Nonlinear Reaction-diffusion Equations Research Articles

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.

Comparative analysis of three memory selection methods for time integration of Fractional Reaction-Diffusion Equations

By discretising in space, a non-linear time fractional reaction-diffusion equations (TFRDEs) can be converted into a system of time-fractional differential equations (TFDEs). The full memory method (FMM) and short memory method (SMM) are well-established memory selection methods used in the time integration of TFDEs. The main drawbacks of FMM and SMM are higher computational cost and uncontrollable error respectively. The only way to increase the accuracy of SMM is by increasing short memory length which causes an increase in computational cost. Especially when we apply these two methods to integrate TFRDEs, we have to solve a large system of TFDEs. Therefore, the drawbacks of these two methods affect seriously, when these are applied to solve TFRDEs. This paper aims to investigate the accuracy and efficiency of the memory selection method, Exponentially Decreasing Random Memory Method (EDRMM), and compare it with FMM and SMM when these methods apply to integrate TFRDEs. Based on these three memory selection methods, three semi-implicit numerical schemes namely semiimplicit scheme with full memory method (SI-FMM), semi-implicit scheme with short memory method (SI-SMM), and semi-implicit scheme with exponentially decreasing random memory method (SI-EDRMM)) are proposed and the accuracy and CPU time (computational time (CT)) of these three numerical schemes are compared. To do this comparison, these three numerical schemes are applied to four TFRDEs whose exact solutions are known. Numerical experiments confirm that the accuracy and efficiency of the SI-EDRMM are better than that of SI-SMM and the efficiency of SI-EDRMM is higher than that of SI-FMM. Therefore, EDRMM is better than SMM and FMM for the integration of TFRDEs.

A hybrid approach for piecewise fractional reaction–diffusion equations

In this paper, the Caputo and Atangana–Baleanu fractional derivatives are handled to introduce a type of piecewise fractional derivative. More precisely, a linear combination of the Caputo and Atangana–Baleanu fractional derivatives are considered in each sub-interval to define this fractional derivative. It is employed to generate another form of nonlinear reaction–diffusion equations. The orthonormal Legendre polynomials together with the orthonormal piecewise Legendre functions are used to make a hybrid algorithm for this new problem. In this way, an explicit formula for computing the piecewise fractional differentiation of the stated piecewise basis functions is obtained and applied in generating the method. The applicability and validity of the adopted procedure are examined through three examples.

Steady state concentration in an amperometric biosensor using mathematical modeling

This paper presents an analytical solution to the model. By solving the system of nonlinear reaction-diffusion equations using Akbari–Ganji’s method (AGM), we can derive the semi-analytical expressions for substrate and oxidized and reduced mediator concentrations. These models provide information about enzyme electrode mechanisms and their kinetics. These modeling results can be helpful for sensor design, optimization, and determining how the electrode will react. Further, we studied the effects of many physical and biological parameters on model prediction. The mathematical equations, which are nonlinear partial differential equations, are solved using AGM. Also, we derived the semi-analytical results for concentrations for saturated and unsaturated kinetics. It is verified that the proposed solution is validated by comparing it with numerical solutions.

The Shigesada–Kawasaki–Teramoto model: Conditional symmetries, exact solutions and their properties

We study a simplification of the well-known Shigesada–Kawasaki–Teramoto model, which consists of two nonlinear reaction–diffusion equations with cross-diffusion. A complete set of Q-conditional (nonclassical) symmetries is derived using an algorithm adopted for the construction of conditional symmetries. The symmetries obtained are applied for finding a wide range of exact solutions, possible biological interpretation of some of which being presented. Moreover, an alternative application of the simplified model related to the polymerization process is suggested and exact solutions are found in this case as well.

Well-posedness results for a class of nonlinear reaction-diffusion equations with memory

In this paper, we are interested to study the initial value problem for nonlinear heat equation with memory. For our problem, we show that the local existence theory related to the finite time blow-up is also obtained for the problem with logarithmic nonlinearity. We also obtain the blowup continuation property of the mild solution. The principal techniques based on some Sobolev embeddings with \(L^p\) spaces.

Mathematical Model of Synaptic Long-Term Potentiation as a Bistability in a Chain of Biochemical Reactions with a Positive Feedback.

Nitric oxide (NO) is involved in synaptic long-term potentiation (LTP) by multiple signaling pathways. Here, we show that LTP of synaptic transmission can be explained as a feature of signal transduction-bistable behavior in a chain of biochemical reactions with positive feedback, formed by diffusion of NO to the presynaptic site and facilitating the release of glutamate (Glu). The dynamics of Glu, calcium (Ca2+) and NO is described by a system of nonlinear reaction-diffusion equations with modified Michaelis-Menten (MM) kinetics. Numerical investigation reveals that the chain of biochemical reactions analyzed can exhibit a bistable behavior under physiological conditions when production of Glu is described by MM kinetics and decay of NO is modeled by means of two enzymatic pathways with different kinetic properties. Our finding extends understanding of the role of NO in LTP: a short high-intensity stimulus is "memorized" as a long-lasting elevation of NO concentration. The conclusions obtained by analysis of the chain of biochemical reactions describing LTP can be generalized to other chains of interactions or for creating the logical elements for biological computers.

Numerical investigations and simulation of calcium distribution in the alpha-cell

The α-cells are a part of islets of Langerhans located in the pancreas and are responsible for glucagon secretion. Calcium signaling is crucial for the regulation of the functions and structure of these α-cells and the same is still not well understood. Here a mathematical model is framed to obtain more insights into calcium signaling in α-cells. The non-linear reaction-diffusion equation for calcium signaling along with boundary conditions is employed to propose the model for a one-dimensional steady-state case. The numerical solutions were obtained using the Newton-Raphson method and the cubic spline method. The combination of Newton-Raphson and cubic spline has proved to be quite effective in numerical simulations and in generating deeper insights into calcium regulation in an α-cell under various conditions. The results provide information about changes in source influx, buffers, ER leak, and SERCA pump leading to disturbances in calcium homeostasis, which can be responsible for the development of diabetes and other metabolic disorders.