Direct Problem And Parameters Estimation Studies Of Time-Fractional Heat Conduct In Two-layer Composite Medium

In recent years,fractional calculus theory has been widely applied in me-chanics and engineering fields.Fractional calculus provides a more accurate way to interpret the memory and non-locality properties of complex phenom-ena or materials.This thesis primarily studies direct problem and parameters estimation of time-fractional heat conduction in two types of two-layer composite medium.Concretely,it concludes:In chapter 1,we introduce a brief history of the origin and development of fractional calculus,as well as the definitions and common properties of Riemamm-Liouville fractional operator and Caputo fractional operator in the first section.In section 2,we introduce some special functions,such as Mittag-Leffler function,Wright function and Mainardi function.We give Laplace transform on fractional operator and special functions in the following section 3.In the last section,we introduce some applications of fractional calculus.In chapter 2,we study on a time fractional heat conduction problem in 2-layer semi-infinite composite medium.In the first two sections,we briefly in-troduce the research backgrounds of time fractional heat conduction problems and propose a mathematical model in such problems with equations below as well as initial and boundary conditions.In section 3,we introduce its analytical solution.This solution depends on the relationship between ordersα and β For α<β we have For α>β we have similar expressions.In section 4,we study on parameters estimation of α and β.We first give the sensitivity analysis,then use least square method and conjugate gradient method to estimate orders.In the last section,we compare the results of two methods and present a summary of this chapter.In chapter 3,we investigate the time fractional heat conduction problem in finite composite medium.In section 1,we propose the mathematical mod-el briefly.In section 2,we give the numerical solution with finite difference method.In section 3,we conduct estimation of orders based on conjugate gradient method as well as Levenberg-Marquardt method.We present the summary in the last section.In chapter 4,we give our conclusion and possible research work in the future.