Applications Of Homotopy Analysis Method In Nonlinear Mechanics And Mathematical Biology

An intensive study on both the theory and the application of Homotopy AnalysisMethod (HAM) is carried out and results of di?erent nonlinear problems are discussedand summarized in this thesis. A lot of literature on the application of HAM havealready been published, still there is room for researchers to find some way to improveHAM or extend its applications.In our study, we focus on two sides. One is the method itself and the other is itsapplication in nonlinear mechanics and mathematical biology. Selecting few problemsfrom nonlinear mechanics and mathematical biology, we test the applicability and?exibility of HAM. This thesis consists of eight chapters. Each chapter is completelyself-contained with its own depth introduction, literature review, methodology, resultanalysis, and conclusion.Firstly, Van der Pol's equation is handled with the help of HAM and some trans-formations are used to improve the convergence of the series solution. We compareour results with that given by Liao [36], also with homotopy-Pad′e approximations.With the help of this simple transformation we have improved the convergence regionfor as compared to the previous work done by the same technique.Secondly, ?exibility and freedom of HAM is being observed with the help oftwo strongly nonlinear problems such as generalized Van der Pol's equation and theRayleigh's equation. We introduce a special kind of linear operator and base functionwith the help of a parameterκthat serves for improving the convergence of the solu-tion. We have observed that convergence of the results can be improved by increasingκ. Only HAM provides such freedom that one can try di?erent linear operator and base function to get better convergent solutions.Thirdly, HAM is applied to Thomas-Fermi equation with new initial guess. Betterresults are obtained as compare to the earlier work done by the same technique. Herewe use one of the advantages of HAM, that is: freedom of the choice of initial guess.In this problem the base function is same as used by Liao [18, 19] but initial guess isdi?erent. We have observed the improvement in the convergence of the result.Fourthly HAM is applied on SIR epidemic model. This model is related to epi-demiology which is a branch of biology that deals with infectious diseases. Themodel is defined with a set of two nonlinear di?erential equations. This model hasbeen shown to predict the behavior of many disease. To the best of our knowledge,it is the first time that analytical results are obtained for these models. We havecompared our results with numerical results and also with the qualitative analysismade by Singh [73].Fifthly HAM is applied on SIS model. This model is more e?ective for diseasescaused by bacteria or helminth agents and also for most sexually transmitted dis-ease. This model is defined by coupled nonlinear di?erential equations with initialconditions. For this model, also we have obtained analytical results for the first time.We have compared our results with numerical results and also with the qualitativeanalysis made by Singh [73].Lastly we investigate the nonlinear delay di?erential equation (DDE) with the helpof HAM, for the first time. To find analytic result for time-delay models, assumed tobe very di?cult. Such models are di?erent from ODEs as these models arises whenthe evolution of a system not only depends on its present state but also on its history.The study of these delay di?erential equations become more complicated when it isnonlinear. Currently, in various fields such as the navigational control of ships andair crafts, electrodynamics, engineering, epidemiology, etc, have shown that delay dif-ferential equations play an important role in explaining many di?erent phenomenon.For example, a time delay compensation algorithm for o?shore platforms is developed by Li [70] based on the prediction of wave force and structural response , the opti-mal prediction of active control force is used to perform time delay compensation,similarly, in epidemiology time-delay appears in many disease such as malaria. Weobserved that time-delay indeed has a great in?uence on the property of the nonlineardynamic system. For this model there exist a certain time-delay parameterτwhichhas in?uence on the convergence of the solution at infinity. Here we have successfullyachieved convergent result with HAM. Our results agree well with that of numericalanalysis. This example illustrates that the analytic approach based on the HAM isalso valid for nonlinear time-delay di?erential equations.