Abul Hasan Siddiqi

Variational Inequalities and Applications

In this chapter, we discuss mathematical models of real-world problems known as variational inequalities introduced systematically by J.L. Lions and S. Stampachia in early seventies. Modeling, discretization, algorithms, and visualization of solutions along with updated references are presented.

Certain remarks on a class of evolution quasi-variational inequalities

We prove two existence theorems, one for evolution quasi-variational inequal- ities and the other for a time-dependent quasi-variational inequality modeling the quasi- static problem of elastoplasticity with combined kinetic-isotropic hardening.

Finite Element and Boundary Element Methods

In this chapter, finite element and boundary element methods are introduced. Functional analysis plays important role to reduce the problem in discrete form amenable to computer analysis. The finite element method is a general technique to construct finite-dimensional spaces of a Hilbert space of some classes of functions such as Sobolev spaces of different orders and their subspaces in order to apply the Ritz and Galerkin methods to a variational problem. The boundary element method comprises transformation of the partial differential equation describing the behavior of an unknown inside and the boundary of the domain into an integral equation relating to any boundary values, and their finding out numerical solution.

Wavelet Method for Partial Differential Equations and Image Processing

In this chapter, applications of wavelet theory to partial differential equations and image processing are discussed.

Wavelets in medical imaging

The aim of this study is to provide emerging applications of wavelet methods to medical signals and images, such as electrocardiogram, electroencephalogram, functional magnetic resonance imaging, computer tomography, X-ray and mammography. Interpretation of these signals and images are quite important. Nowadays wavelet methods have a significant impact on the science of medical imaging and the diagnosis of disease and screening protocols. Based on our initial investigations, future directions include neurosurgical planning and improved assessment of risk for individual patients, improved assessment and strategies for the treatment of chronic pain, improved seizure localization, and improved understanding of the physiology of neurological disorders. We look ahead to these and other emerging applications as the benefits of this technology become incorporated into current and future patient care. In this chapter by applying Fourier transform and wavelet transform, analysis and denoising of one of the important biomedical signals like EEG is carried out. The presence of rhythm, template matching, and correlation is discussed by various method. Energy of EEG signal is used to detect seizure in an epileptic patient. We have also performed denoising of EEG signals by SWT.

Gini Based Learning for the Classification of Alzheimer’s Disease and Features Identification with Automatic RGB Segmentation Algorithm

Magnetic Resonance Image segmentation is the process of partitioning brain data, which is regarded as a highly challenging task for medical applications, particularly in Alzheimer’s Disease (AD). In this study, we have developed a new automatic segmentation algorithm which can be seen as a novel decision making technique that can help diagnose decision rules studying magnetic resonance images of the brain. The proposed work consist of a total of five stages: (i) the preprocessing stage that involves the use of dilation and erosion methods via gray-scale MRI for brain extraction (ii) the application of multi-level thresholding using Otsu’s method with a threshold value of (\({\mu _{i}}> 15\) pixels) to determine the RGB color segment values (iii) the calculation of area detection (RGB segment scores) by applying our newly proposed automatic RGB Color Segment Score Algorithm to the predetermined RGB color segments (iv) creating the AD_dataset using the pixels of the lesion areas calculated via MR imaging (v) the post-processing stage that involves the application of Classification and Regression Tree (CART) algorithm to the AD_dataset. This study aims at contributing to the literature with the decision rules derived from the application of CART algorithm to the calculated RGB segment scores using our newly proposed automatic RGB Color Segment Score Algorithm in terms of the successful classification of AD.

Spectral Theory with Applications

This chapter deals with the introduction of the resolvent and the spectrum set of a bounded linear operator as well as introduction of inverse problems and their regularization.

Banach Contraction Fixed Point Theorem

The main goal of this chapter is to introduce notion of distance between two points in an abstract set. This concept was studied by M. Frechet and it is known as metric. Existence of a fixed point of a mapping on a complete metric into itself was proved by S. Banach around 1920. Application of this theorem for existence of solution of matrix, differential and integral equations is‘ presented in this chapter.

Operator Equations and Variational Methods

In this chapter, existence of solution of some well-known partial differential equations with boundary conditions is studied.

Differential and Integral Calculus in Banach Spaces

In this chapter, differentiation and integration of operators defined on a Banach space into another Banach space are introduced. Basic concepts of distribution theory and Sobolev spaces are discussed, both concepts play very significant role in the theory of partial differential equations. A lucid presentation of these two topics is given.