scholarly journals Mathematical Analysis of Reaction–Diffusion Equations Modeling the Michaelis–Menten Kinetics in a Micro-Disk Biosensor

Molecules ◽  
2021 ◽  
Vol 26 (23) ◽  
pp. 7310
Author(s):  
Naveed Ahmad Khan ◽  
Fahad Sameer Alshammari ◽  
Carlos Andrés Tavera Romero ◽  
Muhammad Sulaiman ◽  
Ghaylen Laouini
Keyword(s):  
Mathematical Model ◽  
Hydrogen Peroxide ◽  
Complexity Analysis ◽  
Enzymatic Reaction ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Supervised Machine Learning ◽  
Reaction Diffusion Equation ◽  
Data Set ◽  
Dimensionless Concentration

In this study, we have investigated the mathematical model of an immobilized enzyme system that follows the Michaelis–Menten (MM) kinetics for a micro-disk biosensor. The film reaction model under steady state conditions is transformed into a couple differential equations which are based on dimensionless concentration of hydrogen peroxide with enzyme reaction (H) and substrate (S) within the biosensor. The model is based on a reaction–diffusion equation which contains highly non-linear terms related to MM kinetics of the enzymatic reaction. Further, to calculate the effect of variations in parameters on the dimensionless concentration of substrate and hydrogen peroxide, we have strengthened the computational ability of neural network (NN) architecture by using a backpropagated Levenberg–Marquardt training (LMT) algorithm. NNs–LMT algorithm is a supervised machine learning for which the initial data set is generated by using MATLAB built in function known as “pdex4”. Furthermore, the data set is validated by the processing of the NNs–LMT algorithm to find the approximate solutions for different scenarios and cases of mathematical model of micro-disk biosensors. Absolute errors, curve fitting, error histograms, regression and complexity analysis further validate the accuracy and robustness of the technique.

Nonlinear solution of the reaction–diffusion equation using a two-step third–fourth-derivative block method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Oluwaseun Adeyeye ◽  
Ali Aldalbahi ◽  
Jawad Raza ◽  
Zurni Omar ◽  
Mostafizur Rahaman ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Block Method ◽  
Wide Range ◽  
Higher Derivatives ◽  
Fourth Derivative ◽  
Improved Accuracy

AbstractThe processes of diffusion and reaction play essential roles in numerous system dynamics. Consequently, the solutions of reaction–diffusion equations have gained much attention because of not only their occurrence in many fields of science but also the existence of important properties and information in the solutions. However, despite the wide range of numerical methods explored for approximating solutions, the adoption of block methods is yet to be investigated. Hence, this article introduces a new two-step third–fourth-derivative block method as a numerical approach to solve the reaction–diffusion equation. In order to ensure improved accuracy, the method introduces the concept of nonlinearity in the solution of the linear model through the presence of higher derivatives. The method obtained accurate solutions for the model at varying values of the dimensionless diffusion parameter and saturation parameter. Furthermore, the solutions are also in good agreement with previous solutions by existing authors.

scholarly journals A Modified Techniques of Fractional-Order Cauchy-Reaction Diffusion Equation via Shehu Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Mounirah Areshi ◽  
A. M. Zidan ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Fractional Order ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Closed Form Solutions ◽  
Homotopy Perturbation ◽  
In Series ◽  
The Given

In this article, the iterative transformation method and homotopy perturbation transformation method are applied to calculate the solution of time-fractional Cauchy-reaction diffusion equations. In this technique, Shehu transformation is combined of the iteration and the homotopy perturbation techniques. Four examples are examined to show validation and the efficacy of the present methods. The approximate solutions achieved by the suggested methods indicate that the approach is easy to apply to the given problems. Moreover, the solution in series form has the desire rate of convergence and provides closed-form solutions. It is noted that the procedure can be modified in other directions of fractional order problems. These solutions show that the current technique is very straightforward and helpful to perform in applied sciences.

A PARALLEL SOLVER FOR REACTION–DIFFUSION SYSTEMS IN COMPUTATIONAL ELECTROCARDIOLOGY

2004 ◽  
Vol 14 (06) ◽  
pp. 883-911 ◽  
Author(s):  
PIERO COLLI FRANZONE ◽  
LUCA F. PAVARINO
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Ionic Currents ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Fiber Direction ◽  
Axially Symmetric ◽  
Reaction Diffusion Systems ◽  
Cardiac Model

In this work, a parallel three-dimensional solver for numerical simulations in computational electrocardiology is introduced and studied. The solver is based on the anisotropic Bidomain cardiac model, consisting of a system of two degenerate parabolic reaction–diffusion equations describing the intra and extracellular potentials of the myocardial tissue. This model includes intramural fiber rotation and anisotropic conductivity coefficients that can be fully orthotropic or axially symmetric around the fiber direction. The solver also includes the simpler anisotropic Monodomain model, consisting of only one reaction–diffusion equation. These cardiac models are coupled with a membrane model for the ionic currents, consisting of a system of ordinary differential equations that can vary from the simple FitzHugh–Nagumo (FHN) model to the more complex phase-I Luo–Rudy model (LR1). The solver employs structured isoparametric Q1finite elements in space and a semi-implicit adaptive method in time. Parallelization and portability are based on the PETSc parallel library. Large-scale computations with up to O(107) unknowns have been run on parallel computers, simulating excitation and repolarization phenomena in three-dimensional domains.

scholarly journals Uniqueness of fast travelling fronts in reaction–diffusion equations with delay

2008 ◽  
Vol 464 (2098) ◽  
pp. 2591-2608 ◽  
Author(s):  
Maitere Aguerrea ◽  
Sergei Trofimchuk ◽  
Gabriel Valenzuela
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Delayed Reaction ◽  
Birth Function ◽  
Scale Of Banach Spaces ◽  
Large Velocity ◽  
Travelling Fronts ◽  
Equations With Delay

We consider positive travelling fronts, u ( t ,  x )= ϕ ( ν . x + ct ), ϕ (−∞)=0, ϕ (∞)= κ , of the equation u t ( t ,  x )=Δ u ( t ,  x )− u ( t ,  x )+ g ( u ( t − h ,  x )), x ∈ m . This equation is assumed to have exactly two non-negative equilibria: u 1 ≡0 and u 2 ≡ κ >0, but the birth function g ∈ C 2 ( ,  ) may be non-monotone on [0, κ ]. We are therefore interested in the so-called monostable case of the time-delayed reaction–diffusion equation. Our main result shows that for every fixed and sufficiently large velocity c , the positive travelling front ϕ ( ν . x + ct ) is unique (modulo translations). Note that ϕ may be non-monotone. To prove uniqueness, we introduce a small parameter ϵ =1/ c and realize a Lyapunov–Schmidt reduction in a scale of Banach spaces.

The evolution of travelling waves in reaction-diffusion equations with monotone decreasing diffusivity. II. Abruptly vanishing diffusivity

1995 ◽  
Vol 350 (1694) ◽  
pp. 361-378 ◽  
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Wave Form ◽  
Initial Development ◽  
Finite Concentration ◽  
Distinct Process ◽  
And Diffusion

In this paper we continue our study of some of the qualitative features of chemical polymerization processes by considering a reaction-diffusion equation for the chemical concentration in which the diffusivity vanishes abruptly at a finite concentration. The effect of this diffusivity cut-off is to create two distinct process zones; in one there is both reaction and diffusion and in the other there is only reaction. These zones are separated by an interface across which there is a jump in concentration gradient. Our analysis is focused on both the initial development of this interface and the large time evolution of the system into a travelling wave form. Some distinct differences from our previous analysis of smoothly vanishing diffusivity are found.

scholarly journals An extension of the principle of spatial averaging for inertial manifolds

1999 ◽  
Vol 66 (1) ◽  
pp. 125-142 ◽  
Author(s):  
Hyukjin Kwean
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Inertial Manifolds ◽  
Inertial Manifold ◽  
Specific Class ◽  
Spatial Averaging ◽  
The Laplace Operator

AbstractIn this paper we extend a theorem of Mallet-Paret and Sell for the existence of an inertial manifold for a scalar-valued reaction diffusion equation to new physical domains ωn ⊂ Rn, n = 2,3. For their result the Principle of Spatial Averaging (PSA), which certain domains may possess, plays a key role for the existence of an inertial manifold. Instead of the PSA, we define a weaker PSA and prove that the domains φn with appropriate boundary conditions for the Laplace operator, δ, satisfy a weaker PSA. This weaker PSA is enough to ensure the existence of an inertial manifold for a specific class of scalar-valued reaction diffusion equations on each domain ωn under suitable conditions.

Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods

2014 ◽  
Vol 6 (2) ◽  
pp. 203-219 ◽  
Author(s):  
Luoping Chen ◽  
Yanping Chen
Keyword(s):  
Finite Element ◽  
Nonlinear Equations ◽  
Finite Element Methods ◽  
Mixed Finite Element ◽  
Reaction Diffusion ◽  
Mixed Finite Element Methods ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Fine Mesh ◽  
Nonlinear Reaction

AbstractIn this paper, we study an efficient scheme for nonlinear reaction-diffusion equations discretized by mixed finite element methods. We mainly concern the case when pressure coefficients and source terms are nonlinear. To linearize the nonlinear mixed equations, we use the two-grid algorithm. We first solve the nonlinear equations on the coarse grid, then, on the fine mesh, we solve a linearized problem using Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfyH=O(h1/2). As a result, solving such a large class of nonlinear equations will not be much more difficult than getting solutions of one linearized system.

scholarly journals Kompiuterinis optinio biojutiklio savybių tyrimas taikant bedimensį modelį

2009 ◽  
Vol 50 ◽  
pp. 306-310
Author(s):  
Evelina Gaidamauskaitė ◽  
Romas Baronas
Keyword(s):  
Mathematical Model ◽  
Digital Simulation ◽  
Reaction Diffusion ◽  
Optical Biosensor ◽  
Reaction Diffusion Equations ◽  
Computational Investigation ◽  
Linear Reaction ◽  
The Mathematical Model ◽  
Enzyme Layer ◽  
Biosensor Response

Šiame darbe, siekiant nustatyti pagrindinius kinetinius peroksidazinio optinio biojutiklio matematinio modelio parametrus, buvo sudarytas bedimensis modelis. Biojutikliui taikomos reakcijos-difuzijos lygtys su netiesiniu nariu, aprašančiu fermentinę reakciją. Biojutiklio veikimas modeliuojamas fermento ir difuzijos sluoksniuose. Ištirta biojutiklio atsako ir jautrio priklausomybė nuo bedimensio biojutiklio modulio. Suformuluotas uždavinys sprendžiamas baigtinių skirtumų metodu. Gauti rezultatai pagrindžia šio modelio pritaikomumą. Atliekami peroksidazinio optinio biojutiklio eksperimentiniai tyrimai leis nustatyti modelio taikymo ribas.A Computational Investigation of the Optical Biosensor by a Dimensionless ModelEvelina Gaidamauskaitė, Romas Baronas SummaryIn order to determine the main governing parameters, a dimensionless mathematical model of a peroxidase-based optical biosensor is derived. The mathematical model of the biosensor is based on a system of non-linear reaction-diffusion equations. The modelled biosensor comprises two compartments, an enzyme layer and an outer diffusion layer. The influence of the dimensionless diffusion modulus on the biosensor response and the sensitivity is investigated. The digital simulation was carried out using a finite difference method.

SPREAD RATES UNDER TEMPORAL VARIABILITY: CALCULATION AND APPLICATION TO BIOLOGICAL INVASIONS

2011 ◽  
Vol 21 (12) ◽  
pp. 2469-2489 ◽  
Author(s):  
GUNOG SEO ◽  
FRITHJOF LUTSCHER
Keyword(s):  
Biological Invasions ◽  
Temporal Variability ◽  
Life Histories ◽  
Wave Speed ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Coefficient Function ◽  
Compartment System ◽  
Minimal Wave Speed

In this paper, we introduce a technique to study the minimal wave speed in reaction-diffusion equations with temporal variability and apply it to two particular models for biological invasions. We use the exponential transform to avoid solving partial differential equations explicitly or finding inverse transforms. In a single reaction-diffusion equation with time-periodic coefficients, the minimal wave speed depends only on time-averages of each coefficient function. In a two-compartment system with mobile and stationary individuals, the invasion speed depends on the precise form of the coefficient functions and their temporal correlations; in some cases, a lower bound can be obtained. Our technique can be extended to more complex life histories of invading organisms.

Analytical Studies of Diffusion by the Dissociative Mechanism in the Case of a Foreign Atom Source

2004 ◽  
Vol 233-234 ◽  
pp. 15-28
Author(s):  
A. Benmakhlouf
Keyword(s):  
Mathematical Model ◽  
Perturbation Method ◽  
Numerical Study ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Analytical Treatment ◽  
Dissociative Mechanism ◽  
Foreign Atom ◽  
Atom Diffusion ◽  
Atom Source

The analytical treatment of dissociative diffusion by using the matched perturbation method given in the literature deals with a virtually infinite foreign-atom source producing a constant^concentration at the boundary. In this paper, a new mathematical model is developed for analysing the dissociative diffusion of the solute atoms in the case of finite-source conditions. The mathematical model combines the reaction-diffusion equations which govern solute atom diffusion by the dissociative mechanism and the boundary condition expressing the fact that the rate at which solute leaves the source is always equal to that at which it enters the sheet over the surface x=0. Solutions obtained by applying the matched perturbation method and their comparison with those of the numerical study are also presented in this paper.

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