The boundary elements method for magneto-hydrodynamic (MHD) channel flows at high Hartmann numbers

In this article the constant and the continuous linear boundary elements methods (BEMs) are given to obtain the numerical solution of the coupled equations in velocity and induced magnetic field for the steady magneto-hydrodynamic (MHD) flow through a pipe of rectangular and circular sections having arbitrary conducting walls. In recent decades, the MHD problem has been solved using some variations of BEM for some special boundary conditions at moderate Hartmann numbers up to 300. In this paper we develop this technique for a general boundary condition (arbitrary wall conductivity) at Hartmann numbers up to 10⁵${10}^5$ by applying some new ideas. Numerical examples show the behavior of velocity and induced magnetic field across the sections. Results are also compared with the exact values and the results of some other numerical methods.     

Solving a laminar boundary layer equation with the rational Gegenbauer functions

In this paper, a collocation method using a new weighted orthogonal system on the half-line, namely the rational Gegenbauer functions, is introduced to solve numerically the third-order nonlinear differential equation, af^?+ff^″=0${\mathit{af}}^{?}+{\mathit{ff}}^{″}=0$, where a   is a constant parameter. This method solves the problems on semi-infinite domain without truncating it to a finite domain and transforming the domain of the problems to a finite domain. For a=2$a=2$, the equation is the well-known Blasius equation, which is a laminar viscous flow over a semi-infinite flat plate. We solve this equation by considering 1?a?2$1?a?2$ and compare the new results with the established results to show the efficiency and accuracy of the new method.     

Nonlinear Kalman Filtering for acoustic emission source localization in anisotropic panels

Nonlinear Kalman Filtering is an established field in applied probability and control systems, which plays an important role in many practical applications from target tracking to weather and climate prediction. However, its application for acoustic emission (AE) source localization has been very limited. In this paper, two well-known nonlinear Kalman Filtering algorithms are presented to estimate the location of AE sources in anisotropic panels: the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF). These algorithms are applied to two cases: velocity profile known (CASE I) and velocity profile unknown (CASE II). The algorithms are compared with a more traditional nonlinear least squares method. Experimental tests are carried out on a carbon-fiber reinforced polymer (CFRP) composite panel instrumented with a sparse array of piezoelectric transducers to validate the proposed approaches. AE sources are simulated using an instrumented miniature impulse hammer. In order to evaluate the performance of the algorithms, two metrics are used: (1) accuracy of the AE source localization and (2) computational cost. Furthermore, it is shown that both EKF and UKF can provide a confidence interval of the estimated AE source location and can account for uncertainty in time of flight measurements.     

The Sinc-collocation and Sinc-Galerkin methods for solving the two-dimensional Schrödinger equation with nonhomogeneous boundary conditions

In the last three decades, Sinc numerical methods have been extensively used for solving differential equations, not only because of their exponential convergence rate, but also due to their desirable behavior toward problems with singularities. This paper illustrates the application of Sinc-collocation and Sinc-Galerkin methods to the approximate solution of the two-dimensional time dependent Schrödinger equation with nonhomogeneous boundary conditions. Some numerical examples are presented and the proposed methods are compared with each other.     

Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method

In this research, we propose a numerical scheme to solve the system of second-order boundary value problems. In this way, we use the Local Radial Basis Function Differential Quadrature (LRBFDQ) method for approximating the derivative. The LRBFDQ method approximates the derivatives by Radial Basis Functions (RBFs) interpolation using a small set of nodes in the support domain of any node. So the new scheme needs much less computational work than the globally supported RBFs collocation method. We use two techniques presented by Bayona et al. (2011, 2012) [29], [30] to determine the optimal shape parameter. Some examples are presented to demonstrate the accuracy and easy implementation of the new technique. The results of numerical experiments are compared with the analytical solution, finite difference (FD) method and some published methods to confirm the accuracy and efficiency of the new scheme presented in this paper.     

Fourth-order compact solution of the nonlinear Klein-Gordon equation

In this work we propose a fourth-order compact method for solving the one-dimensional nonlinear Klein-Gordon equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivative and a fourth-order A-stable diagonally-implicit Runge-Kutta-Nyström (DIRKN) method for the time integration of the resulting nonlinear second-order system of ordinary differential equations. The proposed method has fourth order accuracy in both space and time variables and is unconditionally stable. Numerical results obtained from solving several problems possessing periodic, kinks, single and double-soliton waves show that the combination of a compact finite difference approximation of fourth order and a fourth-order A-stable DIRKN method gives an efficient algorithm for solving these problems.     

A meshless method for numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions

The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional hyperbolic equation that combines classical and integral boundary conditions using collocation points and approximating the solution using radial basis functions (RBFs). The results of numerical experiments are presented, and are compared with analytical solution and finite difference method to confirm the validity and applicability of the presented scheme.     

Application of He’s homotopy perturbation method for non-linear system of second-order boundary value problems

A homotopy perturbation method (HPM) is proposed to solve non-linear systems of second-order boundary value problems. HPM yields solutions in convergent series forms with easily computable terms, and in some cases, yields exact solutions in one iteration. Moreover, this technique does not require any discretization, linearization or small perturbations and therefore reduces the numerical computations a lot. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. The results reveal that the method is very effective, straightforward and simple.     

The use of compact boundary value method for the solution of two-dimensional Schrödinger equation

In this paper, a high-order and accurate method is proposed for solving the unsteady two-dimensional Schrödinger equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives and a boundary value method of fourth-order for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Moreover this method is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature. These results show that the combination of a compact finite difference approximation of fourth-order and a fourth-order boundary value method gives an efficient algorithm for solving the two dimensional Schrödinger equation.