الفهرس 
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Abstract
In the last three decades, considerable interest has been paid to the called fractional calculus, which allows us to consider integration and differentiation of any order, not necessarily integers. For a large extent, this is due to the applications of fractional calculus to problems in different areas of pure and applied sciences, such as physics, chemistry aerodynamics, electrodynamics of complex medium, viscoelasticity, heat conduction electricity mechanics, and control theory. The topic fractional calculus can be measured as an old as well as a new subject. Started from some speculations of Leibniz and Euler, followed by other important mathematicians like Laplace, Fourier, Abel, Liouville, Riemann and Holmgren Applications of PIDEs can be found in various fields such as, heat conduction of materials with memory, viscoelasticity, nuclear reactor dynamics, jump-diffusion models for pricing of derivatives in finance, financial modelling, electricity swaptions and biofluid flow in fractured biomaterials. Most of the research papers deal with the existence of unique solutions to equations for different types of differential and/or integral equations of one or multidimension. The Banach fixed point principle which is established by Polish mathematician Stefan Banach in 1922 is an important approach and it is one of the most powerful, fruitful tools of modern mathematics and may be On the Existence and Uniqueness of Solutions considered as a core subject for nonlinear analysis. It is used by a lot of authors to demonstrate the existence of a unique fixed point of certain self- maps of metric or normed spaces.