A Jacobi Dual-Petrov Galerkin-Jacobi Collocation Method for Solving Korteweg-de Vries Equations

The present paper is devoted to the development of a new scheme to solve the initial-boundary value Korteweg-de Vries equation which models many physical phenomena such as surface water waves in a channel. The scheme consists of Jacobi dual-Petrov Galerkin-Jacobi collocation method in space combined with Crank-Nicholson-leap-frog method in time such that at each time step only a sparse banded linear algebraic system needs to be solved. Numerical results are presented to show that the proposed numerical method is accurate and efficient for Korteweg-de Vries equations and other third-order nonlinear equations

Study on the Solutions of Kawahara, and Complex-valued Burgers and Kdv-burgers Equations

The KdV equation is a nonlinear partial differential equation. The real-valued as well as complex-valued KdV equations have wide physical applications and very rich mathematical theory. The work in this dissertation studies two important problems. First, the initial- and boundary-value problem for the Kawahara equation, a fifth-order KdV type equation, is studied in weighted Sobolev spaces. This functional framework is based on the dual-Petrov-Galerkin algorithm, a numerical method proposed by Shen to solve third and higher odd-order partial differential equations. The theory presented here includes the existence and uniqueness of a local mild solution and of a global strong solution in these weighted spaces. If the L^2-norm of the initial data is sufficiently small, these solutions decay exponentially in time. Numerical computations are performed to complement the theory. Second, spatially periodic complex-valued solutions of the Burgers and KdV-Burgers equations are studied in detail. It is shown that for aDepartment of Mathematic