Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials

• Published in: Journal of computational and applied mathematics Vol. 242; pp. 53 - 69
• Main Authors: Nemati, S., Lima, P.M., Ordokhani, Y.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.04.2013
• Subjects: Approximation; Best approximation; Error analysis; Integral equations; Legendre polynomials; Mathematical analysis; Mathematical models; Nonlinearity; Operational matrix; Polynomial interpolation; Two dimensional; Two-dimensional Volterra integral equation; Volterra integral equations; 45G10; 65R20; Polynomial interpolation; Operational matrix; Best approximation; Legendre polynomials; Two-dimensional Volterra integral equation
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2012.10.021

In this paper, a method for finding an approximate solution of a class of two-dimensional nonlinear Volterra integral equations is discussed. The properties of two-dimensional shifted Legendre functions are presented. The operational matrices of integration and product together with the collocation points are utilized to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations. Some results concerning the error analysis are obtained. We also consider the application of the method to the solution of certain partial differential equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. ► A two-dimensional nonlinear Volterra integral equation of second kind is considered. ► We propose a numerical method based on a basis of bivariate Legendre polynomials. ► The proposed numerical method is easy to implement and provides high accuracy. ► Numerical results are discussed and compared with the ones published before.