Inverse Problems for some Fractional Differential Equations

Abstract

Inverse Problems for Some Fractional Differential Equations Fractional Calculus(FC) is about the investigation of arbitrary order derivatives, integrals, special functions and equations involving these operators. This subject has its roots back to late seventeenth century. In recent years scientists and engineers are using it rigorously as it provides an efficient method to model many well known physical phenomenon when compared with their counterpart (integer order calculus). For example, fractional order diffusion/transport equation has been used to explain anomalies in diffusion/transport process which occurs in many physical situations such as transport in heterogenous or porous media. For a physical process scientists are interested in the investigations of causes and effectsofthatphysicalprocess. Theeffectsofanyphysicalprocess(usuallyknown as direct problems) are easier to study then the causes that forces the system to behave in a particular manner. The mathematical models in which we study the causes are termed as inverse problems(IPs). The field of IPs investigates how to convert measurements into information about a physical process. The field of IPs is of great interest as it has many applications just to mention a few are in medical imaging, acoustic, heat conduction, source identification in a stream, shape optimization etc. In this thesis, we have studied time, space as well as space-time fractional differential equations. Through out our research investigation we have used fractional derivatives defined in the sense of Hilfer and Caputo. It is to be noted that Hilfer fractional derivative (HFD) interpolates both Riemann-Liouville(RLFD) and Caputo fractional derivatives(CFD) for particular choices of parameters. For a fourth order time fractional differential equation(TFDE) with nonlocal boundary conditions(knownasSmaraskii-Ionkinboundaryconditions)involvingHilferfractionalderivative(HFD),twoinversesourceproblems(ISPs)areconsidered. ISPof determining a space dependent source term for a TFDE in two space dimensions is also considered. Existence, uniqueness and stability results for the ISPs are proved under certain regularity conditions on the given data. For a multi-term TFDE involving HFDs ISP of recovering a time dependent source term is studied by using Heaviside-Mikusinski’s operational calculus approach. The spectral problem is non-self-adjoint and a bi-orthogonal system of functions(BSFs) is used toconstructtheseriessolutionoftheISPs. Foraspace-timefractionaldifferential equation(STFDE)withDirichletzeroboundaryconditionsalongwithappropriate over-specified conditions two ISPs of recovering space and time dependent sources are considered. In the last research problem of this thesis inverse coefficient problem(ICP)foraspacefractionaldifferentialequation(SFDE)isstudied. Weproved existence, uniqueness and stability results for the solution of the considered IPs by imposing certain regularity conditions on the given datum.