Qualitative Analysis for Proportional Delay Fractional Differential Wquations

Abstract

This dissertation is committed to the investigation of different types of fractional pantograph differential equations. The concern research is associated with the exis tence theory, stability analysis and numerical approximation of proposed classes of fractional differential equations. We utilized the tools of classical fixed point the ory, degree theory and nonlinear analysis to established the desired results for the existence of solutions and Ulam-Hyers type stability. We investigate anti-periodic boundary value problems and multi-point boundary values problems of fractional pantograph differential equations for the existence of solutions and unique solution via aforementioned techniques. We provided some examples for illustrative purposes of the main work. An important aspect of concerned theory is numerical approxima tion of proposed fractional differential equations. Some times it is too complicated to obtain the exact solution of fractional differential equations. Therefore, the re searchers paid more attention toward the numerical approximation for the proposed type of fractional differential equations. In this connection, we developed the numeri cal schemes for fractional order Benjamin-Bona-Mohany (FBBM) and the dynamical system of an infectious disease known as Measles under nonlocal and nonsingular derivative of Caputo-Fabrizo (CF). Furthermore, we obtained the semi-analytic re sults for aforesaid models via Laplace Adomian decomposition method. At same stages, we also provide graphical presentations for governing results via Matlab and Mathematica. The obtained results are compared with other well known techniques to demonstrate the efficiency and reliability of the concerned technique. With the help of Haar wavelet method (HWM), we compute the numerical solution to Pantograph type integro-differential equations concerning variable fractional order. Furthermore, we extended the concerned scheme to verify the techniques for various examples.