Explicit Finite-difference Methods For A Class Of Biofilm Dynamical Systems

In physics, chemistry and biology, many natural phenomena withreaction-diffusion effects can be expressed by nonlinear reaction-diffusionequations. The analytic solutions of these equations are hard to find. Hence,numerical methods for these equations are studied, which are very importantresearch problems of numerical analysis.In this thesis, we designed some novel explicit finite difference (FD)schemes to solve nonlinear reaction-diffusion equations, which describes thedeveloping process of a microbial colony. Then positivity and boundedness ofthe equations are simulated by using the schemes. This kind of microbial colonyhas the diffusion coefficientWhen the biomass density approaches zero, the diffusion coefficientdegenerates, and when the biomass densities approach one, the diffusioncoefficient becomes singular. So it is essential to design a reasonable, reliableand highly efficient numerical method for this kind of equations. Otherwise, thenumerical results may have singularity or oscillation. The numerical methods inthis thesis conquer these problems successfully.Another significant work in this thesis is that we proved the convergence and stability of our finite-difference schemes. Numerical examples are presentedto illustrate the feasibility and efficiency of our methods.The thesis is organized as follows.In Chapter1, we reviewed the reaction diffusion equations and themicrobial colony, three microbial colony systems being introduced. Then themain works in thesis are illustrated.In Chapter2, we presented the definitions, conceptions and basic theorieswhich are used in the thesis.In Chapter3, we give the explicit finite difference scheme for thehomogeneous microbial colonyThe truncation error is given. The convergence and stability of the methodare proved. Finally, three numerical examples are present to illustrating theperformance of the method.In Chapter4, we give the explicit finite difference scheme for the couplingnonlinear reaction-diffusion microbial colonyThe stability of the method is proved. Non-negativenes and boundnesscondition for the numerical results are given. Finally, numerical examples arepresent to illustrating the reproducing process of the coupling microbial colonyin four different states.In Chapter5, we present the explicit finite-difference schemes for acomplex microbial colony Non-negativeness and boundedness are demonstrated by using thenumerical examples.In Chapter6, we summarized the thesis and forecasted the future researchin this field.