On the Fractional Initial-Boundary Value Problems

Abstract

FractionalCalculus(FC)isthestudyofintegralsandderivativesofarbitraryorder, this subject is as old as integer order calculus and is supposed to be initiated from the question of L’Hôpital to Leibniz when the notion of nth order derivative was coinedfortwo n timesdifferentiablefunctions. FromlastfewdecadesFChasbeen consideredbymanyresearchersduetoitsapplicationsindiversefieldsofsciences, not to mention all some are in Physics, Chemistry, Viscoelasticity, Biology etc. Due to these applications the integral or differential operators of arbitrary order and equations involving these operators are considered by many researchers for mathematical investigations. We intend to consider some Fractional Differential Equations (FDEs) in this dissertation. Indeed, in one part of this dissertation we have considered diffusion equations with fractional derivative in time only. Let us mention that in many physical phenomena, the data obtained from field as well as lab experiments is not in agreement with the integer order Partial Differential Equations (PDEs). The phenomena is usually known as anomalous diffusion/transport. Among several techniques to explain these anomalies one is by considering fractional order operators instead of integer order operators in PDEs. It is important to mention that throughout this dissertation, we have considered the fractional derivatives defined in the sense of Riemann-Liouville, Caputo or Hilfer. The Hilfer fractional derivative is a generalization of the Riemann-Liouville and the Caputo fractional derivatives. The particular choices of the parameters involved in Hilfer fractional derivative give us Riemann-Liouville and Caputo fractional derivatives. We considered direct as well as inverse source problems for FDEs involving time fractionalderivativewithnonlocalboundaryconditions. Theeigenfunctionexpansion method has been used and the spectral problem obtained is non-self-adjoint. The problems considered have initial conditions as in case of integer order deriva x tivesasweconsideredfractionalderivativedefinedinthesenseofCaputo. Forthe case of Hilfer fractional derivative rather than taking a nonlocal initial condition in terms of fractional integral two local conditions are considered. Under certain regularity conditions on the given data, we obtained existence, uniqueness and stability results for the problems. For a space-time fractional diffusion equation with Dirichlet boundary conditions, some inverse problems are also discussed. The spectral problem is generalization of the regular Sturm-Liouville operator. Several properties of the eigenvalues and eigenfunctions of the fractional order Sturm-Liouville operator are used to prove the existence results for the solution of the inverse problems. Some special cases of the inverse problems in the case of space-time differential equations are discussed and results are deduced from the generalized results. In the last part of the dissertation a nonlinear system of fractional differential equations are considered. The results about existence of finite time blowing-up solutions is proved.