Abel's integral equation of order [alpha] is given by

Kumar, "Numerical solution of generalized

Abel's integral equation by variational iteration method," American Journal of Computational Mathematics, vol.

Abel's integral equation, one of the very first integral equations seriously studied, and the corresponding integral operator have never ceased to inspire mathematicians to investigate and to generalize them.

SOHRABI, Comparison Chebyshev wavelets method with BPFs method for solving Abel's integral equation, Ain Shams Engrg.

Rostamy, "Bernstein polynomials for solving

Abel's integral equation," Journal of Mathematics and Computer Science, vol.

Pandey, "A stable numerical inversion of

Abel's integral equation using almost Bernstein operational matrix," Journal of Quantitative Spectroscopy and Radiative Transfer, vol.

Sohrabi, "Comparison Chebyshev wavelets method with BPFs method for solving

Abel's integral equation," Ain Shams Engineering Journal, vol.

In order to use COM for Abel's integral equation of the form (3), we first approximate u(x) by the shifted Chebyshev polynomials as

Consider Abel's integral equation of the first kind [31, 32]

Consider Abel's integral equation of the second kind [32, 33]

There are many different methods and different basis functions have been used to estimate the solution of fractional integro-differential equations or Abel's integral equations, such as Adomian decomposition method [7, 8], fractional differential transform method [9, 10], collocation method [11, 12], homotopy perturbation method [13, 14], homotopy analysis method [15, 16], variational iteration method [17], discrete Galerkin method [18], and Haar wavelet method [19].

Rostamy, "Bernstein polynomials for solving

Abel's integral equation," The Journal of Mathematics and Computer Science, vol.