Approximate Similarity Reduction And Approximate Homotopy Similarity Reduction Of Several Nonlinear Problems

In this dissertation, approximate similarity reduction and approximate ho-motopy similarity reduction for some nonlinear di?erential equations are investi-gated through the symmetry theory.Chapter 1 is devoted to reviewing the background and development of sym-metry theory, perturbation theory and homotopy analysis method, whose gener-alizations or combinations form the foundation for the whole dissertation.Chapter 2 elucidates the classical and nonclassical Lie group method, theLie symmetry method and the direct reduction method which are several basicmethods of the symmetry theory. We manipulate the two-dimensional impress-ible Navier-Stokes equation by the Lie symmetry method and retrieve the specialsolution due to Sen-Yue Lou et al., which can be used to simulate the tropicalcyclones.In chapter 3, we first use the approximate symmetry reduction method tosolve KdV-Burgers equation, singular perturbed Boussinesq equation and two-dimensional impressible Navier-Stokes equation. Inspired by the approximatesymmetry reduction method, we propose the approximate direct method andsolve perturbed mKdV equation and singular perturbed Boussinesq equation.Novel results are obtained for both methods which belong to the approximatesimilarity reduction method: the general formulas for similarity reduction solu-tions and similarity reduction equations of di?erent orders can be summarized forboth methods, educing the related series reduction solutions. Zero-order similar-ity reduction equations are actually similarity reduction equations of unperturbeddi?erential equations. Higher order similarity solutions can be obtained by solv-ing linear variable coe?cients ordinary di?erential equations. Approximate directreduction method produces more results than approximate symmetry reduction method.In chapter 4, approximate homotopy symmetry reduction method is intro-duced and applied to solve KdV-Burgers equation and six-order Boussinesq equa-tion. Approximate homotopy direct reduction method is proposed to solve per-turbed mKdV equation. Both two methods belong to approximate homotopysimilarity reduction method and yield formally coherent similarity reduction so-lutions and similarity reduction equations of di?erent orders for homotopy mod-els. Homotopy series reduction solutions are consequently derived. Zero-ordersimilarity reduction equations are actually similarity reduction equations of thehomotopy models when vanishing the homotopy parameters. Higher order sim-ilarity solutions can be obtained by solving linear variable coe?cients ordinarydi?erential equations. Approximate homotopy similarity reduction method is su-perior to approximate similarity reduction method, since 1) the results obtainedfrom approximate symmetry reduction method and approximate direct reduc-tion method can also be retrieved by approximate homotopy symmetry reductionmethod and approximate homotopy direct reduction method respectively, 2) theconvergence of approximate homotopy symmetry reduction series solutions canbe adjusted by the auxiliary parameters, 3) approximate homotopy similarityreduction method is applicable to nonperturbed di?erential equations.The last chapter concerns the summary and discussion for the whole disser-tation, including the advantages and disadvantages of the approximate similarityreduction method and approximate homotopy similarity reduction method, aswell as the prospect for those methods.