Fractional Calculus of Distributions and Integral Equations

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Fractional calculus is one of the most intensively developing areas of mathematical analysis as a result of its increasing range of applications. Operators for fractional differentiation and integration have been used in various fields, such as hydraulics of dams, potential fields, and waves in liquids and gases. Fractional calculus is found in almost every realm of science and engineering. It is one of the best tools to characterize long-memory processes and materials, anomalous diffusion, long-range interactions, long-term behaviors, power laws and scaling laws, for example. On the other hand, distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense, such as the step function. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional (weak) solutions than classical ones. Studying fractional differential and integral equations distributionally, including Abel's integral equations of the first and second kind, has been a serious challenge. The long term goal of my research proposal is to define fractional calculus of distributions in general by convolution, normalization, and approximation, which is novel in distribution theory. This will require the development of new methods of computing integrals of generalized functions over the Schwartz space or a subspace. Subsequently, I plan to investigate fractional differential and integral equations, particularly Abel's integral equations, in the distributional sense by inverse operators in terms of convolution, and study Mittag-Leffler functions, approximate solutions, their convergence of series, and order of approximation in Big-O notation under topological structures. Additionally, I intend to give meaning to singular distributions and products encountered, while solving fractional differential and integral equations, by delta sequences and complex analysis approaches, as they are also widely needed in physics, quantum field theory and differential equations involving distributions. The significance of the work is the following: Many applied problems from physical, engineering and chemical processes lead to integral equations, which at first glance have nothing in common with Abel's integral equations, and due to this perception additional efforts are undertaken for the development of analytical or numerical procedure for solving these equations. However, through the application of my convolution approach, their transformations to the form of Abel's integral equations will speed up the solution process, or, more significantly, lead to distributional solutions in cases where classical ones do not exist.