A New Application of Gauss Quadrature Method for Solving Systems of Nonlinear Equations

In this paper, we introduce a new three-step Newton method for solving a system of nonlinear equations. This new method based on Gauss quadrature rule has sixth order of convergence (with n=3). The proposed method solves nonlinear boundary-value problems and integral equations in few iterations with good accuracy. Numerical comparison shows that the new method is remarkably effective for solving systems of nonlinear equations.

Rational Transformations for Evaluating Singular Integrals by the Gauss Quadrature Rule

In this work we introduce new rational transformations which are available for numerical evaluation of weakly singular integrals and Cauchy principal value integrals. The proposed rational transformations include parameters playing an important role in accelerating the accuracy of the Gauss quadrature rule used for the singular integrals. Results of some selected numerical examples show the efficiency of the proposed transformation method compared with some existing transformation methods.

Numerical solution for system of nonlinear Fredholm–Hammerstein integral equations based on hybrid Bernstein Block-Pulse functions with the Gauss quadrature rule

This work provides an efficient method for solving system of nonlinear Fredholm–Hammerstein integral equations. The propose method approximates the unknown function with hybrid Bernstein Block-Pulse functions (HBPF). In order to do this, we apply these functions and then using the collocation method for the numerical solutions of this system. Furthermore, we implement this method in conjunction with the quadrature rule for converting the problem to a system of algebraic equations that can be solved easily by applying mathematical programming techniques. The merits of this method lie in the fact that, on the one hand, the problem will be reduced to a system of algebraic equations and on the other hand, the efficiency and accuracy of the hybrid Bernstein Block-Pulse functions with the Gauss quadrature rule (HBPF-GQR) for solving this system are remarkable. The existence and uniqueness of solution have been presented. Moreover, the convergence of this algorithm will be shown by preparing some theorems. Several numerical examples are presented to show the superiority and efficiency of current method in comparison with some other well-known methods.