| Abstrakt: |
Introduction The theory and applications of integral equations have been the subject of many researches during the last decades. Fredholm integral equations have been widely used in applied science such as engineering and physics. In recent years, methods based on orthogonal basis functions, including collocation, Tau and Galerkin methods with Jacobi, Legendre and Chebyshev polynomials have attracted the attention of mathematicians. The most important advantage of using orthogonal basis functions is simplifying the mentioned methods by solving a linear/nonlinear algebraic system. But solving algebraic systems is time-consuming, especially in nonlinear problems. This study presents an efficient iterative semi-analytical method by employing the shifted Legendre polynomials for solving the Fredholm integral equations of the second kind. Material and Methods The proposed method is based on the Picard iteration method, the shifted Legendre polynomials, and the shifted Legendre-Gauss integration rule. According to the orthogonal property of Legendre polynomials, the proposed method uses an iterative scheme to update the coefficients of the series of approximate solution. Also, a vector-matrix form is introduced to increase the efficiency and reduce the computational time. The numerical results clearly indicate the feasibility and the accuracy of the proposed technique. Results and discussion In order to examine the validity of the proposed method, we employed it to find the numerical solution of several Fredholm integral equations of the second kind. The results for six examples are reported in this section. Also, a comparison study between the proposed iterative method and other existing methods is provided. The obtained numerical results show the accuracy and efficiency of the iterative method. Conclusion In this paper, we presented the Legendre-Picard iteration method for the numerical solution of nonlinear Fredholm integral equations of the second kind. To obtain this method, the Picard iteration scheme, the shifted Legendre polynomials and the shifted Legendre-Gauss quadrature formula play a fundamental role. The proposed method basically, is an iterative method, that we can implement easily and attain more accurate approximate solutions with higher iteration. does not require integral computations, because the integration of the shifted Legendre polynomials is calculated analytically with an iterative formula when the coefficients are updated. unlike spectral methods, does not require solving linear or nonlinear systems of algebraic equations and calculating the inverse of a matrix. in vector-matrix form, implemented significantly faster than in the original iterative form. [ABSTRACT FROM AUTHOR] |
