Studies On Two Kinds Of Inverse Problems For Space-fractional Diffusion Equations And The Corresponding Applications In Designing Thermal Protective Clothing

In recent years,fractional diffusion equations,motivated by practical problems,have aroused widespread concern.There is a large number of research achievements about direct problems for fractional diffusion equations,whereas the mathematical study of related inverse problems,especially inverse problems for space-fractional diffusion,is still in its infancy.The purpose of this research is to investigate two categories of inverse problems for spacefractional differential equations,i.e.,backward problem and sideways problem,in which we give theoretical and numerical analysis,and then we consider the corresponding applications in thermal protective clothing.First of all,a brief introduction of research background and significance on fractional diffusion equations and thermal protective clothing is given in chapter 1,including the previous research results,the basic definitions of fractional derivatives,several concepts of operator semigroups and the main content of this paper.In chapter 2,we investigate the implicit-explicit(IMEX)finite difference method for a nonlinear space-fractional diffusion equation.Stability estimation and convergence rate for the numerical scheme are proved,and the numerical experiment validates the theoretical results.Based on the above numerical method,in chapter 3,a modified regularized algorithm for a semilinear space-fractional backward diffusion problem is studied.To derive the gradient of the optimization functional,the variational adjoint method is introduced to reconstruct the unknown initial value.The existence and the uniqueness for the modified regularized solution are proved by a formal analytic way,especially the operator semigroup.Finally,numerical examples are implemented to show the feasibility and effectiveness of the proposed method.As for a sideways problem for space-fractional diffusion,we employ a forward collocation method to transform the inverse problem into a system of well-posed direct problems in chapter 4,and the error estimation of the solution is obtained.The well-posed analysis of the direct problem is also given based on the maximum principle.In chapter 5,we consider a nonlinear fractional heat transfer model within thermal protective clothing in view of the fact that the anomalous diffusion processes can be described by fractional differential equations.The numerical simulations validate the rationality of the proposed heat transfer model,combining with heat transfer law in the air gap and the heat transfer model in skin layer.