Several results of fractional differential and integral equations in distribution

Abstract

This paper is to study certain types of fractional differential and integral equations, such as θ(x-x0) g(x) = 1/Γ(α) ∫0x (x-ζ)α-1 f(ζ) dζ, y(x) + ∫0x y(τ)/√x-τ dτ = x+1√2 + δ(x), and x+k ∫0x y(τ) (x-τ)α-1 dτ = δ(m)(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.