| Problems involving nonlinear differential equations usually come in the field ofscience and engineering. Finding the solution of these equations has great value inmathematics and strong background in applications. In general, these importantsubjects seem difficult to obtain analytic solutions . A method that bypasses themathematical difficulties and also emphasizes those feature of the solution isnecessary. The synergetics analysis method is an analytic tool for nonlinear problemsthat has been successfully applied to many nonlinear problems. In this dissertation, asimple review on analytic techniques for nonlinear problems and a systematicdescription of synergetics is given. The main contents and contributions of thisdissertation may be summarized as follows.Despite much made both theoreticallyand practically, it can be state a full understanding of synergetics analysis methodhas been achieved. These is a need to present synergetics analysis method to a widerfield and more difficult problem.Although both the technology of melt vibration in polymer processing and thekinetic theories of non-Newtonian fluids flowing in transient flow fields have beeninvestigated recently by internal and external researchers, synergetics studies ofmelt vibratory transfer process in single-screw dynamic extrusion processing flowfields are inadequate yet. And one of most important tasks is to build a dynamicconstitutive equation describing motions of macromolecular induced by vibration,and to resolve high non-linear and non-locality order parameter equations ofprocessing flow field dominated by vibration.To predict rheological behaviors of polymer melts under effects of vibrationforce field and simulate kinetic responses of a macromolecular to the vibration, Thephysical model of single-screw dynamic extrusion flow field can be simplified as aflat channel and the combined force field during melt vibratory transfer can beimplemented by superposing vibration of the flat on steady flow. In applying thesynergetics analysis method, viscidity of viscoelastic fluid in several classical casesof flat vibrating shear flows, such as a reciprocating flow, parallel or orthogonalsuperposition of vibration and steady flow, as well as combined flow of the said twoflows, is studied, by use of constitutive equations for polymer melts under effects ofvibration force field, and order parameter equations of the polymer melts in theseflows are deduced, too. as well as kinetic modeling of single-screw vibratingly transfer melt.The mathematical statistics of the stretch rates indicate that vibrationimproves melt mixability and homogenization.At last, also is conducted an experimental study on vibratory transfer melt indynamic capillary rheology experiment,coaxial cylinder dynamic viscometricexperi- ment, a single-screw dynamic plasticizing extruder.using this establishedorder parameter equations for polymer melt in the vibration shear field, it can beused to study the change of shear stress, melt mixing ,We number and the first andthe second normal stress difference coefficient for polymer melt and the effect ofvibration parameters on them in vibration shear field. At the same, it can be used thesynergetics analysis methodin to explain the effects of elasticity stability of polymermelts.The work of this dissertation show the wide applicability of synergetics analysismethodin solving nonlinear problems analytically. Although the nonlinear equationsare different in form, the analytic solution can be obtain by the same procedure.Moreover, compared with other analytic methods, synergetics provides us with greatpotential to contact the physical property of the given problem with method per se.Especially, some of the results indicates that synergetics is an useful tool for thesenonlinear problems with multiple solutions which are not easy to find by otheranalytic methods or numerical methods. All these examples given in this dissertationmight be helpful to keep us an open mind for solving nonlinear problems.
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