| This thesis presents symbolic techniques for analyzing polynomial and differential systems. It consists of three parts.;In Chapter 1, an algorithm is given for the automatic computation of the CRC for a polynomial with complex parametric coefficients. In Chapter 2, an improved algorithm is presented for the automatic computation of the CRC of a real parametric polynomial. The algorithm offers improved efficiency and a new test for non-realizable conditions. In Chapter 3, an algorithm is proposed for the automatic computation of the CRC of a parametric polynomial on an interval. Applications of CRC to real quantifier elimination problems and positive definiteness of polynomials are also presented in these chapters.;In the second part, symbolic methods for finding exact travelling wave solutions to nonlinear partial differential equations (PDEs) are proposed. In Chapter 4, a new algorithm and its implementation for solving single nonlinear PDEs are presented. It turns out that, for PDEs whose balancing numbers are not positive integers, the package works much better than existing packages. Furthermore, the package obtains more solutions than existing packages for most cases. In Chapter 5, the techniques in Chapter 4 are applied for finding exact travelling wave solutions to the modified Camassa-Holm and Degasperis-Procesi equations. It turns out that many new solutions are obtained. Moreover, these solutions are in general forms, and many known solutions to these two equations are only special cases of them.;In the last part, symbolic techniques for finding approximate series solutions to differential equations and boundary value problems are discussed. In Chapter 6, two well-known analytical methods, the homotopy analysis method and the homotopy perturbation method are compared through an evolution equation. Some serious mistakes made in the literature are found. In the last chapter, Chapter 7, based on the homotopy analysis method, an efficient approach is proposed for obtaining approximate series solutions to fourth order two-point boundary value problems. Consequently, an affirmative answer to an open problem is obtained.;In the first part, symbolic techniques for obtaining complete root classifications (CRCs) of parametric polynomials are studied. For a polynomial p(x) with parametric coefficients, the CRC of p(x) is a collection of its all possible root classifications, together with the conditions on the parametric coefficients such that each root classification is realized. As an introductory example, the CRC of p(x) = x2 + bx + c (where b, c are real parameters) consists of the following statements: (1) two distinct real roots, if and only if b2 > 4c; (2) one double real root, if and only if b2 = 4c; (3) no real root, if and only if b2 < 4 c. How to obtain the CRC of a parametric polynomial of any degree? It is an interesting and challenging problem.;Keywords: parametric polynomial, real root, complete root classification, real quantifier elimination, subresultant polynomial, nonlinear partial differential equation, travelling wave solution, tanh method, analytical solution, series solution, homotopy analysis method, boundary value problem.. |
