Study On Time-Fractional Direct And Inverse Heat Conduction Problem In Composite Medium And Their Application

This thesis, which is comprised of three chapters, primarily studies a time-fractional direct and inverse heat conduction problem in composite medi-um. Chapter 1 briefly introduces the history, definitions, properties, integral transform, special function and applications related to fractional calculus. In Chapter 2, a time-fractional heat conduction direct problem in a 3-layer com-posite medium is proposed and its analytical as well as numerical solution are given. And Chapter 3, with the parameter estimation method based on Levenberg-Marquardt (LM) method, the optimal values of Caputo derivative in the sense of simulating the experimental data best are estimated and the re-sults are analyzed, which is considered as the inverse problem of the fractional heat conduction problem proposed in Chapter 2.Chapter 1, provided as the preliminary knowledge, introduces the brief history of the development of fractional calculus and presents the definition-s of Riemann-Liouville (R-L) fractional integral/derivative operator, Caputo fractional derivative operator as well as their related common properties. In §1.2., Laplace transform on fractional integral/derivative operator and a kind of special function:Mittag-Leffler function are introduced. And §1.3 briefly introduces some applications of fractional calculus.In Chapter 2, a time-fractional heat conduction direct problem in a 3-layer composite medium has been studied. In §2.1, a brief introduction of the research status of time-fractional heat conduction direct problem in compos- ite medium is given. In §2.2, the mathematical model of the fractional heat conduction direct problem is proposed, with the governing equation: and boundary as well as initial conditions. In §2.3, employed integral transfor-m, the analytical solution to the fractional heat conduction problem is obtained as: and its expression in the limiting case γ=1 is also provided. Based on the work of predecessors, we also obtain the numerical solution to the present fractional heat conduction direct problem with finite difference method in §2.4. In §2.5, numerical test of the model has been conducted using experimental data and the results have been analyzed. And the conclusion of Chapter 2 is drawn in §2.6.In Chapter 3, we investigate an inverse problem of the fractional heat con-duction problem proposed in Chapter 2. In §3.1, the research status of inverse heat conduction problems and fractional inverse heat conduction problems are briefly introduced. §3.2, based on Levenberg-Marquardt (LM) method, pro-vides a method to estimate the optimal order of Caputo fractional derivative in the sense of simulating the experimental data best. The estimation is con-ducted according to the following iterative scheme of LM method: to implement the scheme, the method to ascertain the damping parameter μk has been given. And in order to ascertain the Jacobian matrix Xk, the sensitivity problem with the following governing equation: is proposed and solved numerically with the finite difference method. In §3.3, using experimental data, the optimal values of Caputo derivative are estimat-ed respectively in two experimental cases. The results show that when the Caputo derivative reaches its optimal value, the proposed time fractional heat conduction model simulates the experimental data very well. And the sensitiv-ity analysis of the estimation process is also conducted. At last, the conclusion of Chapter 3 is drawn in §3.4.