Page - 31 - in Advanced Chemical Kinetics

addition, nonlinear differential equations can also assist to investigate the stability of these solutions aswell as checking the simulation analysis.Nonlinear partial differential equations govern a significant variety of phenomena including physical, chemical, and biological. The development of techniques aimed at exact solutions of nonlinear differential equationswith nonsteady and steady state [35] has been one of the most exciting advances of nonlinear scienceand theoretical physics/chemistry.An important role innonlinear science isplayedby exact solutions of differential equations. Furthermore, this can be especially observed in nonlinear physical chemistry science. This can be attributed to the provision of physical informationaswellasmore insight into thephysical aspectsof theproblem,whichcould lead to further applications. Over the past fewdecades, differentmethods have been reported to solve analytical solutions such as Tanh-sech [36], extended tanh [37], Jacobi elliptic function expansion [39], hyperbolic function [38], F-expansion [40], and theFirst integral [41]. To solve different typesofnonlinear systemsofPDEs, the sine-cosinemethod [42]hasbeenemployed. A variety of powerful analytical methods such as homotopy perturbation method [43–45], homotopyanalysismethod [46, 47],Adomiandecompositionmethod [48, 49],wavelet trans- formmethod[50], etc. areapplied tosolve thenonlinearproblems (e.g.,Eqs. (8)and(13)–(15)) inchemicalkinetics [51]. 6.Numerical solutions Many differential equations cannot be solved analytically. For practical purpose, however, such as in physical engineering sciences, a numerical approximation to the solution is often sufficient. Thenumericalmethod ismainly to solve complexproblemphysicallyorgeometri- cally. It is also used to validate the experimental results. Some of the nonlinear equations in chemicalkineticsweresolvedusingnumericalmethods [52–56]. 7.Summary Mostmathematical models of enzyme kinetics are based on reaction diffusion equations or rate equations containing nonlinear terms related to the kinetics of the enzyme reaction. Powerful and accurate analytical (HPM, HAM, ADM, etc.) and numerical mathematical methods have been employed for their resolution under steady and nonsteady state condi- tions.The theoretical resultsprovideveryuseful insight into theeffectson theperformanceof the thickness and structure of the enzymatic film, the loading of the different species, the diffusivityof themediator, etc.Also, the theoreticalmodelingandsimulationof thesesystems enable us to characterize the enzymatic reactions (i.e., rate constant, turnover rate, and Michaelis-Mentenconstants). In spite of theabove-mentionedbenefits, there areonly limited theoretical studies addressing kinetics of enzyme reaction andmost of them include a number of simplifying assumptions mainly related to themass and charge transport inside and outside the biocatalyst film, the enzymatickineticscheme,andtheelectrodemorphology.Experimentalvalidationofproposed Mathematical Modeling and Simulation of Nonlinear Process in Enzyme Kinetics http://dx.doi.org/10.5772/intechopen.70914 31