M. B. M. Elgindi

Existence and uniqueness of solutions of a class of two-point singular nonlinear boundary value problems

Abstract El-Gebeily, M.A., A. Boumenir and M.B.M. Elgindi, Existence and uniqueness of solutions of a class of two-point singular nonlinear boundary value problems, Journal of Computational and Applied Mathematics 46 (1993) 345–355. This paper is concerned with the existence and uniqueness of solution of a class of two-point singular nonlinear boundary value problems. It is shown that the problem has a unique solution only for certain boundary conditions under the assumption that the range of ∂ f /∂ y has empty intersection with the closure of the spectrum of the singular differential operator, where f denotes the nonlinearity.

Homotopy method for the eigenvalues of symmetric tridiagonal matrices

We will present the homotopy method for finding eigenvalues of symmetric, tridiagonal matrices. This method finds eigenvalues separately, which can be a large advantage on systems with parallel processors. We will introduce the method and establish some bounds that justify the use of Newtons method in constructing the homotopy curves.

Deformation of a thin-walled cylindrical tube submerged in a liquid

Abstract A long, thin, horizontal elastic cylindrical tube either fully or partially submerged in a liquid is considered. The tube supports a centered line load along its bottom and the load-carrying capacity of the tube is analysed. A nonlinear boundary value problem that takes into account the variable hydrostatic pressure is formulated. Perturbation solutions are obtained for the cases of small pressure gradients, especially in the neighborhood of the critical buckling pressure. Numerical solutions based on Newtons iteration and shooting methods are also obtained for general pressure gradients and for both the fully and partially submerged cases. The results show that, given a pressure gradient, there is an upper limit for the maximum load of the partially submerged tube. This maximum load decreases with depth when the shell is fully submerged and thus the corresponding equilibrium solution is unstable.

EXISTENCE OF EQUILIBRIUM STATES OF HOLLOW ELASTIC CYLINDERS SUBMERGED IN A FLUID

This paper is concerned with the existence of equilibrium states of thin-walled elastic, cylindrical shell fully or partially submerged in a fluid. This problem obviously serves as a model for many problems with engineering importance. Previous studies on the deformation of the shell have assumed that the pressure due to the fluid is uniform. This paper takes into consideration the non-uniformity of the pressure by taking into account the effect of gravity. The presence of a pressure gradient brings additional parameters to the problem which in turn lead to the consideration of several boundary value problems.

An iterative finite difference scheme for buckling of graphene beam subject to axial compressive load

In this paper buckling loads and modes of an Euler beam made of graphene are considered. An eigenvalue problem is formulated with an additional parameter representing materials elastoplasticity. Analytical approximate solution using the perturbation method is presented. This approximate solution is used as a basis for an iterative finite difference scheme to develop a more accurate solution. Deflection at points along the beam are calculated using this iterative finite difference scheme. Convergence with good accuracy is achieved.

ON THE APPLICATION OF NEWTON'S AND CHORD METHODS TO BIFURCATION PROBLEMS

Thispaper isconcernedwith the applicationsofNewtons andchordmethods in thecomputations of the bifurcation solutions in a neighborhood of a simple bifurcation point for prescribed values of the