Helmi Temimi

A semi-analytical iterative technique for solving nonlinear problems

We present a semi-analytical iterative method for solving nonlinear differential equations. To demonstrate the working of the method we consider some nonlinear ordinary differential equations with appropriate initial/boundary conditions. In each of the examples we demonstrate the accuracy and convergence of the method to the solution. We demonstrate clearly that the method is accurate, fast and has a high order of convergence.

A new iterative technique for solving nonlinear second order multi-point boundary value problems

Abstract We present a semi-analytical iterative method for solving nonlinear second order multi-point boundary value problems. To demonstrate the working of the method we consider a particular example of this class of problems. In this example, we demonstrate the accuracy and convergence of the method to the solution. We demonstrate clearly that the method is accurate, fast and has a reasonable order of convergence.

An iterative finite difference method for solving Bratu's problem

In this paper we propose a new iterative finite difference (IFD) scheme based on the Newton-Raphson-Kantorovich approximation method in function space to solve the classical one-dimensional Bratus problem. This new numerical method produces accurate solutions with low computational cost. The effectiveness and accuracy of the IFD method are confirmed through several numerical examples and compared to some existing numerical solvers.

A discontinuous Galerkin finite element method for solving the Troesch’s problem

Abstract In this paper we propose a new discontinuous Galerkin finite element (DG) method to solve Troesch’s problem, which is highly sensitive for large values of the parameter. This two-point boundary value problem has been heavily studied since 1960, however, only a few papers have provided a reliable solution for high sensitivity. Therefore, we developed the DG method which has proved its efficiency for many decades to be a new numerical solver. We demonstrate through computational results compared with those computed by other methods, that the discontinuous Galerkin method provides a quite efficient, accurate and reliable solution. Thus, the DG method is an attractive and competitive alternative to other numerical and semi-analytical techniques to solve highly sensitive nonlinear problems.

Time-Delay Effects on Controlled Seismically Excited Linear and Nonlinear Structures

The main purpose of this paper is to examine the influence of time delay associated with a semi-active variable viscous (SAVV) damper on the response of seismically excited linear and nonlinear structures. The maximum time delay is estimated on the basis of stability criteria, which consist of analyses of structural modal properties. Numerical computation of the critical time delay is performed by using dichotomic approach, which is based on multiple solving of the eigenvalue problem. Simulation results indicate that variable dampers can be effective in reducing the seismic response of structures, and that time-delay effects are important factors in control design of seismically excited structures. Furthermore, simulation results show degradation of performance whenever the actual delay exceeds the calculated critical time delay, which shows the accuracy and reliability of the proposed approach.

Error analysis of a discontinuous Galerkin method for systems of higher-order differential equations

In this paper, we provide an error analysis of the discontinuous Galerkin (DG) method applied to the system of first-order ordinary differential equations (ODEs) arising from the transformation of an mth-order ODE. We compare this DG method with the DG method introduced in [4], which applies DG directly to the mth-order ODE, and present the advantages and disadvantages of each approach based on certain metrics, such as computational time, L^2 norm of the approximation error, L^2 norm of the derivatives error, and maximum approximation error at the endpoints of each timestep. We generalize the two approaches by introducing a DG method applied to the system of @w-order ODEs arising from an mth-order ODE, where 1=<@w=

An approximate solution for the static beam problem and nonlinear integro-differential equations

We consider a Kirchhoff type nonlinear static beam and an integro-differential convolution type problem, and investigate the effectiveness of the Optimal Homotopy Asymptotic Method (OHAM), in solving nonlinear integro-differential equations. We compare our solutions via the OHAM, with bench mark solutions obtained via a finite element method, to show the accuracy and effectiveness of the OHAM in each of these problems. We show that our solutions are accurate and the OHAM is a stable accurate method for the problems considered.

An Iterative Numerical Method for Solving the Lane–Emden Initial and Boundary Value Problems

In this paper, we propose fast iterative methods based on the Newton–Raphson–Kantorovich approximation in function space [Bellman and Kalaba, (1965)] to solve three kinds of the Lane–Emden type problems. First, a reformulation of the problem is performed using a quasilinearization technique which leads to an iterative scheme. Such scheme consists in an ordinary differential equation that uses the approximate solution from the previous iteration to yield the unknown solution of the current iteration. At every iteration, a further discretization of the problem is achieved which provides the numerical solution with low computational cost. Numerical simulation shows the accuracy as well as the efficiency of the method.

Superconvergence of Discontinuous Galerkin Method to Nonlinear Differential Equations

In this paper, we investigate the superconvergence criteria of the discontinuous Galerkin (DG) method applied to one-dimensional nonlinear differential equations. We show numerically that the p-degree finite element (DG) solution is \(O(\Delta x^{p+2})\) superconvergent at the roots of specific combined Jacobi polynomials. Moreover, we used these results to construct efficient and asymptotically exact a posteriori error estimates.