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Title: The Qualitative Theory of Fractional Difference Equations
Authors: Haider, Syed Sabyel
Keywords: Physical Sciences
Issue Date: 2021
Publisher: National University of Science & Technology, Islamabad.
Abstract: Focus of this study is to cover up some gaps and further build up the theory of discrete fractional calculus. This dissertation starts with brief introduction and definitions to discrete fractional calculus. Two new definitions of generalized fractional difference operator are intro duced namely Hilfer fractional difference operator and substantial fractional difference operator. A missing property in the literature for delta Laplace transform i.e. delta exponential shift is established. The delta Laplace transform is presented for the newly introduced Hilfer and substantial fractional differences. The double Laplace transform in a delta discrete setting is introduced, and its existence, uniqueness and basic properties are discussed. The delta double Laplace transform is presented for integer and non-integer order partial differences. Another goal of this study is to establish the existence and UHR stability for various classes of fractional difference equations. Conditions are acquired for RL, Caputo, Hilfer and substantial type fractional difference equations. Moreover we establish a technique to transforming arbitrary real order delta difference equations with impulses to corresponding summation equations. Existence results are built up for impulsive delta fractional difference equation with nonlocal initial condition and two-point and four-point boundary conditions. The conditions for existence and UHR stability of the solution to multi-point summation boundary value problem are established.
Gov't Doc #: 23242
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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