M.S. El-Azab

A numerical algorithm for the solution of telegraph equations

Abstract In this paper, we present a new competitive numerical scheme to solve nonlinear telegraph equations. The method is based on Rothe’s approximation in time discretization and on the Wavelet–Galerkin in the spatial discretization. The approximate solutions converge in the space C ( 0 , T ) ; L 2 ( Ω ) ∩ L 2 ( 0 , T ) ; W 0 1 , 2 ( Ω ) to the variational solution. A full error analysis is performed and a numerical experiment is given to illustrate the good convergence behavior of the approximate solution.

An approximation scheme for a nonlinear diffusion Fisher's equation

Abstract A nonlinear diffusion Fisher’s equation is solved by fully different numerical scheme. The equation is discretized in time by Rothe’s method and in space by wavelet-Galerkin method. We prove the convergence of the approximate solution to the solution of the continuous problem. A full error analysis is performed. A numerical experiment is presented.

A computer aided diagnostic system for detecting diabetic retinopathy in optical coherence tomography images

Purpose: Detection (diagnosis) of diabetic retinopathy (DR) in optical coherence tomography (OCT) images for patients with type 2 diabetes, but almost clinically normal retina appearances. Methods: The proposed computer‐aided diagnostic (CAD) system detects the DR in three steps: (a) localizing and segmenting 12 distinct retinal layers on the OCT image; (b) deriving features of the segmented layers, and (c) learning most discriminative features and classifying each subject as normal or diabetic. To localise and segment the retinal layers, signals (intensities) of the OCT image are described with a joint Markov‐Gibbs random field (MGRF) model of intensities and shape descriptors. Each segmented layer is characterized with cumulative probability distribution functions (CDF) of its locally extracted features, such as reflectivity, curvature, and thickness. A multistage deep fusion classification network (DFCN) with a stack of non‐negativity‐constrained autoencoders (NCAE) is trained to select the most discriminative retinal layers’ features and use their CDFs for detecting the DR. A training atlas was built using the OCT scans for 12 normal subjects and their maps of layers hand‐drawn by retina experts. Results: Preliminary experiments on 52 clinical OCT scans (26 normal and 26 with early‐stage DR, balanced between 40–79 yr old males and females; 40 training and 12 test subjects) gave the DR detection accuracy, sensitivity, and specificity of 92%; 83%, and 100%, respectively. The 100% accuracy, sensitivity, and specificity have been obtained in the leave‐one‐out cross‐validation test for all the 52 subjects. Conclusion: Both the quantitative and visual assessments confirmed the high accuracy of the proposed computer‐assisted diagnostic system for early DR detection using the OCT retinal images.

Legendre-Galerkin method for the linear Fredholm integro-differential equations

This paper presents a study of the performance of the Galerkin method using Legendre basis functions for solving linear Fredholm integro-differential problems. Convergence and error estimation of the method were discussed. The method is then tested on several examples. Numerical results are included and comparisons with other methods are made to confirm the efficiency and accuracy of the method.

A 2N order compact finite difference method for solving the generalized regularized long wave (GRLW) equation

The generalized regularized long wave (GRLW) equation is solved by fully different numerical scheme. The equation is discretized in space by 2N order compact finite difference method and in time by a backward finite difference method. At the inner and the boundary nodes, the first and the second order derivatives with 2N order of accuracy are obtained. To determine the conservation properties of the GRLW equation three invariants of motion are evaluated. The single solitary wave and the interaction of two and three solitary waves are presented to validate the efficiency and the accuracy of the proposed scheme.

Solution of nonlinear transport–diffusion problems by linearization

We construct and analyze a fully discretization scheme for approximating the solution of a class of nonlinear degenerate parabolic problems with a nonlinear Neumann boundary conditions. The method is based on Rothe type discretization in time and on wavelet-Galerkin discretization in space. A proof of convergence of the approximate solution is given and error estimates are proved.

2N order compact finite difference scheme with collocation method for solving the generalized Burger's-Huxley and Burger's-Fisher equations

The generalized Burgers-Huxley and Burgers-Fisher equations are solved by fully different numerical scheme. The equations are discretized in time by a new linear approximation scheme and in space by 2N order compact finite difference scheme, after that a collocation method is applied. Also, the two-dimensional unsteady Burgers equation is described by our proposed scheme. Numerical experiments and numerical comparisons are presented to show the efficiency and the accuracy of the proposed scheme.

Chebyshev-Chebyshev spectral collocation method for solving the generalized regularized long wave (GRLW) equation

In this research, a Chebyshev-Chebyshev spectral collocation method based on Kronecker and Hadamard products is proposed for solving the generalized regularized long wave (GRLW) equation. Chebyshev-Gauss-Lobatto collocation points are used in both time and space directions. Three invariants of motion: mass, momentum and energy are evaluated to determine the conservation properties of the GRLW equation. The single solitary wave and the interaction of two and three solitary waves are presented to validate the efficiency and the accuracy of the proposed scheme.

Level sets-based image segmentation approach using statistical shape priors

Abstract A robust 3-D segmentation technique incorporated with the level sets concept and based on both shape and intensity constraints is introduced. A partial differential equation (PDE) is derived to describe the evolution of the level set contours. This PDE does not contain weighting parameters that need to be tuned, which overcomes the drawbacks of other PDE approaches. The shape information is collected from a set of co-aligned manually segmented contours of the training data. A promising statistical approach is used to get the distribution of the intensity gray values. The introduced statistical approach is built by modeling the empirical PDF (normalized histogram of occurrences) for the intensity level distribution with a linear combination of Gaussians (LCG) incorporating both negative and positive components. An Expectation-Maximization (EM) algorithm is modified to deal with the LCGs, and we also proposed an EM-based sequential technique to acquire a close initial LCG approximation for the modified EM algorithm to start with. The PDF of the intensity levels is incorporated in the speed function of the moving level set to specify the evolution direction. Experimental results show how accurately the approach is in segmenting various types of 2-D and 3-D datasets comprising medical images.

Finite element solution of nonlinear diffusion problems

Abstract In this paper we describe the Rothe-finite element numerical scheme to find an approximate solution of a nonlinear diffusion problem modeled as a parabolic partial differential equation of even order. This scheme is based on the Rothe’s approximation in time and on the finite element method (FEM) approximation in the spatial discretization. A proof of convergence of the approximate solution is given and error estimates are shown.