Application of modified hat functions for solving nonlinear quadratic integral equations

Document Type : Research Article

Authors

Malayer University

Abstract

A numerical method to solve nonlinear quadratic integral equations (QIE) is presented in this work. The method is based upon modification of hat functions (MHFs) and their operational matrices. By using this approach and the collocation points, solving the nonlinear QIE reduces to solve a nonlinear system of algebraic equations. The proposed method does not need any integration for obtaining the constant coefficients. Hence, it can be applied in a simple and fast technique. Convergence analysis and associated theorems are considered. Some numerical examples illustrate the accuracy and computational efficiency of the proposed method.

Keywords


1. Argyros, I.K. On a class of quadratic integral equations with perturbations, Funct. Approx., 20 (1992) 51-63.
2. Argyros, I.K. Quadratic equations and applications to Chandrasekhar's and related equations, Bull. Aust. Math. Soc., 32 (1985) 275-292.
3. Atkinson, K.E. The numerical solution of integral equations of the second kind, Cambridge University Press, Cambridge, 1997.
4. Bana_s, J., Caballero, J., Rocha, J. and Sadarangani, K Monotonic solutions of a class of quadratic integral equations of Volterra type, Comput. Math. Appl., 49 (2005) 943-952.
5. Bana_s, J., Lecko, M. and El-Sayed, W.G. Existence theorems of some quadratic integral equation, J. Math. Anal. Appl., 227 (1998) 276-279.
6. Bana_s, J. and Martinon, A. Monotonic solutions of a quadratic integral equation of Volterra type, Comput. Math. Appl., 47 (2004) 271-279.
7. Bana_s, J., Rocha Martin, J. and Sadarangani, K. On the solution of a quadratic integral equation of Hammerstein type, Math. Comp. Model., 43 (2006) 97-104.
8. Bana_s, J. and Rzepka, B.Monotonic solutions of a quadratic integral equations of fractional order, J. Math. Anal. Appl., 332 (2007) 1370-1378.
9. Bana_s, J. and Rzepka, B.Nondecreasing solutions of a quadratic singular Volterra integral equation, Math. Comput. Model., 49 (2009) 488-496.
10. Benchohra, M. and Darwish, M.A. On quadratic integral equations of urysohn type in fr_echet Spaces, Acta Math. Univ. Comenianae, 79(1) (2010) 105-110.
11. Curtain, R.F. and Pritchard, A.J. Functional analysis in modern applied mathematics, Vol. 132. London: Academic press, 1977.
12. Darwish, M.A.On monotonic solutions of a singular quadratic integral equation with supremum, Dyn. Syst. Appl., 17 (2008) 539-550.
13. El-Borai, M.M., El-Sayed, W.G. and Abbas, M.I.Monotonic solutions of a class of quadratic singular integral equations of Volterra type, Int. J. Contemp. Math. Sci., 2(2) (2007) 89-102.
14. El-Sayed, A.M.A. and Hashem, H.H.G. Carath_eodory type theorem for a nonlinear quadratic integral equation, Math. Sci. Res. J., 12(4) (2008) 71-95.
15. El-Sayed, A.M.A. and Hashem, H.H.G. Integrable and continuous solutions of a nonlinear quadratic integral equation, EJQTDE, 25 (2008) 1-10.
16. El-Sayed, A.M.A. and Hashem, H.H.G. Monotonic positive solution of nonlinear quadratic Hammerstein and Urysohn functional integral equations, Commentationes Math., 48(2) (2008) 199-207.
17. El-Sayed, A.M.A. and Hashem, H.H.G. Monotonic solutions of functional integral and di_erential equations of fractional order, EJQTDE, 7 (2009) 1-8.
18. El-Sayed, A.M.A. and Hashem, H.H.G. Monotonic positive solution of a nonlinear quadratic functional integral equation, Appl. Math. Comput., 216 (2010) 2576-2580.
19. El-Sayed, A.M.A. and Hashem, H.H.G. Existence results for nonlinear quadratic functional integral equations of fractional orders, Miskolc Math. Notes, 14(1) (2013) 79-88.
20. El-Sayed, A.M.A. and Hashem, H.H.G. Existence results for coupled systems of quadratic integral equations of fractional orders, Optim. Lett., 7(2013) 1251-1260.
21. El-Sayed, A.M.A., Hashem, H.H.G. and Omar, Y.M.Y. Positive continuous solution of a quadratic integral equation of fractional orders, Math. Sci. Lett., 2(1) (2013) 19-27.
22. El-Sayed, A.M.A., Hashem, H.H.G. and Ziada, E.A.A. Picard and Adomian decomposition methods for a quadratic integral equation of fractional order, Comp. Appl. Math., 33 (2014) 95-109.
23. El-Sayed, A.M.A., Hashem, H.H.G. and Ziada, E.A.A. Picard and Adomian Methods for quadratic integral equation, Comp. Appl. Math., 29(3) (2010) 447-463.
24. El-Sayed, A.M.A., Mohamed, M.Sh. and Mohamed, F.F.S. Existence of positive continuous solution of a quadratic integral equation of fractional orders, J. Fract. Calc. Appl., 1(9) (2011) 1-7.
25. El-Sayed, W.G. and Rzepka, B. Nondecreasing solutions of a quadratic
integral equation of Urysohn Type, Comput. Math. Appl., 51 (2006) 1065 -1074.
26. El-Sayed, A.M.A., Saleh, M.M. and Ziada, E.A.A. Numerical and analytic solution for a nonlinear quadratic integral equation, Math. Sci. Res. J., 12(8) (2008) 183-191.
27. Gasca, M. and Sauer, T. On the history of multivariate polynomial interpolation, J. Comput. Appl. Math., 122 (2000) 23-35.
28. Mirzaee, F. and Hadadiyan, E. Application of two-dimensional hat functions for solving space-time integral equations, J. Appl. Math. Comput., (2015) In press.
29. Mirzaee, F. and Hadadiyan, E.Numerical solution of linear Fredholm integral equations via two-dimensional modi_cation of hat functions, Appl. Math. Comput., 250 (2015) 805-816.
30. Mirzaee, F., Hadadiyan, E. and Bimesl, S. Numerical solution for three-dimensional nonlinear mixed Volterra-Fredholm integral equations via three-dimensional block-pulse functions, Appl. Math. Comput., 237(2014) 168-175.
31. Nemati, S., Lima, P.M. and Ordokhani, Y. Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials, J. Comput. Appl. Math., 242 (2013) 53-69.
32. Salem, H.A.H. On the quadratic integral equations and their applications, Comput. Math. Appl., 62(8) (2011) 2931-2943.
33. Ziada, E.A.A. Adomian solution of a nonlinear quadratic integral equation, J. Egy. Math. Soci., 21 (2013) 52-56.