Integral Equation Methods For The Source Term Inversion Of A Class Of Time Fractional Heat Conduction Equation

The research field of inverse problem has a wide range of applications,which covers medical imaging,the difficult problem of seismic wave detection in natural disaster,the weather forecast,the textile industry and so on.The fractional heat conduction equation can be used to describe the anomalous diffusion phenomena in some fields of science and technology the rapid development in the field,this paper considered a class of inverse unknown time-dependent source term function in the right hand side of time fractional heat conduction equation,which is solved with an nonlocal over-determined data.Then,we also proved the conditional stability to reconstruct a stable source term.Finally,the numerical examples show that the proposed method is efficient with respect to data noise.The research results of this paper are organized as follows:In the first chapter,the significance of researching the inverse time fractional heat conduction problem,the research trends and the main contents of this paper are introduced.In the second chapter,we mainly study the inverse source term with Dirichlet boundary condition in a general region.The inverse problem reconstruct an unknown time-dependent source term function at the right side of the governing equation with non-local over-determined data.Firstly,the inverse problem is formulated into a Volterra integral equation of the first kind.Then,using frictional derivative on the both sides the integral equation of the first kind is transformed into the second kind,and the conditional stability result of this problem is discussed.Thereby,the conditional stability of the inverse problem is estimated through introducing the mollification method to reconstruct a stable source term.In the third chapter,we consider an inverse source problem in a time fractional diffusion equation with Neumann boundary condition.The inverse problem mainly study the source terms in the sense of classical solutions,we carry out the study under the assumption of the source functionf(x)?C~2(0,l),and the weight function?(x)?L~2(0,l).By use of fractional derivative the inverse source problem is transformed into the second kind Volterra integral equation,then the conditional stability and error estimation are established.The error estimation of the inverse source term is given with utilizing the mollification method.Results of numerical experiments verify the effectiveness of the inversion algorithm.