This paper proposes a new paradigm for solving systems of nonlinear equations through using Genetic Algorithm (GA) techniques. So, a great attention was presented to illustrate how genetic algorithms (GA) techniques can be used in finding the solution of a system described by nonlinear equations. To achieve this purpose, we apply Gauss–Legendre integration as a technique to solve the system of ...

In this paper, we consider numerical differentiation of bivariate functions when a set of noisy data is given. A mollification method based on spanned by Legendre polynomials is proposed and the mollification parameter is chosen by a discrepancy principle. The theoretical analyses show that the smoother the genuine solution, the higher the convergence rates of the numerical solution. To get a p...

This paper proposes a direct approach for solving optimal control problems. The time domain is divided into multiple subdomains, and a Lagrange interpolating polynomial using the Legendre–Gauss– Lobatto points is used to approximate the states and controls. The state equations are enforced at the Legendre–Gauss–Lobatto nodes in a nonlinear programming implementation by partial Gauss–Lobatto qua...

The Falkner-Skan equation is a nonlinear third-order boundary value problem defined on the semi-infinite interval [0,∞). This equation plays an important role to illustrate the main physical features of boundary layer phenomena. This paper presents a new collocation method for solving the Falkner-Skan equation. The proposed approach is equipped by the orthogonal Chebyshev polynomials that have ...

In the context of numerical methods for conservation laws, not only the preservation of the primary conserved quantities can be of interest, but also the balance of secondary ones such kinetic energy in case of the Euler equations of gas dynamics. In this work, we construct a kinetic energy preserving discontinuous Galerkin method on Gauss-Legendre nodes based on the framework of summation-by-p...

This proceeding is intended to be a first introduction to spectral methods. It is written around some simple problems that are solved explicitly and in details and that aim at demonstrating the power of those methods. The mathematical foundation of the spectral approximation is first introduced, based on the Gauss quadratures. The two usual basis of Legendre and Chebyshev polynomials are then p...

We generalize existing Jacobi–Gauss–Lobatto collocation methods for variable-order fractional differential equations using a singular approximation basis in terms of weighted Jacobi polynomials of the form (1 ± x)μP a,b j (x), where μ > −1. In order to derive the differentiation matrices of the variable-order fractional derivatives, we develop a three-term recurrence relation for both integrals...

In the present paper, a solution for the boundary value problem over a semi-infinite interval has been obtained by transforming the two-dimensional laminar boundary equations into a nonlinear ordinary equation using similarity variables. Moreover, a collocation method is proposed to solve the Falkner-Skan equation. This method is based on rational Legendre functions and convert the Falkner-Skan...

In this paper, An effective and simple numerical method is proposed for solving systems of integral equations using radial basis functions (RBFs). We present an algorithm based on interpolation by radial basis functions including multiquadratics (MQs), using Legendre-Gauss-Lobatto nodes and weights. Also a theorem is proved for convergence of the algorithm. Some numerical examples are presented...

Abstract. In this paper, a method is employed to approximate the solution of two-dimensional nonlinear Volterra integro-differential equations (2DNVIDEs) with supplementary conditions. First, we introduce twodimensional Legendre polynomials, then convert 2DNVIDEs to the two-dimensional linear Volterra integrodifferential equations (2DLVIDEs). Using this properties and collocation points, reduce...