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Browsing Phd by Author "Asmat Batool"

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    Existence and approximation of solutions of fractional boundary value problems
    (UMT Lahore, 2024-08-07) Asmat Batool
    This thesis deals with the fascinating and rapidly evolving field of fractional calculus, focusing specifically on the existence and approximation of solutions of fractional boundary value problems (FBVPs). These FBVPs, characterized by the presence of fractional derivatives in the governing equations and specified boundary conditions, have emerged as powerful tool for modeling complex phenomena across diverse scientific and engineering disciplines. Unlike traditional integer-order derivative, fractional-order derivative (FOD) possess a remarkable "memory property", enabling them to capture the history-dependent behavior often observed in physical systems. This inherent ability to model nonlocal effects has led to the widespread application of FBVPs in areas such as viscoelasticity, anomalous diffusion, control theory, signal processing, and image analysis. Despite their growing significance, FBVPs present unique mathematical challenges, particularly concerning the existence and computation of solutions. The nonlocal nature of FOD necessitates the development of specialized analytical and numerical techniques that differ significantly from those employed for classical boundary value problems (BVPs). This thesis aims to contribute significantly to both the theoretical understanding and practical solvability of FBVPs. A significant part of this thesis is dedicated for a comprehensive investigation into the existence of solutions for various classes of FBVPs. This analysis is very important for ensuring that the mathematical models based on FBVPs are well-posed and possess physically meaningful solutions. The thesis employs the powerful and versatile lower and upper solutions (LUSs) approach that stands out as a strong method for establishing the existence of solutions. This approach involves transforming the original problem into a logically modified problem. By using the theory of differential inequalities and combining it with established existence results, this method effectively establishes the existence of a solution to the modified problem which leads to the solution of the original problem.