In this paper, we define fractional derivative of arbitrary complex order of the distributions concentrated on R + , based on convolutions of generalized functions with the supports bounded on the same side. Using distributional derivatives, which are generalizations of classical derivatives, we present a few interesting results of fractional derivatives in D (R + ), as well as the symbolic solution for the following differential equation by Babenko's methodwhere Re α > 0.MSC 2010 : 46F10, 26A33

This paper is to study the following generalized Abel's integral equationand its variant in the distributional (Schwartz) sense based on fractional calculus of distributions. We obtain a number of interesting and new results which are not achievable in the classical sense.

Abstract:The goal of this paper is to investigate the following Abel's integral equation of the second kind:and its variants by fractional calculus. Applying Babenko's approach and fractional integrals, we provide a general method for solving Abel's integral equation and others with a demonstration of different types of examples by showing convergence of series. In particular, we extend this equation to a distributional space for any arbitrary α ∈ R by fractional operations of generalized functions for the first time and obtain several new and interesting results that cannot be realized in the classical sense or by the Laplace transform.

This paper is to study certain types of fractional differential and integral equations, such (x) in the distributional sense by Babenko's approach and fractional calculus. Applying convolutions and products of distributions in the Schwartz sense, we obtain generalized solutions for integral and differential equations of fractional order by using the Mittag-Leffler function, which cannot be achieved in the classical sense including numerical analysis methods, or by the Laplace transform.

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