Application Of The Homotopy Analysis Method In The Nonlinear Mechanics And Finance

The nonlinear world attracts people because of its abundant diversity. Particu-larly, there are an array of nonlinear problems in ?uid mechanics. No doubt, it mustbe a meaningful work to explore a mathematical method for these nonlinear prob-lems. The homotopy analysis method (HAM) which is proposed and developed inrecent years provides us with such a means to obtain series approximation solutionsto the nonlinear problems. Based on the concept of'homotopy'in algebraic topology,the homotopy analysis method transfers a nonlinear problem into a series of linearsub-problems by the construction of the zero-th order deformation equation and thehigh order deformation equations. Di?erent from its counterparts, such as pertur-bation methods, the homotopy analysis method is independent of any small physicalparameters and hence, it is applicable for not only weakly nonlinear problems but alsostrongly nonlinear problems. Besides, the homotopy analysis method has greet free-doms which can be exploited to combine itself closely with physical problems. Thanksto these advantages, the homotopy analysis method has been widely applied to manyaspects of natural and social sciences, such as mechanics, heat transfer, physics, ap-plied mathematics and finance. In this dissertation, we use the homotopy analysismethod to solve several problems in ?uid mechanics and financial mathematics whichare of both practical and theoretical interest.Firstly, the problem of wave-current interaction is considered. A train of deep wa-ter waves propagates on a steady current which possesses an exponential distributionof velocity. From the point of view of equations, this is an elliptic partial di?erentialequation with free boundary conditions. By means of a coordinate transform, the freeboundary is fixed, resulting to a governing equation with nonlinear terms dominated.Physically, we utilize many freedoms of the homotopy analysis method to choose thesolution expression, the linear operator and etc. Obtaining a convergent high orderapproximation solution, we carefully examine each physical quantitative that may beconcerned in the wave-current field, analyzing the changes resulting from the wave-current interaction such as the wave speed, the wave height, the velocity profile, thewave shape, the kinetic energy of the ?ow particle, the amplitude decaying and so on.It is pointed out that the key point of the wave-current interaction is the vorticity. Auniform current and a train of deep water waves do not produce nonlinear interaction.The results show that (1) waves propagate faster on an aiding shear current but more slowly on an opposing one, compared with waves in still water; (2) the aiding shearcurrent tends to sharpen the crest but smoothen the trough, while the opposing shearcurrent has the opposite e?ects; (3) the amplitude of waves and ?uid velocity decayover the depth more quickly on an aiding shear current but more slowly on an oppos-ing shear current than that of waves in still water. Also, we extend Stokes'kinematiccriteria of wave breaking in still water to waves on a shear current. The highest waveon an opposing shear current is even higher and steeper than that of waves in stillwater. Especially, we consider two limiting cases, that is, the pure waves situationand the pure current situation. In these two limiting cases, our series approximationagrees well with previous theoretical results, hence verifying the validity of the presentmethod.Secondly, some examples of viscous boundary-layer ?ows are considered (includ-ing Section 3 to Section 6 in this dissertation). In Section 3, the unsteady stagnation?ow on a stretching surface is considered. The unsteady boundary-layer ?ow equa-tion is a parabolic partial di?erential equation containing nonlinear terms. It is foundthat such kind of equation has much attribution of an ordinary di?erential equation.Therefore, we apply the homotopy analysis method almost the same way as we applyit to ordinary di?erential equations. Comparing with perturbation solutions, our highorder series approximation is uniformly valid in all time. We also discuss the impactof the velocity of the external potential ?ow and the stretching surface on the ?owvelocity profile and the skin friction coe?cient. In Section 4, the unsteady mixedconvection boundary-layer ?ow near the stagnation point on a vertical surface in aporous medium is investigated. A high order series approximation which is uniformlyvalid in all time is also obtained. Then the impact of the mixed convection parameteron the heat transfer is investigated. In Section 5, the nano boundary-layer ?ows areinvestigated. Experiment and theory show that the condition of no-slip in the classicalboundary-layer theory is no longer true for ?uid ?ows at the micro and nano scale. In-stead, a certain degree of tangential slip must be allowed. Adopting nonlinear Navierslip boundary condition, we investigate three common types of boundary-layer ?ows.They are: (1) the ?ow past a wedge; (2) the ?ow in a convergent channel; and (3)the ?ow driven by an exponentially-varying outer ?ows. Proceeding to a high orderto obtain a convergent series approximation, we discuss the impact of the slip lengthand other parameters on the velocity profile and the tangential stress. In Section 6,two common types of the heat transfer of nano boundary-layer ?ows are considered. They are: (1) two dimensional stagnation ?ow; and (2) three dimensional axisymmet-ric ?ow. We discuss the characteristics of di?erent kinds of ?ows when there exist avelocity slip and a thermal jump at solid interfaces.Finally, a common financial derivative, the American put option, is investigated.The American put option is such a contract that the holder has the right to sell theunderlying asset on or before the expiry date. From the point of view of equations,the American put option problem is a parabolic partial di?erential equation withmoving boundary conditions. Unlike the free boundary problem of the wave-currentinteraction in Section 2, the moving boundary of the American put option has singu-larity on the expiry date. Within the framework of the homotopy analysis method, weuse a complicated linear operator, and obtain a formal solution by means of Laplacetransform. Then Taylor series approximation is applied to the formal solution. Thisidea is rather inspiring for later application of the homotopy analysis method to somecomplicated nonlinear problems. The present work di?ers from the previous analyticones from at least three aspects: (1) we do not use any transform, such as the Landautransform, to reformulate the problem into a fix domain. Instead, the original movingboundary problem is directly solved within the framework of the homotopy analysismethod. This provides us with a more general approach for the moving boundaryproblems; (2) a series approximation is explicitly given. No numerical methods areused during our derivation. No extra-parameters or functions to be determined areinvolved; (3) our approximation converges quickly. It needs only a few seconds to getan accurate result by the present explicit formula; (4) di?erent from some previousasymptotic formulas that can only be applicable for options with very short life time(for example, one or two months), our approximation is accurate for options withrather long life time (for example, a couple of years). Adopted Pad′e method, ourapproximation can be valid for options with much longer life time. To the best of theauthors'knowledge, such an accurate and simple-in-form approximation is given forthe first time.In general, from the point of view of equations, this dissertation focuses on partialdi?erential equations with free (moving) boundary conditions. They are the waveproblem equation with a free surface and the Black-Sholes equation with an optimalexercise boundary, respectively. From the point of view of physical problems, thisdissertation focuses on the problems of wave-current interaction, the nano boundary-layer ?ows, the unsteady boundary-layer ?ows and etc in ?uid mechanics, and the American put options in finance.The innovation points of this dissertation can be concluded mechanically, finan-cially and mathematically, respectively.I. Mechanically:(1) We propose, for the first time, an analytic method applicalbe for not onlyweakly but also strongly nonlinear wave-current interaction. The wave is not neces-sarily of small amplitude and the current is not necessarily weak. We also point outthat the key point of the wave-current interaction is the vorticity of the current.(2) We derive series solutions to some kinds of unsteady boundary-layer ?owswhich are uniformly valid for all time.II. Financially:(3) We derive, for the first time, an analytic approximation for the optimal ex-ercise boundary of an American put option which is very accurate in the practicalfinancial market. Di?erent from some previous analytic approximations, our approx-imation is explicitly expressed, with no extra parameters to be determined. Thee?ective region of the approximation is enlarged at least 15 times than previous ones.III. Mathematically:(4) We use two di?erent methods to solve free (moving) boundary problems. Thewave-current problem is reformulated into a fixed domain by means of a coordinatetransform. The American put options problem is directly solved within the frameworkof the homotopy analysis method. The latter one is a more general approach.(5) We propose to choose the linear operator for the wave-current problem fromthe point of view of physics instead of the equations themselves. The linear operatormay have no relationship with the form of the original equations. The idea of refer-ring to the physical background greatly simplifies the procedure of solving nonlinearequations.(6) We use, for the first time, a complicated linear partial di?erential operatorto solve the American put options within the framework of the homotopy analysismethod. A formal solution is obtained by means of the Laplace transform. ThenTaylor series approximation is applied to the formal solution. This approach enrichesthe homotopy analysis method. It provides us with a new idea to solve complicated partial di?erential equations in future.In conclusion, this dissertation not only applies the homotopy analysis methodto wider fields and more types of problems, but also provides us with a consultablereference in theory for mechanics, engineering and finance.