Reaction–Diffusion Equations and Their Application on Bacterial Communication

Kuttler, Christina

doi:10.1016/bs.host.2017.07.003

Cited by 18 publications

(18 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For locally integrable functions φ (t) on [0, T ) define the operator B by Bφ (t) = g(t) t 0 (t − τ) β −1 φ (τ) dτ. Then for each integer n by successively applying (19) we obtain…”

Section: Using Young's Inequality For Convolutionmentioning

confidence: 99%

On the identification of a nonlinear term in a reaction–diffusion equation

Kaltenbacher

Rundell

2019

Inverse Problems

View full text Add to dashboard Cite

Reaction-diffusion equations are one of the most common partial differential equations used to model physical phenomenon. They arise as the combination of two physical processes: a driving force f (u) that depends on the state variable u and a diffusive mechanism that spreads this effect over a spatial domain. The canonical form is u t − △u = f (u). Application areas include chemical processes, heat flow models and population dynamics. As part of the model building, assumptions are made about the form of f (u) and these inevitably contain various physical constants. The direct or forwards problem for such equations is now very well-developed and understood, especially when the diffusive mechanism is governed by Brownian motion resulting in an equation of parabolic type.However, our interest lies in the inverse problem of recovering the reaction term f (u) not just at the level of determining a few parameters in a known functional form, but recovering the complete functional form itself. To achieve this we set up the usual paradigm for the parabolic equation where u is subject to both given initial and boundary data, then prescribe overposed data consisting of the solution at a later time T . For example, in the case of a population model this amounts to census data at a fixed time. Our approach will be two-fold. First we will transform the inverse problem into an equivalent nonlinear mapping from which we seek a fixed point. We will be able to prove important features of this map such as a self-mapping property and give conditions under which it is contractive. Second, we consider the direct map from f through the partial differential operator to the overposed data. We will investigate Newton schemes for this case.Classical, Brownian motion diffusion is not the only version and in recent decades various anomalous processes have been used to generalize this case. Amongst the most popular is one that replaces the usual time derivative by a subdiffusion process based on a fractional derivative of order α ≤ 1. We will also include this model in our analysis. The final section of the paper will show numerical reconstructions that demonstrate the viability of the suggested approaches. This will also include the dependence of the inverse problem on both T

show abstract

Section: Using Young's Inequality For Convolutionmentioning

confidence: 99%

On the identification of a nonlinear term in a reaction–diffusion equation

Kaltenbacher

Rundell

2019

Inverse Problems

View full text Add to dashboard Cite

show abstract

“…We will now consider convergence of the fixed point scheme defined by the operator T defined by (17) for reconstructing the reaction and interaction functions f i in (9). To some extent we can here build on previous work for the case of one scalar equation and a single function f to be reconstructed in [16].…”

Section: Convergence Of a Fixed Point Scheme For Final Time Datamentioning

confidence: 99%

“…We will show the results of numerical experiments using the basic versions of the iterative schemes defined by (17) and (21) for each of the two data types: time trace data consisting of the value of h(t) := u(x 0 ,t) for t ∈ [0, T ]; final time data g(x) := u(x, T ) for some chosen value of T . The numerical results presented will be set in one space dimension although there is no limitation in this regard (other than computational complexity of the direct solvers) as our unknowns are functions of a single variable.…”

Section: Reconstructionsmentioning

confidence: 99%

The Inverse Problem of Reconstructing Reaction-Diffusion Systems

Kaltenbacher,

Rundell

2020

Preprint

View full text Add to dashboard Cite

This paper considers the inverse problem of recovering state-dependent source terms in a reaction-diffusion system from overposed data consisting of the values of the state variables either at a fixed finite time (census-type data) or a time trace of their values at a fixed point on the boundary of the spatial domain. We show both uniqueness results and the convergence of an iteration scheme designed to recover these sources. This leads to a reconstructive method and we shall demonstrate its effectiveness by several illustrative examples.

show abstract

“…In addition, taking into account heterogeneous space distributions of AHL and Lactonase concentrations for the Gram-negative bacteria species Pseudomonas putida the reaction-diffusion models of quorum sensing have been proposed in the form of initial-boundary value problems for a system of parabolic partial differential equations (PDEs) [18,19], time-lagging parabolic partial differential equations [20] as well as fractional partial differential equations [21,22]. In particular, the study [19] has reported the numerical simulation results with a focus on AHL and Lactonase concentrations changing under the external addition of substances.…”

Section: Introductionmentioning

confidence: 99%

Optimal multiplicative control of bacterial quorum sensing under external enzyme impact

Maslovskaya¹,

Kuttler²,

Chebotarëv³

et al. 2022

Math. Model. Nat. Phenom.

Self Cite

View full text Add to dashboard Cite

The use of external enzymes provides an alternative way of reducing communication in pathogenic bacteria that may lead to the degradation of their signal and the loss of their pathogeneity. The present study considers an optimal control problem for the semilinear reaction-diffusion model of bacterial quorum sensing under the impact of external enzymes. Estimates of the solution of the controlled system are obtained, on the basis of which the solvability of the extremal problem is proved and the necessary optimality conditions of the first-order are derived. A numerical algorithm to find a solution of the optimal control problem is constructed and implemented. The conducted numerical experiments demonstrate an opportunity to build an effective strategy of the enzymes impact for treatment.

show abstract

Reaction–Diffusion Equations and Their Application on Bacterial Communication

Cited by 18 publications

References 24 publications

On the identification of a nonlinear term in a reaction–diffusion equation

On the identification of a nonlinear term in a reaction–diffusion equation

The Inverse Problem of Reconstructing Reaction-Diffusion Systems

Optimal multiplicative control of bacterial quorum sensing under external enzyme impact

Contact Info

Product

Resources

About