Estimating Bounds On Linear Functional Deifned On Solution Of Linear Differential And Integral Equations

Semidefinite programming is a very important programming. Because it can be solved efcientlyby interior point algorithm with polynomial complexity, it has been widely used to solve important prob-lems, such as combinatorial optimization and eigenvalue optimization. Recently, semidefinite program-ming is applied to estimate bounds on linear functionals defined on solutions of diferential equationswith polynomial coefcients. In this paper semidefinite programming is generalized to estimate boundson linear functionals defined on solutions of linear diferential equations with generally smooth functioncoefcients, and for various types linear integral equations and linear integral-diferential equations, wealso present semidefinite programming estimating bounds on linear functional of their solutions.In Chapter1, we introduce the history of the emergence and development of semidefinite program-ming, followed by the description of applying semidefinite programming to estimate bounds on linearfunctionals defined on solutions of linear diferential equations with polynomial coefcients. Moreover,we briefly introduce package SeDuMi solving semidefinite programming. Finally, we list the maincontents of this paper.In Chapter2, we first put forward semidefinite programming estimating bounds on linear func-tionals of solutions defined on diferential equations with generally smooth coefcients. Then for lineardiferential equations with infinitely smooth coefcients, we prove the theory that the bounds series onlinear functionals defined on solutions of the equations can approach the exact linear functionals of thesolution of the problem. Finally, we test the numerical efects of the proposed semidefinite program-ming, and all numerical results show that this method can estimate the bounds on linear functionals ofthe solutions of the testing problem. In order to reduce the exponents of testing function base, and toreduce calculation amount under the premise of achieving the same accuracy, we replace Taylor poly-nomial by Chebyshev polynomial.In Chapter3, for linear integral equations and linear integral-diferential equations with polyno-mial kernel and generally smooth function kernel, we present semidefinite programming to estimatebounds on linear functionals of their solutions, respectively. For the equations with polynomial kernel,we first choose monomial as testing function, and transform the primal problem to semidefinite pro-gramming, then apply package SeDuMi to obtain the solution of the programming. The result is just thebounds on the linear functional of the solution of the primal problem. For the equations with generallysmooth function kernel, we first apply polynomial to approximate the smooth kernel, and then constructsemidefinite programming to get the approximating bounds on the linear functional defined on the so-lution of the primal problem. In most cases, we also transform the primal integral equation to the otherequivalent one or diferential equation by diferentiating technique, and then use semidefinite program-ming to solve the equivalent problem. Numerical experiments show that semidefinite programming canestimate bounds on the linear functionals defined on solutions of various types linear integral equationsand linear integral-diferential equations. Finally, we propose some problems, for example how to estimate bounds on the linear or nonlinearfunctionals defined on solutions of the nonlinear diferential and integral equations.