Analytical Solution For The Time-fractional Heat Conduction Equation In The Spherical Coordinate System

This master thesis consists of three parts. In chapter 1, there are some preliminary knowledge about theory of fractional calculus and kings of spe-cial functions. In section§1.1, we introduce the history of fractional calculus and the recent achievements in many fields. We also give the definitions and some important properties of the Riemman-Liouville fractional integral oper-ator 0D-βt,Rienman-Liouville fractional differential operator oDαt and Caputo fractional differential operator C0Dαt. In section§1.2, main properties and for-mulae of Mittag-Lefficr function, Legcndre function and associated Legendre function are presented. These functions are powerful tool in course of solving fractional equations and will be used in the subsequent chapters.In chapter 2 and 3, we mainly talk about the application of fractional calculus to heat conduction field.In chapter 2, the time-fractional heat conduction equation in the spherical coordinate system is built, and the solution of the problem for time-fractional heat conduction describing the heat transport in spatially three-dimensional sphere has been founded.In section§2.3, the Laplace transform and the method of separating variables are employed to find the solution. The inverse Laplace transform is expressed in terms of the Mittag-Leffler type functions.Especially, In the case of the valueα=1, the solution is well known as the solution correspond to the classical heat conduction. Finally, numerical results are illustrated graphically.In chapter 3, fractional calculus is applied to the central-symmetric heat conduction problem for an finite space (r0≤r≤r1) in spherical coordinates. In section§3.2, we build the heat conduction equation with the third Boundary conditions,Finally, the computions which arc carried out according to the values ofαare reflecting the characteristic features of the curves for various order of the time-fractional derivative, and the influences of fractional derivative are also clearly illustrated graphically.