Document Type : Research Paper
Author
Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran.
Abstract
In this study, the Bernoulli polynomials are used to obtain an approximate solution of a class of nonlinear two-dimensional integral equations. To this aim, the operational matrices of integration and the product for Bernoulli polynomials are derived and utilized to reduce the considered problem to a system of nonlinear algebraic equations. Some examples are presented to illustrate the efficiency and accuracy of the method.
Keywords
[1] E. Babolian, S. Bazm, and P. Lima, Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions, Commun. Nonl. Sci. Numer. Simul. 16(3) (2011) 1164{1175.
[2] S. Bazm, Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations, J. Comput. Appl. Math. 275 (2015) 44-60.
[3] A. H. Bhrawy, E. Tohidi, and F. Soleymani, A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-di erential equations with piecewise intervals, Appl. Math. Comput. 219(2) (2012) 482-497.
[4] A. Erdelyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcen-dental Functions, Vol. III, McGraw-Hill, New York, 1955.
[5] H. Guoqiang, K. Hayami, K. Sugihara, and W. Jiong, Extrapolation method of iterated collocation solution for two-dimensional nonlinear Volterra integral equations, App. Math. Comput. 112 (2009) 70-76.
[6] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1989.
[7] P. Lancaster, The Theory of Matrices: With Applications, second ed., Academic Press, New York, 1984.
[8] Y.L. Luke, The Special Functions and Their Approximations, Vol. I, Academic Press, New York, 1969.
[9] K. Maleknejad, S. Sohrabi, and B. Baranji, Application of 2D-BPFs to nonlinear integral equations, Commun. Nonl. Sci. Numer. Simul. 15 (2010) 527-535.
[10] S. Nemati, P.M. Lima, and Y. Ordokhani, Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials, J. Comput. Appl. Math. 242 (2013) 53{69.
[11] S. Nemati, and Y. Ordokhani, Solving Nonlinear Two-Dimensional Volterra Integral Equations of the First-kind Using the Bivariate Shifted Legendre Functions, International Journal of Mathematical Modelling & Computations 5(3)
(2015) 1-12.
(2015) 1-12.
[12] A. Tari, M.Y. Rahimi, S. Shahmorad, and F. Talati, Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the di erential transform method, J. Comput. Appl. Math. 228 (2000) 49-61.
[13] F. Toutounian and E. Tohidi, A new Bernoulli matrix method for solving second order linear partial di erential equations with the convergence analysis, Appl. Math. Comput. 223 (2013) 298-310.
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