A. Mohsen

On the Galerkin and collocation methods for two-point boundary value problems using sinc bases

This paper presents a study of the performance of the collocation and Galerkin methods using sinc basis functions for solving linear and nonlinear second-order two-point boundary value problems. The two methods have the linear systems solved by the Q-R method and have the nonlinear systems solved by Newtons method. This study shows that the collocation method performs better than the Galerkin method for the cases considered.

A simple solution of the Bratu problem

A brief survey of the properties and different treatments of the one-dimensional (1D) and (2D) Bratu problems is presented. Different iterative treatments of the resulting nonlinear system of equations are discussed. The finite-difference treatment of the problem is considered. Nonstandard finite-difference methods with a simple sinusoidal starting function having an appropriate amplitude are recommended. Bounds on the amplitude for yielding both lower and upper solutions are given.

On the numerical solution of linear and nonlinear volterra integral and integro-differential equations

Abstract Sinc bases are developed to approximate the solutions of linear and nonlinear Volterra integral and integro-differential equations. Properties of these sinc bases and some operational matrices are first presented. These properties are then used to reduce the integral and integro-differential equations to systems of linear or nonlinear algebraic equations. Numerical examples illustrate the pertinent features of the method and its applicability to a large variety of problems. The examples include convolution type, singular as well as singularly-perturbed problems.

New smoother to enhance multigrid-based methods for Bratu problem

Abstract In this paper, we present more investigations of the numerical solution of the 2D Bratu equation to obtain the second solution on the upper branch by Multigrid. Classical smoothers such as Gauss–Seidel and weighted Jacobi have proven ineffective for obtaining the second solution due to the loss of diagonal dominance and the presence of indefinite Jacobian system at some parameter values. In this paper, we modify the Multigrid algorithms by adding and combining some Krylov methods as smoothers to enhance the multigrid efficiency. Though the idea is not new but we could get new enhanced results compared to that presented by Hackbusch [W. Hackbusch, Comparison of different multi-grid variants for nonlinear equations, ZAMM Z. Angew. Math. Mech. 72 (1992) 148–151] and Washio and Oosterlee [T. Washio, C.W. Oosterlee, Krylov subspace acceleration for nonlinear multigrid schemes, Electron. Trans. Numer. Anal. 6 (1997) 271–290].

The corrected induced surface current for arbitrary conducting objects at resonance frequencies

It is well known that using the method of moments in conjunction with either the electric or magnetic field surface integral equations (EFIE or MFIE) produces inaccurate surface currents on conducting bodies at resonance frequencies. A new technique (based on singular value decomposition) is developed to correct the computed current by adding a correction factor term. This term is seen to be the resonant mode current, obtained by employing the power method in the moment method matrix, multiplied by an unknown complex factor. Applying the condition of vanishing field inside the conducting object results in obtaining the unknown complex factor. Therefore, this technique is hereafter referred to as correction factor technique (CFT). When the computed surface current on a conducting sphere, proposed technique, is compared with the exact one, the numerical results show excellent agreement. >

Sinc-Galerkin method for solving biharmonic problems

There are many techniques available to numerically solve the biharmonic equation. In this paper we show that the sinc-Galerkin method is a very effective tool in numerically solving this equation. Hermite interpolation is used to treat the nonhomogeneous boundary conditions. Our method is tested on examples and comparisons with other methods are made. It is shown that the sinc-Galerkin method yields good results even when singularities occur at the boundaries.

Letter to the editor: On the integral solution of the one-dimensional Bratu problem

A brief survey of the properties and different treatments of the one-dimensional (1D) planar Bratu problem is presented. The treatment using Greens function is considered. Different iterative treatments of the resulting nonlinear system of equations are discussed. Starting the solution with simple sinusoidal function having appropriate amplitude is recommended. Then the resulting system is solved using a nonlinear conjugate-gradient (NL-CG) method. The importance of the proper choice of the starting function as well as the stopping criterion of the iterative solution are stressed.

Two Dimensional Long Transmission Line-Frequency Domain (Ltl-Fd) Treatment of Waveguides

In this paper, two dimensional transmission line circuits are used to determine cutoff wavenumbers of homogeneous arbitrarily shaped cylindrical waveguides. The derived circuits depend mainly on long transmission line (LTL) circuits. The resonance frequencies of these circuits are the required cutoff frequencies of the guide. The problem is reduced to a simple eigenvalue problem, which is solved to get the required solution. Numerical results are given to compare results with available analytical solutions and results given by other reported methods.

Magnetic field integral equation for electromagnetic scattering by conducting bodies of revolution in layered media - Abstract

This work presents a new formulation for conducting bodies of revolution (BOR) in layered media. It is derived using magnetic field integral equation (MFIE). The discrete complex image method is used to transform the Sommerfeld integrals into closed form solutions. For large closed objects, the use of the magnetic field formulation is shown to be more advantageous than the electric field integral equation (EFIE) concerning the convergence of the solution. The obtained formulation is applied to the contiguous half-spaces case. Results of both the magnetic and electric field integral equation formulations are presented.

The source simulation technique for exterior problems in acoustics

A study of the source simulation technique is presented for the solution of the exterior Dirichlet and Neumann problems for the Helmholtz equation. The solution is in terms of a single-layer potential extended over an internal circle. The advantages of the method in treating the nonuniqueness problem and in computing the scattered and the far fields are illustrated.