Please use this identifier to cite or link to this item:

`http://prr.hec.gov.pk/jspui/handle/123456789/10822`

Title: | On the Fractional Initial-Boundary Value Problems |

Authors: | Ali, Muhammad |

Keywords: | Physical Sciences Mathematics |

Issue Date: | 2018 |

Publisher: | COMSATS Institute of Information Technology, Islamabad |

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. |

Gov't Doc #: | 18275 |

URI: | http://prr.hec.gov.pk/jspui/handle/123456789/10822 |

Appears in Collections: | PhD Thesis of All Public / Private Sector Universities / DAIs. |

Files in This Item:

File | Description | Size | Format | |
---|---|---|---|---|

Muhammad Ali_Maths_2018_Comsats_PRR.pdf | 3.22 MB | Adobe PDF | View/Open |

Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.