup.

Bifurcation and stability analysis of a Leslie–Gower diffusion predator–prey model with prey refuge and Beddington–DeAngelis functional response

In this paper, we consider the spatiotemporal dynamics behaviors of a Leslie–Gower diffusion predator–prey system with prey refuge and Beddington–DeAnglis (B‐D) functional response. By using the Poincaré inequality and topological degree theory, we first investigate the Turing instability of the reaction–diffusion system and prove the existence of nonconstant positive steady‐state solutions. Then we discuss the steady‐state bifurcation and the direction to Hopf bifurcation of the PDE model by the local bifurcation theorem and center manifold theory. Finally, some numerical simulations are presented to supplement the analytic results in one dimension which indicates that changes in prey refuge and diffusion coefficient can increase the complexity of the system.

Effects of Anisotropy, Convection, and Relaxation on Nonlinear Reaction-Diffusion Systems

The effects of relaxation, convection, and anisotropy on a two-dimensional, two-equation system of nonlinearly coupled, second-order hyperbolic, advection–reaction–diffusion equations are studied numerically by means of a three-time-level linearized finite difference method. The formulation utilizes a frame-indifferent constitutive equation for the heat and mass diffusion fluxes, taking into account the tensorial character of the thermal diffusivity of heat and mass diffusion. This approach results in a large system of linear algebraic equations at each time level. It is shown that the effects of relaxation are small although they may be noticeable initially if the relaxation times are smaller than the characteristic residence, diffusion, and reaction times. It is also shown that the anisotropy associated with one of the dependent variables does not have an important role in the reaction wave dynamics, whereas the anisotropy of the other dependent variable results in transitions from spiral waves to either large or small curvature reaction fronts. Convection is found to play an important role in the reaction front dynamics depending on the vortex circulation and radius and the anisotropy of the two dependent variables. For clockwise-rotating vortices of large diameter, patterns similar to those observed in planar mixing layers have been found for anisotropic diffusion tensors.

Galerkin approximation of Burgers-Huxley equation with fractional order arising in reaction mechanism and diffusion transport

Abstract Burgers-Huxley model depicts a prototype model of the interaction of convection effect, reaction mechanisms and diffusion transport, used to study the liquid crystal and nerve fibers. This study introduces Galerkin approximation for time-fractional Burgers-Huxley equation (TFBHE) . The Caputo derivative is used to evaluate the temporal part using the $L_1$ formula. The Galerkin approach employs cubic B-spline as a shape and test function, resulting in a symmetric matrix that is easily convergent. In addition, the three-point quadrature rule is implemented to evaluate the integration of complex function . The Von Neumann analysis is used to discuss stability of the scheme. The performance and robustness of the technique is measured using various error norms. The results are compared with the exact solution, demonstrating effectiveness of the proposed method.

A Space Distributed Model and Its Application for Modeling the COVID-19 Pandemic in Ukraine

A space distributed model based on reaction–diffusion equations, which was previously developed, is generalized and applied to COVID-19 pandemic modeling in Ukraine. Theoretical analysis and a wide range of numerical simulations demonstrate that the model adequately describes the second wave of the COVID-19 pandemic in Ukraine. In particular, comparison of the numerical results obtained with the official data shows that the model produces very plausible total numbers of the COVID-19 cases and deaths. An extensive analysis of the impact of the parameters arising from the model is presented as well. It is shown that a well-founded choice of parameters plays a crucial role in the applicability of the model.

Dynamics behaviours of N-kink solitons in conformable Fisher–Kolmogorov–Petrovskii–Piskunov equation

PurposeThis manuscript is related to compute $N$-kink soliton solutions for conformable Fisher–Kolmogorov equation (CFKE) by using the generalized extended direct algebraic method (EDAM). The considered problem has important applications in mathematical biology and reaction diffusion processes. Also, the mentioned problem has significant applications in population dynamics. The fractional order conformable derivative has many features as compared to the other fractional order differential operators. For instance, the chain, product and quotient procedures do not satisfy by other fractional differential operators, but conformable operators obey the mentioned rules. Hence, we compute the soliton solutions for the mentioned problem and present its various dynamical behaviours graphically.Design/methodology/approachThe generalized EDAM is used in this article to examine the calculation of N-kink soliton solutions for the CFKE. In mathematical biology and reaction-diffusion processes, the topic under consideration holds great significance, especially when considering population dynamics.FindingsThe results highlight the benefits of utilising conformable derivatives in mathematical modelling and further our understanding of fractional differential equations and their applications.Research limitations/implicationsThe work focuses primarily on N-kink soliton solutions, which may limit the examination of alternative types of solutions (e.g., multi-soliton or periodic solutions) that might give new insights into the dynamics of the CFKE.Practical implicationsThe generated N N-kink soliton solutions can enhance mathematical models in biological contexts, notably in modelling population dynamics, disease propagation and ecological interactions, leading to better forecasts and interventions.Social implicationsPublic health initiatives can benefit from the understanding of disease transmission and intervention efficacy that comes from modelling population dynamics and reaction-diffusion processes.Originality/valueThe use of the generalized EDAM to obtain solutions for N-kink soliton problems is an innovative method for solving the conformable Fisher–Kolmogorov equation, demonstrating the power of this mathematical tool.

Publisher Correction: Numerical Study of the Reaction Diffusion Prey–Predator Model Having Holling II Increasing Function in the Predator Under Noisy Environment

Publisher Correction: Numerical Study of the Reaction Diffusion Prey–Predator Model Having Holling II Increasing Function in the Predator Under Noisy Environment