The Abel Integral Equations in Distribution

Abstract

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.

Several Results of Fractional Differential and Integral Equations in Distribution

“…However, their transformations to the form of Abel's integral equations will speed up the solution process [24], or, more significantly, lead to distributional solutions in cases where classical ones do not exist [22].…”

“…There have been lots of approaches, including numerical analysis, thus far to studying fractional differential and integral equations, including Abel's equations, with many applications [8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Recently, Li et al [22,23] studied integral equations associated with Abel's types in the distributional (Schwartz) sense, based on new fractional calculus of distributions and derived fresh results which are not achievable in the classical sense. On the other hand, the development of science has led to formation of many physical and engineering problems that can be mathematically represented by differential equations.…”

Babenko’s Approach to Abel’s Integral Equations

“…The main advance of this proposal is its capability of obtaining rapid convergent series for singular integral equations of Abel type. Recently, Li et al [23] studied the following Abel's integral equation:…”

Solutions to Abel’s Integral Equations in Distributions

“…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].…”