Hardback
ISBN 9781032491370
384 Pages 5 B/W Illustrations
June 27, 2023 by Chapman & Hall
Multiplicative Differential Equations: Volume I is the first part of a comprehensive approach to the subject. It continues a series of books written by the authors on multiplicative, geometric approaches to key mathematical topics. This volume begins with a basic introduction to multiplicative differential equations and then moves on to first and second order equations, as well as the question of existence and unique of solutions. Each chapter ends with a section of practical problems. The book is accessible to graduate students and researchers in mathematics, physics, engineering and biology.
Preface
Introduction
Definition of MDE
Order of MDE
Solution of MDE
Classification of MDE
Basic Problems for MDE
Advanced Practical Problems
Elementary First Order MDEs
Separable First Order MDEs
Multiplicative Homogeneous Functions
Multiplicative Homogeneous MDE
Exact Multiplicative Differential Equations
Multiplicative Integrating Factor
Advanced Practical Problems
First Order Multiplicative Linear Differential Equations
Definition. General Solutions
The Multiplicative Bernoulli Equation
The Multiplicative Riccati Equation
Applications
Multiplicative Initial Value Problems
Some Multiplicative Nonlinear Differential Equations
Advanced Practical Problems
4. Second Order Linear MDEs
General Properties
Multiplicative Linear Dependence
The Multiplicative Abel Theorem
A Particular Case
The Multiplicative Constant Case
The Method of Variation of Parameters
The Multiplicative Cauchy-Euler Equation
Advanced Practical Problems
5. Existence and Uniqueness of Solutions
Introduction
The Multiplicative Gronwall Type Integral Inequalities
Picard’s Method of Successive Approximations and Existence Theorems
Uniqueness
Continuous Dependence on Initial Data
Advanced Practical Problems
Bibliography
Index
Hardback
ISBN 9781032495842
376 Pages
Multiplicative Differential Equations: Volume 2 is the second part of a comprehensive approach to the subject. It continues a series of books written by the authors on multiplicative, geometric approaches to key mathematical topics. This volume is devoted to the theory of multiplicative differential systems. The asymptotic behavior of the solutions of such systems is studied. Stability theory for multiplicative linear and nonlinear systems is introduced and boundary value problems for second order multiplicative linear and nonlinear equations are explored. The authors also present first order multiplicative partial differential equations. Each chapter ends with a section of practical problems. The book is accessible to graduate students and researchers in mathematics, physics, engineering and biology.
Preface
1 Systems Multiplicative Differential Equations
1.1 Systems of Multiplicative Linear Differential Equations
1.2 nth Order MDE
1.3 Multiplicative Homogeneous Systems
1.4 Multiplicative Fundamental Matrix Solutions
1.5 Multiplicative Adjoint Systems
1.6 The Method of Variation of Constants
1.7 Systems with Multiplicative Constant Coefficients
1.8 Advanced Practical Exercises
2 Qualitative Analysis of Multiplicative Differential Systems
2.1 Periodic Multiplicative Linear Systems
2.2 Asymptotic Behaviour of Solutions I
2.3 Asymptotic Behaviour of Solutions II
2.4 Asymptotic Behaviour of Solutions III
2.5 Asymptotic Behaviour of Solutions IV
2.6 Asymptotic Behaviour of Solutions V
2.7 Asymptotic Behaviour of Solutions VI
2.8 Advanced Practical Problems
3 Stability Theory
3.1 Definition. Examples
3.2 Criteria for Stability
3.3 Uniform Stability
3.4 Stability of Quasi-Multiplicative Linear Systems
3.5 Two-Dimensional Autonomous Systems
3.6 Advanced Practical Problems
4 Multiplicative Linear Boundary Value Problems
4.1 Introduction
4.2 Existence of Solution
4.3 Multiplicative Green Functions
4.4 Advanced Practical Problems
5 Multiplicative Nonlinear MDEs
5.1 Multiplicative Lyapunov Direct Method I
5.2 Multiplicative Lyapunov Direct Method II
5.3 Multiplicative Nonlinear BVPs
5.4 Advanced Practical Problems
6 First Order MPDE
6.1 Classification
6.2 Solvability
6.3 The Cauchy Problem
6.4 The Multiplicative Pfaff Equation
6.5 Some Special Systems
6.6 Advanced Practical Problems
Bibliography
Index
Copyright 2023
Hardback
ISBN 9781032526898
330 Pages
This book is a compilation of all basic topics on functions of Several Variables and is primarily meant for undergraduate and post graduate students.
Topics covered are:
Limits, continuities and differentiabilities of functions of several variables.
Properties of Implicit functions and Jacobians.
Extreme values of multivariate functions.
Various types of integrals in planes and surfaces and their related theorems including Dirichlet and Liouville’s extension to Dirichlet.
Print edition not for sale in South Asia (India, Sri Lanka, Nepal, Bangladesh, Pakistan or Bhutan)
Preface
Preliminaries
1. Functions of Several Variables: Limits & Continuity
2. Functions of Several Variables: Differentiation ? I
3. Functions of Several Variables: Differentiation ? II
4. Jacobians, Functional Dependence and Implicit Functions
5. Extrema of Functions of Several Variables
6. Multiple Integrals
7. Line, Surface and Volume Integrals
8. Dirichlet’s Theorem and Liouville’s Extension
Bibliography
Index
Copyright 2023
Hardback
ISBN 9781032435008
322 Pages 80 B/W Illustrations
June 29, 2023 by CRC Press
Special functions play a very important role in solving various families of ordinary and partial differential equations as well as their fractional-order analogs, which model real-life situations. Owing to the non-local nature and memory effect, fractional calculus is capable of modeling many situations which arise in engineering. This book includes a collection of related topics associated with such equations and their relevance and significance in engineering.
Special Functions in Fractional Calculus and Engineering highlights the significance and applicability of special functions in solving fractional-order differential equations with engineering applications. This book focuses on the non-local nature and memory effect of fractional calculus in modeling relevant to engineering science and covers a variety of important and useful methods using special functions for solving various types of fractional-order models relevant to engineering science. This book goes on to illustrate the applicability and usefulness of special functions by justifying their numerous and widespread occurrences in the solution of fractional-order differential, integral, and integrodifferential equations.
This book holds a wide variety of interconnected fundamental and advanced topics with interdisciplinary applications that combine applied mathematics and engineering sciences, which are useful to graduate students, Ph.D. scholars, researchers, and educators interested in special functions, fractional calculus, mathematical modeling, and engineering.
Chapter 1. An Introductory Overview of Special Functions and Their Associated Operators of Fractional Calculus
H. M. Srivastava
Chapter 2. Analytical Solutions for Fluid Model Described by Fractional Derivative Operators Using Special Functions in Fractional Calculus
Ndolane Sene
Chapter 3. Special Functions and Exact Solutions for Fractional Diffusion Equations with Reaction Terms
E. K. Lenzi and M. K. Lenzi
Chapter 4. Computable Solution of Fractional Kinetic Equations Associated with Incomplete ?-Functions and M-series
Nidhi Jolly and Manish Kumar Bansal
Chapter 5. Legendre Collocation Method for Generalized Fractional Advection Diffusion Equation.
Sandeep Kumar, Rajesh K. Pandey, Shiva Sharma, Harendra Singh
Chapter 6. The Incomplete Generalized Mittag-Le?er Function and Fractional Calculus Operators
Rakesh K. Parmar and Purnima Chopra
Chapter 7. Numerical Solution of Fractional Order Diffusion Equation Using Fibonacci Neural Network
Kushal Dhar Dwivedi
Chapter 8. Analysis of a Class of Reaction-Diffusion Equation Using Spectral Scheme
Prashant Pandey and Priya Kumari
Chapter 9. New Fractional Calculus Results for the Families of Extended Hurwitz-Lerch Zeta Function
Rakesh K. Parmar, Arjun K. Rathie and S. D. Purohit
Chapter 10. Compact Di?erence Schemes for Solving the Equation of Oscillator Motion with Viscoelastic Damping
A. M. Elsayed and T. S. Aleroev
Chapter 11. Dynamics of the Dadras-Momeni System in the Frame of the Caputo-Fabrizio Fractional Derivative
Chandrali Baishya and P. Veeresha
Chapter 12. A Fractional Order Model with Non-Singular Mittag-Leffler Kernel
Ali Akgul
Copyright 2023
Hardback
ISBN 9781032516783
160 Pages 2 Color & 7 B/W Illustrations
July 20, 2023 by Chapman & Hall
There is no clear sense of when the continued fraction was originally conceived of. It is likely that one of the first authors who, indirectly, suggested this notion was Euclid (c. 300 BC) via his famous algorithm (the oldest nontrivial algorithm that has survived to the present day) in the seventh book of his Elements. Since then, Aryabhata, Fibonacci, Bombelli, Wallis, Huygens, and Euler have developed this theory, and it continues to evolve today, especially as a means of linking different areas of mathematics.
This book, whose primary audience is graduate students and senior researchers, is motivated by the fascinating interrelations between ergodic theory and number theory (as established since the 1950s). It examines several generalizations and extensions of classical continued fractions, including generalized Lehner, simple, and Hirzebruch-Jung continued fractions. After deriving invariant ergodic measures for each of the underlying transformations on [0,1] it is shown that any of the famous formulas, going back to Khintchine and Levy, carry over to more general settings. Complementing these results, the entropy of the transformations is calculated and the natural extensions of the dynamical systems to [0,1]2 are analyzed.
Suitable for graduate students and senior researchers.
Written by international senior experts in number theory.
Contains the basic background, including some elementary results, that the reader may need to know before hand, making it a self-contained volume.
1. Generalized Lehner Continued Fractions. 2. a-modified Farey Series. 3. Ergodic Aspects of the Generalized Lehner Continued Fractions. 4. The a-simple Continued Fraction. 5. The Generalized Khintchine Constant. 6. The Entropy of the System ([0; 1]; s; μa; Ta). 7. The Natural Extension of ([0; 1]; s; μa; Ta). 8. The Dynamical System ([0; 1]; s; va;Qa). 9. Generalized Hirzebruch-Jung Continued Fractions. 10. The Entropy of ([0; 1]; s; νa;Ha). 11. The Natural Extension of ([0; 1]; s; νa;Ha). 12. A New Generalization of the Farey Series.