Solving Abel’s equations with the shifted Legendre polynomials

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P. O. Box 35195-363, Semnan, Iran.


In this article, a numerical method is presented to solve Abel’s equations. In the given method, the solution of the equation is found as a finite expansion of the shifted Legendre polynomials. To this end, the integral and differential parts of the equation are converted to vector-matrix representations. Therefore, the equation is converted to an algebraic system of the equations and by solving it, the solution of the equation is obtained. Further, the numerical example is given to illustrate the method’s efficiency.


  • [1] S. Alavi, A. Haghighi, A. Yari, and F. Soltanian, A numerical method for solving fractional optimal control problems using the operational matrix of Mott polynomials, Comput. methods differ. equ., 10(3) (2022), 755–773.
  • [2] K. E. Atkinson, The Numerical Solution of Integral equations of Second Kind, Cambridge University Press, Cambridge, 1997.
  • [3] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Dif- ferential Operators of Caputo Type, Springer-Verlag Berlin, Heidelberg, 2010.
  • [4] S. Dixit, R. K. Pandey, S. Kumar, and O. P. Singh, Solution of the generalized Abel integral equation by using almost operational matrix, Am. J. Comput. Math., 1 (2011), 226–234.
  • [5] R. B. Guenther, Linear Integral Equations (Ram P. Kanwal), SIAM Review, 14 (1972), 669.
  • [6] M. Gulsu, Y. Ozturk, and M. Sezer, On the solution of the Abel equation of the second kind by the shifted Chebyshev polynomials, Appl. Math. Comput., 217 (2011), 4827–4833.
  • [7] K. Issa, B. Yisa, and J. Biazar, Numerical solution of space fractional diffusion equation using shifted Gegenbauer polynomials, Comput. methods differ. equ., 10(2) (2022), 431–444.
  • [8] K. Sadri, A. Amini, and C. Cheng, A new operational method to solve Abel’s and generalized Abel’s integral equations, Appl. Math. Comput., 317 (2018), 49–67.
  • [9] J. Shen, T. Tang, and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag Berlin, Heidelberg 2011.
  • [10] C. S. Singh, H. Singh, V. K. Singh, and O. P. Singh, Fractional order operational matrix methods for fractional singular integro-differential equation, Appl. Math. Model., 40 (2016), 10705–10718.
  • [11] O. P. Singh, V. K. Singh, and R. K. Pandey, A stable numerical inversion of Abel’s integral equation using almost Bernstein operational matrix, Appl. Numer. Math., 62 (2012), 567–579.
  • [12] S. Sohrabi, Comparison Chebyshev wavelets method with BPFs method for solving Abel’s integral equation, Ain Shams Eng. J., 2 (2011), 249–254.