Solving Abel’s equations with the shifted Legendre polynomials. (English) Zbl 07809638
Summary: 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.
MSC:
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 65R20 | Numerical methods for integral equations |
Keywords:
Abel’s equation; integral equation; Caputo differential operator; shifted Legendre polynomialReferences:
| [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. · Zbl 1524.65247 |
| [2] | K. E. Atkinson, The Numerical Solution of Integral equations of Second Kind, Cambridge University Press, Cambridge, 1997. · Zbl 0899.65077 |
| [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. · Zbl 1215.34001 |
| [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. · Zbl 1207.65098 |
| [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. · Zbl 1499.35653 |
| [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. · Zbl 1426.65218 |
| [9] | J. Shen, T. Tang, and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag Berlin, Heidelberg 2011. · Zbl 1227.65117 |
| [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. · Zbl 1443.65446 |
| [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. · Zbl 1241.65122 |
| [12] | S. Sohrabi, Comparison Chebyshev wavelets method with BPFs method for solving Abel’s integral equation, Ain Shams Eng. J., 2 (2011), 249-254. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
