scholarly journals Exact solutions of linear reaction-diffusion processes on a uniformly growing domain: Criteria for successful colonization

2014 ◽  
Author(s):  
Matthew J Simpson
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Diffusion Processes ◽  
Reaction Diffusion ◽  
Domain Growth ◽  
Initial Profile ◽  
One Dimensional ◽  
Growing Domain ◽  
Linear Reaction ◽  
Successful Colonization

Many processes during embryonic development involve transport and reaction of molecules, or transport and proliferation of cells, within growing tissues. Mathematical models of such processes usually take the form of a reaction?diffusion partial differential equation (PDE) on a growing domain. Previous analyses of such models have mainly involved solving the PDEs numerically. Here, we present a framework for calculating the exact solution of a linear reaction?diffusion PDE on a growing domain. We derive an exact solution for a general class of one-dimensional linear reaction?diffusion process on $0 \le x \le L(t)$, where $L(t)$ is the length of the growing domain. Comparing our exact solutions with numerical approximations confirms the accuracy of the method. Furthermore, our examples illustrate a delicate interplay between: (i) the rate at which the domain elongates, (ii) the diffusivity associated with the spreading density profile, (iii) the reaction rate, and (iv) the initial condition. Altering the balance between these four features leads to different outcomes in terms of whether an initial profile, located near $x=0$, eventually overcomes the domain growth and colonizes the entire length of the domain by reaching the boundary where $x = L(t)$.

scholarly journals Exact solutions of coupled multispecies linear reaction-diffusion equations on a uniformly growing domain

2015 ◽  
Author(s):  
Matthew Simpson ◽  
Jesse Sharp ◽  
Liam Morrow ◽  
Ruth Baker
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Moving Boundary ◽  
Reaction Diffusion ◽  
Proliferation Rate ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Growing Domain ◽  
Linear Reaction

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction-diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction-diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction-diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The frst example calculation corresponds to a situation where the initially-confned population diffuses suffciently slowly that it isunable to reach the moving boundary at x=L(t). In contrast, the second example calculation corresponds to a situation where the initially-confned population is able to overcome the domain growth and reach the moving boundary at x=L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.

scholarly journals Exact Solutions of Coupled Multispecies Linear Reaction–Diffusion Equations on a Uniformly Growing Domain

PLoS ONE ◽  
2015 ◽  
Vol 10 (9) ◽  
pp. e0138894 ◽  
Author(s):  
Matthew J. Simpson ◽  
Jesse A. Sharp ◽  
Liam C. Morrow ◽  
Ruth E. Baker
Keyword(s):  
Exact Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Growing Domain ◽  
Linear Reaction

scholarly journals New Classes of Solutions of Dynamical Problems of Plasticity

2020 ◽  
pp. 792-796
Author(s):  
Sergei I. Senashov ◽  
Olga V. Gomonova ◽  
Irina L. Savostyanova ◽  
Olga N. Cherepanova
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Plastic Layer ◽  
Two Dimensional ◽  
Science And Engineering ◽  
Spatial Variables ◽  
One Dimensional ◽  
New Exact Solutions

Dynamical problems of the theory of plasticity have not been adequately studied. Dynamical problems arise in various fields of science and engineering but the complexity of original differential equations does not allow one to construct new exact solutions and to solve boundary value problems correctly. One-dimensional dynamical problems are studied rather well but two-dimensional problems cause major difficulties associated with nonlinearity of the main equations. Application of symmetries to the equations of plasticity allow one to construct some exact solutions. The best known exact solution is the solution obtained by B.D. Annin. It describes non-steady compression of a plastic layer by two rigid plates. This solution is a linear one in spatial variables but includes various functions of time. Symmetries are also considered in this paper. These symmetries allow transforming exact solutions of steady equations into solutions of non-steady equations. The obtained solution contains 5 arbitrary functions

Reaction-diffusion processes in a one-dimensional array of active non-linear circuits

1993 ◽  
Vol 14 (2) ◽  
pp. 74-79 ◽  
Author(s):  
T Sobrino ◽  
M Alonso ◽  
V Pérez-Muñuzuri ◽  
V Pérez-Villar
Keyword(s):  
Diffusion Processes ◽  
Reaction Diffusion ◽  
One Dimensional ◽  
Dimensional Array ◽  
Non Linear ◽  
Linear Circuits

scholarly journals The effect of domain growth on spatial correlations

2016 ◽  
Author(s):  
Robert JH Ross ◽  
Christian A Yates ◽  
Ruth E Baker
Keyword(s):  
Differential Equations ◽  
Cell Motility ◽  
Ordinary Differential Equations ◽  
Cell Movement ◽  
Domain Growth ◽  
Continuum Approximation ◽  
Spatial Correlations ◽  
One Dimensional ◽  
Growing Domain ◽  
Pair Density

Mathematical models describing cell movement and proliferation are important research tools for the understanding of many biological processes. In this work we present methods to include the effects of domain growth on the evolution of spatial correlations between agent locations in a continuum approximation of a one-dimensional lattice-based model of cell motility and proliferation. This is important as the inclusion of spatial correlations in continuum models of cell motility and proliferation without domain growth has previously been shown to be essential for their accuracy in certain scenarios. We include the effect of spatial correlations by deriving a system of ordinary differential equations that describe the expected evolution of individual and pair density functions for agents on a growing domain. We then demonstrate how to simplify this system of ordinary differential equations by using an appropriate approximation. This simplification allows domain growth to be included in models describing the evolution of spatial correlations between agents in a tractable manner.

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