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.
Let {τn} be a certain sequence of functions in D converging to 1 in D. The commutative neutrix convolution f ♦ * g of two distributions f and g in D is defined to be the neutrix limit of the sequence 1 2 {(f τn) * g + f * (gτn)} , provided the limit exists. We present relations between this new convolution and other existing distributional convolutions, and demonstrate its strong computational power in evaluating convolutions as well as applications to defining new fractional derivatives and integrals of generalized functions in the new space H which contains D (R +). The neutrix convolutions x λ − ♦ * x µ + for λ, µ, λ + µ = 0, ±1, ±2, • • • and x λ − ♦ * x s + for λ = 0, ±1, ±2, • • • and s = 0, 1, 2, • • • are evaluated, from which other neutrix convolutions are deduced.
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