Fractional and Multivariable Calculus

1. Front Matter

   Pages i-xiii

   PDF

2. Essentials of Fractional Calculus

   • A. M. Mathai, H. J. Haubold

   Pages 1-37

3. Multivariable Calculus

   • A. M. Mathai, H. J. Haubold

   Pages 39-88

4. Deterministic Models and Optimization

   • A. M. Mathai, H. J. Haubold

   Pages 89-106

5. Non-deterministic Models and Optimization

   • A. M. Mathai, H. J. Haubold

   Pages 107-181

6. Optimal Regression Designs

   • A. M. Mathai, H. J. Haubold

   Pages 183-208

7. Pathway Models

   • A. M. Mathai, H. J. Haubold

   Pages 209-229

8. Back Matter

   Pages 231-234

   PDF

This textbook presents a rigorous approach to multivariable calculus in the context of model building and optimization problems. This comprehensive overview is based on lectures given at five SERC Schools from 2008 to 2012 and covers a broad range of topics that will enable readers to understand and create deterministic and nondeterministic models.  Researchers, advanced undergraduate, and graduate students in mathematics, statistics, physics, engineering, and biological sciences will find this book to be a valuable resource for finding appropriate models to describe real-life situations.

The first chapter begins with an introduction to fractional calculus moving on to discuss fractional integrals, fractional derivatives, fractional differential equations and their solutions.  Multivariable calculus is covered in the second chapter and introduces the fundamentals of multivariable calculus (multivariable functions, limits and continuity, differentiability, directionalderivatives and expansions of multivariable functions). Illustrative examples, input-output process, optimal recovery of functions and approximations are given; each section lists an ample number of exercises to heighten understanding of the material. Chapter three  discusses deterministic/mathematical and optimization models evolving from differential equations, difference equations, algebraic models, power function models, input-output models and pathway models. Fractional integral and derivative models are examined.  Chapter four covers non-deterministic/stochastic models. The random walk model, branching process model, birth and death process model, time series models, and regression type models are examined. The fifth chapter covers optimal design. General linear models from a statistical point of view are introduced; the Gauss–Markov theorem, quadratic forms, and generalized inverses of matrices are covered. Pathway, symmetric, and asymmetric models are covered in chapter six, the concepts are illustrated with graphs.

• Mittag-Leer and Wright's functions
• fractional order dierential and integral equations
• fractional calculus
• multivariable calculus
• deterministic models
• linear and non-linear analysis
• non-deterministic models
• regression type prediction models
• prediction models
• matrix-variate calculus
• Canonical Correlation Analysis
• Analysis of Variance

“This book presents the readers with a basic overview of the rigorous approach to the study of fractional and multivariate calculus in the context of model building and optimization problems. This book covers a broad range of topics that will enable readers to fully understand and create deterministic and nondeterministic models. … book is well presented and would be very useful for researchers and graduate students working in several areas including mathematics, physics, chemistry, engineering, finance and even social sciences.” (James Adedayo Oguntuase, zbMATH 1376.26001, 2018)

• Centre for Mathematical and Statistical Sciences, Peechi Campus, India

  A.M. Mathai

• Vienna International Centre, Office for Outer Space Affairs, United Nations, Vienna, Austria

  H.J. Haubold

A.M. Mathai is Emeritus Professor of Mathematics and Statistics, McGill University, Canada, and Director of the Centre for Mathematical and Statistical Sciences, India. He has published over 300 research papers and 25 books on topics in mathematics, statistics, physics, astrophysics, chemistry, and biology. He is a Fellow of the Institute of Mathematical Statistics, National Academy of Sciences of India, served as President of the Mathematical Society of India, and a Member of the International Statistical Institute. 

H.J. Haubold is Professor of Theoretical Astrophysics. He has published over 200 research papers and 10 books in physics, astrophysics, and the development of basic space science worldwide. The United Nations Basic Space Science Initiative (UN BSSI) for the worldwide development of astronomy, physics, and mathematics was implemented in 1991 through the joint work of H.J Haubold and A.M. Mathai.

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