Page - 23 - in Advanced Chemical Kinetics

chemical reaction. The substances are transformed into each other in local chemical reaction, whereas the substances are spread out over a surface in space in diffusion. Reaction-diffusion (RD) systemsarise inmanybranchesofphysics, chemistry,biology, ecology, etc.Reviewsof the theory and applications of reaction-diffusion systems can be found in books and numerous articles (see, forexample[20–23]).Thesearise ina largevarietyofapplicationareas, suchas flow inporousmedia [24], heat conduction inplasma [25], combustionproblems [26], liquid evapo- ration[27],andofmorerecent interest, imageprocessing[28].Agreateffort isbeingmadeinthe development of themathematical theory of nonlinear diffusion equations and to obtain exact solutions for special cases. Their significancenot only relies on thehugenumberof their appli- cationsbutalsoon the fact that theyprovidewitha rathergeneral classof linearandnonlinear differential operators. Inmathematical analysis, it has showntobeamilestone for thedevelop- mentofapplied,abstract,andnumericalanalysisaswellas foralgebra,geometry,andtopology. 3.Nonlinearphenomena Themodern theory of the nonlinear reactiondiffusionprocess is an important field in today’s science. The nonlinear systemand coherent structures represent an interdisciplinaryareawith manynonlinear applications invarious fields. Those applications canbedivided into sixdisci- plines: chemistry (autocatalytic chemical and enzyme reactions), physics (nonlinear optics and electric circuits, plasmas and states of solid, condensed atomic gases, hydrodynamics, galaxy dynamics and cosmology, fluid dynamics, and celestialmechanics), general relativity, biology (biofuel cell, bioreactor andbiosensor, atmosphere andoceans, and animal dispersal), random media,andmoderntelecommunications.Agreatvarietyofphenomenainphysics,chemistry,or biology canbedescribedbynonlinearODE/PDEs andparticularly by reaction-diffusion equa- tions.For thesereasons, the theoryof theanalytical solutionsof thereaction-diffusionequations isconsidered. Inreactiondiffusionsystems,nonlinearphenomenaplayacrucial role inappliedmathematics andchemistry.Exact (closed-form) solutionofnonlinear reactiondiffusionequationsplaysan important role in the proper understanding of qualitative features ofmanyphenomena and processes in various areas of natural science. The main result obtained from reaction and diffusion systems is that nonlinear phenomena include diversity of stationary and spatio- temporarydissipativepatterns, oscillations, different types ofwaves, excitability, biostability, etc.But it isdifficult forus toobtain theexact solution for theseproblems.The investigationof exact solutionofnonlinearequation is interestingand important. Ingeneral, this results in the needtosolve linearandnonlinear reactiondiffusionequationswithcomplexboundarycondi- tions. The enzymekinetics inbiochemical systemshaveusually beenmodeledbydifferential equations,which are based only on reactionwithout spatial dependence of the various con- centrations.Thedimensionlessnonlinear reactiondiffusionequationsaredescribedbelow: ∂S ∂τ ¼∇2S� f R;τ;S;Pð Þ (1) ∂P ∂τ ¼∇2Pþg R;τ;S;Pð Þ (2) Mathematical Modeling and Simulation of Nonlinear Process in Enzyme Kinetics http://dx.doi.org/10.5772/intechopen.70914 23