We suggest a convenient method based on the Fibonacci polynomials and the collocation points for solving approximately the Abel’s integral equation of second kind. Initially, the solution is supposed in the form of the Fibonacci polynomials truncated series with the unknown coefficients. Then, by placing this series into the main problem and collocating the resulting equation at some points, a system of algebraic equations is obtained. After solving it, the unknown coefficients and so the solution of main problem are determined. The error analysis is discussed elaborately. Also, the reliability of the method is quantified through numerical examples.
Abel’s integral equation;, Fibonacci polynomials;, Collocation points;, Error analysis.
 Alipour, M. and Rostamy, D. Bernstein polynomials for solving Abel’s integral equation, J. Math. Computer Sci. 3(4) (2011), 403–412.
 Andanson, P., Cheminat B. and Halbique, A.M. Numerical solution of the Abel integral equation: application to plasma spectroscopy, J. Phys. D Appl. Phys. 11(3) (1978), 209.
 Azodi, H.D. Euler polynomials approach to the system of nonlinear fractional differential equations, Punjab Univ. J. Math. (Lahore) 51(7) (2019), 71–87.
 Azodi, H.D. and Yaghouti, M.R. Bernoulli polynomials collocation for weakly singular Volterra integro-differential equations of fractional order, Filomat 32(10) (2018), 3623–3635.
 Başi, A.K. and Yilçinbaş, S. Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method, Springerplus 5 (2016) 1375.
 Bicknell, M. A primer for the Fibonacci numbers VII Fibonacci Quart. 8 (1970), 407–420.
 Cimatti, G. Application of the Abel integral equation to an inverse problem in thermoelectricity, Eur. J. Appl. Math. 20(6) (2009), 519–529.
 Cremers, C.J. and Birkebak, R.C. Application of the Abel integral equation to spectrographic data, Appl. Opt. 5(6) (1966), 1057–1064.
 Huang, L., Huang, Y. and Li, X.F. Approximate solution of Abel integral equation, Comput. Math. Appl. 56(7) (2008), 1748–1757.
 Liu, Y.P. and Tao, L. High accuracy combination algorithm and a posteriori error estimation for solving the first kind Abel integral equations, Appl. Math. Comput. 178(2) (2006), 441–451.
 Liu, Y.P. and Tao, L. Mechanical quadrature methods and their extrapolation for solving first kind Abel integral equations, J. Comput. Appl. Math. 201(1) (2007), 300–313.
 Lucas, E. Theorie de fonctions numeriques simplement periodiques Amer. J. Math. 1 (1878), 184–240; 289–321.
 Loh, J.R., Phang, C. and Isah, A. New operational matrix via Genocchi polynomials for solving Fredholm-Volterra fractional integro-differential equations, Adv. Math. Phys. 2017, ID 3821870, 12 pages.
 Koç, A.B., Çakmak, M. and Kurnaz, A. and Uslu, K. A new Fibonacci type collocation procedure for boundary value problems, Adv. Differ. Equ. (2013) 2013: 262.
 Kumar, S., Kumar, A., Kumar, D., Singh, J. and Singh, A. Analytical solution of Abel integral equation arising in astrophysics via Laplace transform, J. Egypt. Math. Soc. 23(1) (2015), 102–107.
 Mirzaee, M. and Hoseini, S.F. Solving singularly perturbed differential difference equations arising in science and engineering with Fibonacci polynomials, Results Phys. 3 (2013), 134–141.
 Mirzaee, M. and Hoseini, S.F. Solving systems of linear Fredholm integro differential equations with Fibonacci polynomials, Ain Shams Eng. J. 5(1) (2014), 271–283.
 Mirzaee, M. and Hoseini, S.F. A Fibonacci collocation method for solving a class of Fredholm-Volterra integral equations in two-dimensional spaces, Beni-Suef Univ. J. Basic Appl. Sci. 3(2) (2014), 157–163.
 Mirzaee, M. and Hoseini, S.F. Numerical approach for solving nonlinear stochastic Itô-Volterra integral equations using Fibonacci operational matrices, Sci. Iran. 22(6) (2015), 2472–2481.
 Mirzaee, M. and Hoseini, S.F. A new collocation approach for solving systems of high-order linear Volterra integro-differential equations with variable coefficients, Appl. Math. Comput. 273 (2016), 637–644.
 Mirzaee, M. and Hoseini, S.F. A new collocation approach for solving systems of high-order linear Volterra integro-differential equations with variable coefficients, Appl. Math. Comput. 311 (2017), 272–282.
 Mirzaee, M. and Samadyar, S. Numerical Solution of Weakly Singular Ito-Volterra Integral Equations via Operational Matrix Method based on Euler Polynomials, Math. Res. 4(1) (2018), 91–104
 Nosrati Sahlan, M., Marasi, H.R. and Ghahramani, F. Block-pulse functions approach to numerical solution of Abel’s integral equation, Cogent Math. 2(1) (2015) Art. ID 1047111, 9 pp.
 Pandey, R.K., Singh, O.P. and Singh, V.K. Efficient algorithms to solve singular integral equations of Abel type, Comput Math Appl. 57(4) (2009), 664–676.
 Prajapati, R.N. Mohan, R. and Kumar, P. Numerical solution of generalized Abel’s integral equation by variational iteration method, Am. J. Comput. Math. 2(4) (2012), 312–315.
 Saadatmandi, A. and Dehghan, M. A collocation method for solving Abel’s integral equations of first and second kinds, Z. Naturforsch. A 63(12) (2008), 752–756.
 Setia A. and Pandey, R.K. Laguerre polynomials based numerical method to solve a system of generalized Abel integral equations, Procedia Eng. 38 (2012), 1675–1682.
 Yousefi, S.A. Numerical solution of Abel’s integral equation by using Legendre wavelets, Appl. Math. Comput. 175(1) (2006), 574–580.
 Zhu, L. and Wang, Y. Numerical solutions of Volterra integral equation with weakly singular kernel using SCW method, Appl. Math. Comput. 260 (2015), 63–70.
This work is licensed under a Creative Commons Attribution 4.0 International License.