Computability Analysis Of Initial Boundary Value Problem Of Non-linear Differential Equations

The study of numerical algorithms for solutions of differential equations is central in numerical analysis. The numerical technology for solving differential equations includes finite-difference methods and finite-elements methods. The aim is to find numerically stable algorithms that rapidly converge to the correct solution. But these algorithms can not be suitable to every differential equation. Computable analysis is studying how to compute physical processes modeled by differential equations. It is the study of computability and complexity of continuous problems based on Turing machines. In the context of computable analysis, the solution of a differential equation is computable if there is a Turing machine that computes approximations converging to the solution from approximations to given parameters. So the existence of convergent algorithms for numerical solution is guaranteed.In this paper, we study the computability of the solution operators of the variable-coefficient KdV-Burgers equation, and the Schrodinger equation. This whole work is presented in 5 chapters. Firstly, an overview is given on the historical and present studies of computability theory. Chapter 2 introduces Turing machine, the framework of TTE, and proposes several basic concepts and computability properties on computability space. Chapter 3 discusses the variable coefficient KdV-Burgers equation in Bourgain-type spaces, we prove its solution operator is Turing computable by making use of TTE framework and semi-group theories. Chapter 4 studies the linear Schrodinger equations with initial boundary value conditions, we get its equivalent integral equation by taking the Laplace transform with respect to the variable t, then we prove that its solution is Turing computable. Chapter 5 studies the nonlinear Schr(o|¨)ddinger equation with initial condition, we prove that its solution is Turing computable in Sobolev space.The proofs of computability give rise to Turing algorithms, which may possibly be translated into numerical algorithms. These results extend the application of digital computers to solve differential equations.