Nonlinear Reaction-diffusion Equations 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.

Nonlinear transport and kinetics in electrocatalytic thin film of arbitrary shape for non-Michaelis-Menten reaction kinetics

The mathematical model for non-Michael-Menten kinetics, which describes a substrate that produces a complex with the immobilised catalyst, is discussed. This paper analytically solves the nonlinear reaction–diffusion equation in an electrocatalytic thin film of arbitrary shape. We provide a mathematical process that enables a comprehensive analytical solution to the nonlinear boundary value problem. Closed and simple forms of the approximate expression for substrate concentration profiles for general geometry (planar, cylindrical, and spherical) and corresponding steady-state amperometric current response are presented. The results obtained by three analytical methods are then compared with numerical solutions. The comparison has been represented graphically, and the Tabler form shows the effectiveness and advantages of the featured techniques. The effect of the parameters on concentration is also discussed.

The dimension reduction method of two-grid Crank–Nicolson mixed finite element solution coefficient vectors for nonlinear fourth-order reaction diffusion equation with temporal fractional derivative

Herein, we mainly resort to a proper orthogonal decomposition (POD) to study the dimension reduction of unknown solution coefficient vectors in the two-grid Crank–Nicolson mixed finite element (CNMFE) (TGCNMFE) method for the nonlinear fourth-order reaction diffusion equation with temporal fractional derivative and establish a new reduced-dimension extrapolated TGCNMFE (RDETGCNMFE) method. For this purpose, we firstly retrospect the TGCNMFE method for the nonlinear fourth-order reaction diffusion equation and provide the existence, unconditional stability, and error estimates of the TGCNMFE solutions. Thereafter, we use the POD method to reduce the dimensionality of the unknown TGCNMFE solution coefficient vectors of the nonlinear fourth-order reaction diffusion equation and develop the new RDETGCNMFE method, and use matrix analysis to analyze the existence, unconditional stability, and errors of the RDETGCNMFE solutions. Finally, we perform numerical experiments to verify our theoretical results and show the advantages of the RDETGCNMFE method. It is worth noting that the RDETGCNMFE method herein is completely distinguished from the existing ones. Hence, the work in this paper is brand-new.

A Harnack inequality for a class of 1D nonlinear reaction–diffusion equations and applications to wave solutions

In this paper, a differential-geometric method is applied to build some Li–Yau–Hamilton-type Harnack inequalities for the positive solutions to a one spatial dimensional nonlinear reaction–diffusion equation in a plane geometry. The class of reaction–diffusion equation that is considered here contains several important equations some of which are Newel–Whitehead–Segel, Allen–Cahn and Fisher–KPP equations. The Harnack inequalities so derived are used to discuss some other important properties of positive solutions and in the characterization of positive wave solutions.

Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaces

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

Mixed finite element methods for nonlinear reaction–diffusion equations with interfaces

We develop mixed finite element methods for nonlinear reaction–diffusion equations with interfaces which have Robin-type interface conditions. We introduce the velocity of chemicals as new variables and reformulate the governing equations. The stability of semidiscrete solutions, existence and the a priori error estimates of fully discrete solutions are proved by fixed point theorem and continuous/discrete Grönwall inequalities. Numerical results illustrating our theoretical analysis are included.

A kinetic non-steady state analysis of immobilized enzyme systems with external mass transfer resistance

<abstract><p>The goal of this paper is to utilize the homotopy perturbation method (HPM) and Laplace transform to provide an approximate analytical expression to the non-linear time-dependent reaction diffusion equation arising in a mathematical model of an immobilized enzyme system with external mass transfer resistance. This mathematical model is a non-steady, non-linear reaction diffusion equation based on Michaelis–Menten kinetics. Approximate analytical expressions are also provided for various geometries of the enzyme catalytic pellets, namely, planar, cylindrical, and spherical. Obtained semi-analytical expressions are proven to fit for all the parameters appearing in the system and for all the geometries of enzyme catalytic pellets. When comparing the numerical and approximate analytical solutions, satisfactory results are obtained. Also, approximate analytical expressions of the effectiveness factor (EF) of the immobilized system are presented, and the effect of parameters on the EF is also analyzed.</p></abstract>

Mathematical analysis of urea amperometric biosensor with Non-Competitive inhibition for Non-Linear Reaction-Diffusion equations with Michaelis-Menten kinetics

This article examines a mathematical model for stable urea amperometric biosensors with non-competitive inhibition under homogenous conditions. The system is based on reaction and diffusion equations with a non-linear component associated with the Michaelis-Menten kinetics of the enzyme reaction. Theoretical findings in this study can be used to examine the impact of various factors, including the Michaelis-Menten constant and the Thiele modulus. The steady-state non-linear reaction–diffusion equations were solved using two effective and widely used analytical approaches, the Akbari-Ganji method (AGM) and the differential transform method (DTM). The generalized approximation analytical solution for the concentrations of the substrate, inhibitor, and product, as well as the current for the experimental values of the parameter, is developed. The current can also be used to analyze the resistance and sensitivity of biosensors. The digital simulation was carried out utilizing MATLAB software. The numerical results help to validate the analytical results. The impact of membrane thickness, maximum enzymatic rate, and substrate concentration on biosensor response was evaluated.

A first-order fractional–steps–type method to approximate a nonlinear reaction–diffusion equation with homogeneous Cauchy–Neumann boundary conditions

A first-order fractional–steps–type method to approximate a nonlinear reaction–diffusion equation with homogeneous Cauchy–Neumann boundary conditions

The Holby–Morgan Model of Platinum Catalyst Degradation in PEM Fuel Cells: Range of Feasible Parameters Achieved Using Voltage Cycling

Loss of electrochemical surface area in proton-exchange membrane is of large practical importance, since membrane degradation largely affects the durability and life of fuel cells. In this paper, the electrokinetic model developed by Holby and Morgan is considered. The paper describes degradation mechanisms in membrane catalyst presented by platinum dissolution, platinum diffusion, and platinum oxide formation. A one-dimensional model is governed by nonlinear reaction–diffusion equations given in a cathodic catalyst layer using Butler–Volmer relationships for reaction rates. The governing system is endowed with initial conditions, mixed no-flux boundary condition at the interface with gas diffusion layer, and a perfectly absorbing condition at the membrane boundary. In cyclic voltammetry tests, a non-symmetric square waveform is applied for the electric potential difference between 0.6 and 0.9 V held for 10 and 30 s, respectively, according to the protocol of European Fuel Cell and Hydrogen Joint Undertaking. Aimed at mitigation strategies, the impact of cycling operating conditions and model parameters on the loss rate of active area is investigated. The global behavior with respect to variation of parameters is performed using the method of sensitivity analysis. Finding feasible and unfeasible values helps to determine the range of test parameters employed in the model. Comprehensive results of numerical simulation tests are presented and discussed.

Very high-order finite difference method on arbitrary geometries with Cartesian grids for non-linear convection diffusion reaction equations

An arbitrary order finite difference method for solving non-linear convection Diffusion Reaction equations in curved boundary domains with Cartesian grid is proposed. Ghost points' values are determined with the Reconstruction Off-Site Data based on a polynomial interpolation using the least square method with constraints to enforce the boundary conditions. We propose a second-, fourth-, and sixth-order schemes for linear non-constant coefficients problem in both the conservative and non-conservative scalar equations. Extensions to non-linear scalar problems and systems are then implemented while preserving the optimal orders. Numerical simulations are carried out to provide evidence about the convergence order and the stability of the method.

Analytical expression for the model that describes the heterogeneous reaction-diffusion process with immobilized enzyme (penicillin G acylase)

This research article examines the reaction-diffusion process in an immobilized enzyme batch reactor. The model incorporates strongly non-linear factors that are associated with standard Michaelis-Menten kinetics. The non-linear reaction-diffusion equations for substrate and product concentrations have been approximated analytically. Employing two different semi-analytical methods, Akbari-Ganji's method (AGM) and the modified Adomian decomposition method (MADM), to compute the dimensionless steady-state solutions to the system of non-linear differential equations for all values of reaction parameters. In addition, the dynamics of the mean integrated effectiveness factor of penicillin acylase in porous spherical particles have been presented for the determination of the local effectiveness factor. In order to gauge the potency of our proposed solution, we compare two semi-analytical results with a numerical result that are in good agreement across the whole concentration range. The proposed formulation aims to simulate the dynamic performance of the system utilizing the parameters and would enhance the determination of the optimum particle size for enzyme catalysts.

Traveling waves for discrete reaction-diffusion equations in the general monostable case

We consider general fully nonlinear discrete reaction-diffusion equations ut=F[u], described by some function F. In the positively monostable case, we study monotone traveling waves of velocity c, connecting the unstable state 0 to a stable state 1. Under Lipschitz regularity of F, we show that there is a minimal velocity cF+ such that there is a branch of traveling waves with velocities c≥cF+ and no traveling waves for c<cF+. We also show that the map F↦cF+ is not continuous for the L∞ norm on F. Assuming more regularity of F close to the unstable state 0, we show that cF+≥cF⁎, where the velocity cF⁎ can be computed from the linearization of the equation around the unstable state 0. In addition, we show that the inequality can be strict for certain nonlinearities F. On the contrary, under a KPP condition on F, we show the equality cF+=cF⁎. Finally, we provide an example in which cF+ is negative.

Second-order efficient algorithm for coupled nonlinear model of groundwater transport system

In this paper, an efficient modified two-grid block-centered finite difference algorithm with second-order backward difference formula is presented for coupled nonlinear reaction-diffusion equations of the solutes in the groundwater flow. The main steps of the modified two-grid algorithm include dealing with a small-scale nonlinear problem on the coarse grid with a large time step size and then dealing with a large-scale linear problem on the fine grid with a smaller time step size base on the second-order backward difference formula. Moreover, the theoretical analysis illustrates that the modified two-grid algorithm is able to maintain second convergence order in space-time domain. Several numerical experiments including a practical example are carried out to support the theoretical results and verify that the modified two-grid block-centered finite difference with second-order backward difference formula algorithm can achieve high efficiency while maintaining the second-order accuracy, meanwhile, has a certain applicable value in actual problem.

A hybrid fourth order time stepping method for space distributed order nonlinear reaction-diffusion equations

A new fourth order highly efficient Exponential Time Differencing Runge-Kutta type hybrid time stepping method is developed by hybriding two fourth order methods. The new hybrid method takes advantages of a positivity preserving L-stable method to damp unnatural oscillations due to low regularity of the initial data and of a computationally efficient A-stable method to achieve optimal order convergence. Computational efficiency and stability of the new method is further enhanced by applying a splitting technique which makes it possible to implement the method on parallel processors. A hybriding criteria is presented and an algorithm based on the hybrid method is developed. The method is implemented to solve two two-dimensional problems having Riesz space distributed order diffusion and nonlinear reaction terms, an Allen-Cahn equation with a cubic nonlinearity and an Enzyme Kinetics equation with a rational nonlinearity. Fourth-order temporal convergence is obtained through numerical experiments. A third test problem with exact solution available is also considered and both spatial as well as temporal orders of convergence are computed. High computational efficiency of the method is shown by recording the CPU time. Numerical solutions using different order strength distribution functions are also presented.

Semi-analytical Expressions of a Non-linear Reaction-Diffusion (R-D) Equations in a Mixed Quadratic-Cubic Autocatalytic Reactions

The semi-analytical expressions for the autocatalytic reactions with the mixed cubic and quadratic terms are derived. The kinetic model is associated with the diffusion, which is considered in a one-dimensional reactor. The semi-analytical solutions are derived for the concentrations of dimensionless reactant and dimensionless autocatalyst in the cubic autocatalytic reaction-diffusion equations for the steady-state and non-steady state by using the Homotopy analysis method (HAM). The derived approximate analytical solutions are compared with the numerical simulation and found to be very good fit for all values of the dimensionless parameters.

Exact solutions of hyperbolic reaction-diffusion equations in two dimensions

Exact solutions are constructed for a class of nonlinear hyperbolic reaction-diffusion equations in two-space dimensions. Reduction of variables and subsequent solutions follow from a special nonclassical symmetry that uncovers a conditionally integrable system, equivalent to the linear Helmholtz equation. The hyperbolicity is commonly associated with a speed limit due to a delay, \(\tau\), between gradients and fluxes. With lethal boundary conditions on a circular domain wherein a species population exhibits logistic growth of Fisher–KPP type with equal time lag, the critical domain size for avoidance of extinction does not depend on \(\tau\). A diminishing exact solution within a circular domain is also constructed, when the reaction represents a weak Allee effect of Huxley type. For a combustion reaction of Arrhenius type, the only known exact solution that is finite but unbounded is extended to allow for a positive \(\tau\). doi: 10.1017/S1446181123000093

MODEL TO STUDY INTERDEPENDENT CALCIUM AND IP3 DISTRIBUTION REGULATING NFAT PRODUCTION IN T LYMPHOCYTE

The cooperative system of calcium (Ca[Formula: see text]) and inositol 1,4,5-trisphosphate (IP3) plays a crucial role in maintaining structure and functions of T lymphocyte cells including the production of nuclear factor of activated T cells (NFAT) as an immunological response. NFAT regulates several mechanisms and dysregulation in this process can lead to various disorders like inflammatory bowel diseases, etc. In this study, the two-way feedback model of Ca[Formula: see text] and IP3 dynamics regulating NFAT production in T lymphocytes has been constructed. A system of nonlinear reaction-diffusion equations for calcium and IP3 for a one-dimensional case has been employed in this proposed model. The numerical simulation is performed using finite element and the Crank–Nicholson method. The effects of the parameters like Ryanodine Receptor (RyR), source amplitude and buffers on active NFAT production have been analyzed. The novel insights about regulatory and dysregulation events and mechanisms of Ca[Formula: see text], IP3 and NFAT are concluded from these results.

On Finite-Time Blow-Up Problem for Nonlinear Fractional Reaction Diffusion Equation: Analytical Results and Numerical Simulations

The study of the blow-up phenomenon for fractional reaction–diffusion problems is generally deemed of great importance in dealing with several situations that impact our daily lives, and it is applied in many areas such as finance and economics. In this article, we expand on some previous blow-up results for the explicit values and numerical simulation of finite-time blow-up solutions for a semilinear fractional partial differential problem involving a positive power of the solution. We show the behavior solution of the fractional problem, and the numerical solution of the finite-time blow-up solution is also considered. Finally, some illustrative examples and comparisons with the classical problem with integer order are presented, and the validity of the results is demonstrated.

Theoretical analysis and steady-state responses of the multienzyme amperometric biosensor system for nonlinear reaction-diffusion equations

In this study, a mathematical model for a multienzyme biosensor system is presented. This system identifies analytes by chemical conversion into the signal response. It is based on nonlinear reaction-diffusion equations for a bisubstrate amperometric system under steady-state conditions. A precise parametric analysis is crucial for a better understanding of the nonlinear multienzyme biosensor. The Akbari-Ganji method is used to solve the system of nonlinear differential equations. AGM is already utilized to solve differential equations in heat transfer science. In this study, it is employed for the first time to approximate the solution of a nonlinear multienzyme biosensor. An approximate analytical expression for the concentration of substrate, inhibitor, co-substrate, product 1, product 3 and current are obtained. Furthermore, the current is used to determine the sensitivity and resistance of a biosensor. This study uses the MATLAB tool to describe the numerical simulation of the problem. A satisfactory level of agreement was found between the analytical and simulation results for the experimental parameter values. The influence of the parameters on sensitivity, resistance and current are also investigated using sensitivity analysis. Current and sensitivity are strongly influenced by the substrate concentration in the bulk solution and the glucose reaction constant, respectively. The most significant factor influencing biosensor resistance is membrane thickness. This work serves as a resource for further research into concentration models and sheds light on relevant applications, such as food safety, biomedicine, and environmental research.