الفهرس 
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Abstract
n inverse problem is a general framework that is used to convert observed measurements
into information a bout a physical object or system that we are interested in.For example, if
we have measurements of the Earth’S gravity field, then we might ask the question:” given
the data that we have available, what can we say a bout the density distribution of the
Earth in that area?” The solution to this problem (i.e the density distribution that best,
matches that data) is useful because it generally tells us something a bout a physical param-
eter that we cannot directly observe. Thus, inverse problems are one of the most important,
and well-studied mathematical problems in science and mathematics.
Inverse problems arise in many branches of science and mathematics, including: machine
learning. statistics, statistical inference, geophysics, medical imaging such as computed ax-
ial tomography and ERP(Enterprise Resource Planning), remote sensing, ocean acoustic
tomography, nondestructive testing, astronomy, physics and many other fields.
The material of this thesis is divided into three chapters as follows :
Chapter (1) collects some preliminaries, notations and known results which will be used
in the other chapters. Also we introduce the main concepts of fractional-order integration,
differentiation, and we give some recent results for the existence of solution of some an
inverse problems.
An inverse problems was studied by many authors, for examples
In 2003 Giivenilir, et al. [21] studied that, under some conditions ,all solutions of the
inverse problem for a class of first-order and second-order linear differential operator equa-
tions in a Hilbert space.
First-order equation
In a Hilbcrt space H with the inner product (.,.) and the norm 11.11 they consider the
following problem (see [21]).