Semidefinite Programming Estimating Linear Functionals Defined On Solutions Of Differential And Integral Equations

Semidefinite programming (SDP) is an important branch of mathematical programming. It has been used in many important areas. Recently this method has been successfully applied to solving moment problem, as well as dealing with global optimization over polynomial and estimating the linear functionals defined on solutions of differential equations. In this paper SDP is proposed for estimating linear functionals defined on solutions of the linear integral equations with non-differentiable kernels and the system of differential equations. We use SDP to estimate the linear functionals defined on solutions of differential equations in their any sub-interval and the value of solution at an arbitrary point in the interval, not solving the integral and differential equations directly.In Chapter1, we briefly describe the developments, applications and common solvers of semidefi-nite programming.In Chapter2, we propose SDP to estimate the solution of linear functionals defined on solutions of the linear integral equations with non-differentiable kernels. Numerical results show that the method can obtain good bounds on linear functionals of the corresponding problems.In Chapter3, SDP is presented to estimate the bounds on linear functionals of solution of the system of linear differential equations with polynomial coefficients and generally smooth coefficients. Numerical results illustrate that the bounds on the linear functionals of solutions of the system of linear differential equations gotten by SDP can achieve high accuracy.In Chapter4, we put forward SDP for estimating the bounds on the linear functionals of solutions of the linear differential equations in any sub-interval. In addition, the method also can obtain the value of solutions at arbitrary point in the interval without directly solving differential equations. Numerical results demonstrate that SDP is effective.At last, some undiscussed problems are proposed, for instance how to use SDP to estimate the functionals of solutions defined on the linear integral equations with generally non-differentiable ker-nels, smooth nonlinear differential and integral equations.