Siddiqi Abul Hasan - Modern Engineering Mathematics

1. A.2 Intervals, Absolute Value
   and Inequalities

Modern Engineering Mathematics

Modern Engineering Mathematics

Abul Hasan Siddiqi

Mohamed Al-Lawati

Messaoud Boulbrachene

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Prof. A. H. Siddiqi would like to dedicate the book to his wife, Prof. Azra H.Siddiqi

Prof. M. Al-Lawati dedicates to his spouse.

Prof. M. Messaoud dedicates to his family.

Contents

A.2 Intervals, Absolute Value
and Inequalities

This book entitled Modern Engineering Mathematics is a compendium of fundamental mathematical concepts, methods, models and their wide range of applications in diverse fields of engineering. It comprises essentially a contemporary coverage of those areas of mathematics which provide foundation to electronic, electrical, communication, petroleum, chemical, civil, mechanical, biomedical, software and financial engineering. The rapid growth of technology and engineering science based on mathematical concepts is driving force for enlarging the domain of modern engineering mathematics.

The book gives a fairly extensive treatment to some of the recent developments in mathematics which have found very significant applications to engineering problems. Rich illustrative engineering examples and the integration of MATLAB and MATHEMATICA further support readers of the book. Essential topics of mathematics required by all types of engineers are covered clearly and concisely and fundamental ideas are highlighted and developed using illustrative examples. Extensive exercises and self-assessment questions create confidence among students to apply mathematics to engineering problems.

Contemporary topics used by engineers to solve their problems are included along with well thought applications appropriate for students of undergraduate engineering. Unlike most engineering mathematics books, the presentation, solved examples and unsolved examples are tailored (designed) particularly for engineering students. A special feature of this book is the assumption of bare minimum knowledge of pre-engineering mathematics along with evidence of theoretic results not required by engineers. Theoretic results are discussed in Appendices. The following special features are the hallmarks of the book which make it distinct from the existing books on the theme.

Some new areas relevant to exploration and production of hydrocarbons (petroleum engineering), biomedical engineering (biomedical images, particularly, CT scans, MRI, EEG), financial engineering (Black-Sholes model), biometric for identification of persons, brain-machine interface, software engineering and information technology.

Foundations and updated special mathematical techniques most appropriate to students of electrical, electronic, communication, computer, software, petroleum, chemical and financial engineering.

Lucid presentation of topics like wavelet transforms, inverse problems, Radon transform, fractals, compression sensing and besides well non-classical topic and their applications is an attractive feature. Another interesting feature is a wealth of examples to provide clues for tackling engineering problems.

Visualization of ODEs (ordinary differential equations) and PDEs (partial differential equations) by MATLAB and MATHEMATICA are presented. We also cover topics like matrices, ODEs, PDEs, and complex analyses along with their applications. Suggestions for further reading are given at the ends of each chapter. Such lists are useful for those who intend to pursue specific topics at more advanced levels.

The book comprises eleven chapters and five appendices. Each chapter is divided into sections and subsections. begins with motivation for studying differential equations by future engineers. Applications of MATLAB and MATHEMATICA are also presented in this chapter along with modeling of physical phenomena such as population dynamics, radioactive decay, Newtons law of cooling, spread of disease, series circuits, falling bodies, draining a tank, carbon dating and spring systems by differential equations. Classical methods of solutions of these models, namely underdetermined coefficients, variation of parameters and Cauchy-Euler methods are presented. The Laplace transform and its applications to solve linear differential equations as well as series solutions of two well known differential equations: Legendre and Bessel are discussed.

deals with vector calculus providing basic properties including three fundamental theorems: Green theorem, divergence theorem of Gauss, Stokes theorem and application of these results to engineering problems.

Concepts of orthonormal systems along with several concrete examples, Fourier expansion, Fourier transform, Fourier integral and fast Fourier transform are explained in .

. Several case studies from diverse fields are presented.

Basic results of complex analysis essential for understanding and solving engineering problems are discussed in is devoted to wavelets a topic introduced in the early 1980 s and quite popular among engineers. It is a refinement of Fourier series. Introduction of this field is presented along with various applications to engineering problems.