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 (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.

Uniqueness of the Hadamard-type integral equations

“…We begin by showing the solution for equation ( 2 ) as a convergent series in the space by Babenko’s approach [ 15 ], which is a powerful tool in solving differential and integral equations. The method itself is close to the Laplace transform method in the ordinary sense, but it can be used in more cases [ 16 , 17 ], such as solving integral or fractional differential equations with distributions whose Laplace transforms do not exist in the classical sense. Clearly, it is always necessary to show the convergence of the series obtained as solutions.…”

Solutions to Abel’s Integral Equations in Distributions

“…Due to this perception, additional efforts are undertaken for the development of analytical or numerical procedure for solving these equations. However, their transformations to the form of Abel's integral equations will speed up the solution process [20], or, more significantly, lead to distributional solutions in cases where classical ones do not exist [40,41].…”

“…The development of integral equations has led to the construction of many real world problems, such as mathematical physics models [23,24], scattering in quantum mechanics and water waves. There have been lots of techniques, such as numerical analysis and integral transforms [25][26][27], thus far to studying fractional differential and integral equations, including Abel's equations, with many applications [1,20,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42].…”

Solutions of the Generalized Abel’s Integral Equations of the Second Kind with Variable Coefficients

“…and convolution. Li et al [25][26][27] recently studied Abel's integral Equation (1) for any arbitrary α ∈ R in the generalized sense based on fractional calculus of distributions, inverse convolutional operators and Babenko's approach [28]. They obtained several new and interesting results that cannot be realized in the classical sense or by the Laplace transform.…”