Numerical solution of Fredholm and Volterra integral equations using the normalized Müntz−Legendre polynomials
Subject Areas : Numerical Analysis
Fereshteh Saemi
1
,
Hamideh Ebrahimi
2
,
Mahmoud Shafiee
3
1 -
2 -
3 -
Keywords: Operational matrix, Spectral method, Nonlinear Fredholm and Volterra integral equations, Müntz-Legendre polynomials, Stability, error bound, and convergence analysis,
Abstract :
The current research approximates the unknown function based on the normalized Müntz−Legendre polynomials (NMLPs) in conjunction with a spectral method for the solution of nonlinear Fredholm and Volterra integral equations. In this method, by using operational matrices, a system of algebraic equations is derived that can be readily handled through the use of the Newton scheme. The stability, error bound, and convergence analysis of the method are discussed in detail by preparing some theorems. Several illustrative examples are provided formally to show the efficiency of the proposed method.
Bloom, F. (1979). Asymptotic Bounds for Solutions to a System of Damped Integrodifferential Equations of Electromagnetic Theory. South Carolina Univ Columbia Dept of Mathematics Computer Science and Statistics.
Abdou, M. A. (2002). Fredholm–Volterra integral equation of the first kind and contact problem. Applied Mathematics and Computation, 125(2-3), 177-193.
Isaacson, S. A., & Kirby, R. M. (2011). Numerical solution of linear Volterra integral equations of the second kind with sharp gradients. Journal of Computational and Applied Mathematics, 235(14), 4283-4301.Math., 53 (2) (1995) 245–258.
AGARWAL, R. P., & O'REGAN, D. (2004). Fredholm and Volterra integral equations with integrable singularities. Hokkaido mathematical journal, 33(2), 443-456.
Mokhtary, P., Ghoreishi, F., & Srivastava, H. M. (2016). The Müntz-Legendre Tau method for fractional differential equations. Applied Mathematical Modelling, 40(2), 671-684.
Esmaeili, S., Shamsi, M., & Luchko, Y. (2011). Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials. Computers & Mathematics with Applications, 62(3), 918-929.
Mokhtary, P. (2016). Operational Müntz-Galerkin approximation for Abel-Hammerstein integral equations of the second kind. Electron. Trans. Numer. Anal, 45, 183-200.
Yüzbaşı, Ş., Gök, E., & Sezer, M. (2013). Müntz-Legendre polynomial solutions of linear delay Fredholm integro-differential equations and residual correction. Mathematical and Computational Applications, 18(3), 476-485.
Aghashahi, M., Gandomani, M. R., Shahr, S., & Branch, I. K. (2017). Numerical solution of fractional differential equation system using the Müntz–Legendre polynomials. Int. J. Pure Appl. Math, 115(3), 467-475.
Rahimkhani, P., & Ordokhani, Y. (2018). Application of Müntz–Legendre polynomials for solving the Bagley–Torvik equation in a large interval. SeMA Journal, 75(3), 517-533.
Safaie, E., Farahi, M. H., & Farmani Ardehaie, M. (2015). An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials. Computational and applied mathematics, 34(3), 831-846.
Borwein, P., Erdélyi, T., & Zhang, J. (1994). Müntz systems and orthogonal Müntz-Legendre polynomials. Transactions of the American Mathematical Society, 342(2), 523-542.
Stefánsson, Ú. F. (2010). Asymptotic behavior of Müntz orthogonal polynomials. Constructive Approximation, 32(2), 193-220.
Bernstein, P. S. (1912). SUR LES RECHERCHES RECENTES RELATIVES A LA MEILLEURE APPROXIMATION DES FONCTIONS CON-TINUES PAR DES POLYNÔMES.
Müntz, C. H. (1914). Über den approximationssatz von Weierstrass. In Mathematische Abhandlungen Hermann Amandus Schwarz (pp. 303-312). Springer, Berlin, Heidelberg.
Ortiz, E. L., & Pinkus, A. (2005). Herman Müntz: a mathematician’s odyssey. The Mathematical Intelligencer, 27(1), 22-31.
Stefánsson, U. F. (2010). Asymptotic properties of Müntz orthogonal polynomials. Georgia Institute of Technology.
Esmaeili, S., Shamsi, M., & Luchko, Y. (2011). Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials. Computers & Mathematics with Applications, 62(3), 918-929.
Jackson, D. (1941). Fourier Series and Orthogonal Polynomials, The Mathematical Association of America.
Babolian, E., & Mordad, M. (2011). A numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions. Computers & Mathematics with Applications, 62(1), 187-198.
Mirzaee, F., & Hadadiyan, E. (2016). Numerical solution of Volterra–Fredholm integral equations via modification of hat functions. Applied Mathematics and Computation, 280, 110-123.
Maleknejad, K., Almasieh, H., & Roodaki, M. (2010). Triangular functions (TF) method for the solution of nonlinear Volterra–Fredholm integral equations. Communications in Nonlinear Science and Numerical Simulation, 15(11), 3293-3298.
Soleymanpour Bakefayat, A., & Karamseraji, S. (2017). Solving Second Kind Volterra-Fredholm Integral Equations by Using Triangular Functions (TF) and Dynamical Systems. Control and Optimization in Applied Mathematics, 2(1), 43-63.
Shali, J. A., Akbarfam, A. J., & Ebadi, G. (2012). Approximate solutions of nonlinear Volterra-Fredholm integral equations. International Journal of Nonlinear Science, 14(4), 425-433.
