A previous paper by our research group introduced the nonlinear multitime reaction–diffusion PDE (with oblique derivative) as a generalized version of the single-time model. This paper states and uses some hypotheses that allow the finding of s...
In this paper, we derive a delayed reaction-diffusion equation to describe a two-species predator-prey system with diffusion terms and stage structure. By coupling the uniformly approximate approach with the method of upper and lower solutions, we pr...
In this paper, we concentrate on the propagation dynamics of stochastic reaction–diffusion equations, including the existence of travelling wave solution and asymptotic wave speed. Based on the stochastic Feynman–Kac formula and compariso...
This paper considers the dynamical properties of a space and time discrete fractional reaction–diffusion epidemic model, introducing a novel generalized incidence rate. The linear stability of the equilibrium solutions of the considered discret...
In this work, we study the global existence of solutions reaction-diffusion systems with control of mass on multiple domains. Some of these domains overlap, and as a result, an unknown defined on one subdomain can impact another unknown defined on a...
This study is devoted to reaction–diffusion equations with spatially anisotropic time delay. Reaction–diffusion PDEs with either constant or variable transfer coefficients are considered. Nonlinear equations of a fairly general form conta...
In this paper we propose and compare two methods to optimize the numerical computations for the diffusion term in a nonlocal formulation for a reaction-diffusion equation. The diffusion term is particularly computationally intensive due to the integr...
Reaction–diffusion equations can model complex systems where randomness plays a role, capturing the interaction between diffusion processes and random fluctuations. The Kolmogorov equations associated with these systems play an important role i...
This work is devoted to deriving a more accurate reaction–diffusion equation for an A/B binary system by summing over microscopic trajectories. By noting that an originally simple physical trajectory might be much more complicated when the reac...
The long-term behavior of the weak solution of a fractional delayed reaction–diffusion equation with a generalized Caputo derivative is investigated. By using the classic Galerkin approximation method and comparison principal, the existence and...
In this paper, we study a fractional Brusselator reaction–diffusion model with the help of the residual power series transform method. Specific reaction–diffusion chemical processes are modeled by applying the fractional Brusselator react...
A wide range of reaction–diffusion systems with constant diffusivities that are invariant under Q-conditional operators is found. Using the symmetries obtained, the reductions of the corresponding systems to the systems of ODEs are conducted in order...
DNA nanotechnology offers a fine control over biochemistry by programming chemical reactions in DNA templates. Coupled to microfluidics, it has enabled DNA-based reaction-diffusion microsystems with advanced spatio-temporal dynamics such as traveling...
This work analyzes the effect of wall geometry when a reaction–diffusion system is confined to a narrow channel. In particular, we study the entropy production density in the reversible Gray–Scott system. Using an effective diffusion equa...
This paper delves into a comprehensive analysis of a generalized impulsive discrete reaction–diffusion system under periodic boundary conditions. It investigates the behavior of reactant concentrations through a model governed by partial differ...
In this article, we study how the rates of diffusion in a reaction-diffusion model for a stage structured population in a heterogeneous environment affect the model’s predictions of persistence or extinction for the population. In the case of a...
In this paper, we study the local stability of an incommensurate fractional reaction–diffusion glycolysis model. The glycolysis process, fundamental to cellular metabolism, exhibits complex dynamical behaviors when formulated as a nonlinear rea...
This paper addresses the asymptotic synchronization problem for networked nonlinear reaction–diffusion systems under a novel hybrid control strategy. The hybrid controller consists of two components: impulsive control and continuous feedback co...
The method of linear determining equations is constructed to study conditional Lie–Bäcklund symmetry and the differential constraint of a two-component second-order evolution system, which generalize the determining equations used in the search for c...
In this article, a new competitive neural network (CNN) with reaction-diffusion terms and mixed delays is proposed. Because this network system contains reaction-diffusion terms, it belongs to a partial differential system, which is different from th...
The review is devoted to an analysis of mathematical models used for describing epidemic processes. Our main focus is on the models that are based on partial differential equations (PDEs), especially those that were developed and used for the COVID-1...
In this article, we discuss the relationships between thermodynamic quantities and the spatial pattern in a reaction-diffusion model based on the reversible Gray-Scott model. This model is introduced for calculation of the entropy production in a rea...
Because they are useful for both enabling numerical simulations and containing well-defined physical phenomena, discrete fractional reaction–diffusion models have attracted a great deal of interest from academics. Within the family of fractiona...
The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of ef...
This study investigates the application of physics-informed neural networks (PINNs) to two-dimensional reaction–diffusion biological models with singular terms. Motivated by the classical prey–predator framework, we propose an improved model that inc...
The architecture of hydrogel composites results in not only synergistic property enhancement but also superior functionality. The reaction–diffusion (RD) process is acommon phenomenon throughout nature which induced ordered structure on a lengt...
In this paper, we consider a reaction–diffusion system governed by incommensurate fractional time derivatives based on the Degn–Harrison model. Its formulation incorporates various memory effects on axial position through Caputo derivativ...
In this paper, we investigated the fixed-time synchronization problem of a type of reaction-diffusion fuzzy neural networks with stochastic perturbations by developing simple control schemes. First, some generalized fixed-time stability results are i...
The purpose of this work is to introduce a new kind of finite difference formulation inspired from Fourier analysis, for reaction-diffusion equations. Compared to classical schemes, the proposed scheme is much more accurate and has interesting stabil...
In this paper, a complete analysis is presented to study a reaction–diffusion system of general gene expression with two time delays and with Neumann boundary conditions. The global existence of a unique strong solution and the existence of an...
A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction–diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through por...
In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to...
In this short review article, two atherosclerosis models are presented, one as a scalar equation and the other one as a system of two equations. They are given in terms of reaction-diffusion equations in an infinite strip with nonlinear boundary cond...
In the last few years, reaction–diffusion models associated with discrete fractional calculus have risen in prominence in scientific fields, not just due to the requirement for numerical simulation but also due to the described biological pheno...
In the work, a nonlinear reaction-diffusion model in a class of delayed differential equations on the hexagonal lattice is considered. The system includes a spatial operator of diffusion between hexagonal pixels. The main results deal with the qualit...
The Selkov system, which is typically employed to model glycolysis phenomena, unveils some rich dynamics and some other complex formations in biochemical reactions. In the present work, the synchronization problem of the glycolysis reaction-diffusion...
Approximate analytical solutions to doubly degenerate reaction-diffusion models pertinent to population dynamics and chemical kinetics have been developed. The double integral-balance method applied to preliminary transformed models and by a direct i...
The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study d...
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...
This work develops and analyzes an incommensurate fractional FitzHugh–Nagumo (FHN) reaction–diffusion system in which each state variable evolves with a distinct fractional order. The formulation extends the classical and commensurate fra...
In this work, we aim to explore new exact traveling wave solutions for the reaction–diffusion equation, which describes complex nonlinear phenomena such as cell growth and chemical reactions in nature. Obtaining exact solutions to this equation...
In this paper, we investigate a reaction–diffusion equation with a Caputo fractional derivative in time and with boundary conditions. According to the principle of contraction mapping, we first prove the existence and uniqueness of local soluti...
Due to the facts that epidemic-related parameters vary significantly in different stages of infectious diseases and are relatively stable within the same stage, infectious disease models should be switch-type models. However, research on switch-type...
Despite the fact the Laplace transform has an appreciable efficiency in solving many equations, it cannot be employed to nonlinear equations of any type. This paper presents a modern technique for employing the Laplace transform LT in solving the non...
In this work, we explore the state feedback regulator problem (SFRP) in order to achieve the goal for trajectory tracking with harmonic disturbance rejection to one-dimensional (1-D) reaction-diffusion (R-D) equation, namely, a partial differential e...
This paper deals with the effects of partial differential equations on the development of spatiotemporal patterns in reaction–diffusion systems. These systems describe how the concentration of a certain substance is distributed in space or time...
This paper uses the attached flow method for solving nonlinear second-order differential equations of the reaction–diffusion type. The key steps of the method consist of the following: (i) reducing the differentiability order by defining the fi...
Diffusion within porous media, such as biological tissues, often deviates from conventional Fick’s laws that may be described by space-fractional diffusion equations. Microscale tissue heterogeneity can be represented by the space-fractional Ri...
This study presents a higher-order uniformly convergent numerical method for a singularly perturbed nonlinear reaction–diffusion equation. The quasilinearization technique is used to transform a nonlinear term into a linear boundary value probl...
This paper aims to present a robust computational technique utilizing finite difference schemes for accurately solving time fractional reaction–diffusion models, which are prevalent in chemical and biological phenomena. The time-fractional deri...
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