The fractional Laplacian, also known as the Riesz fractional derivative operator, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighbouring or nearby sites, as well as perform excursions to remote sites by way of Lévy flights. The fractional Laplacian has many applications in the boundary behaviours of solutions to differential equations. The goal of this paper is to investigate the half-order Laplacian operator ( − Δ ) 1 2 in the distributional sense, based on the generalized convolution and Temple’s delta sequence. Several interesting examples related to the fractional Laplacian operator of order 1 / 2 are presented with applications to differential equations, some of which cannot be obtained in the classical sense by the standard definition of the fractional Laplacian via Fourier transform.

Applying Babenko’s approach, we construct solutions for the generalized Abel’s integral equations of the second kind with variable coefficients on R and R n , and show their convergence and stability in the spaces of Lebesgue integrable functions, with several illustrative examples.

The goal of this paper is to study fractional calculus of distributions, the generalized Abel's integral equations, as well as fractional differential equations in the distributional space D (R + ) based on inverse convolutional operators and Babenko's approach. Furthermore, we provide interesting applications of Abel's integral equations in viscoelastic systems, as well as solving other integral equations, such as π/2 θ y(ϕ) cos β ϕ(cos θ−cos ϕ) α dϕ = f (θ) , and ∞ 0 x 1/2 g(x)y(x + t)dx = f (t).