Generalized Quantum Calculus with Applications,

Edition 1

Editors: By Svetlin G. Georgiev and Sanket Tikare

Publication Date: 07 May 2025

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Generalized Quantum Calculus with Applications is devoted to the qualitative theory of general quantum calculus and its applications to general quantum differential equations and inequalities. The book is aimed at upper-level undergraduate students and beginning graduate students in a range of interdisciplinary courses including physical sciences and engineering, from quantum mechanics to differential equations, with pedagogically organized chapters that each concludes with a section of practical problems. Generalized quantum calculus includes a generalization of the q-quantum calculus and the time scale calculus. There are many open problems and difficulties in q-quantum calculus and time-scale calculus, and this book explores how to use the generalized quantum operators to solve difficulties arising in q-quantum calculus and time-scale calculus, including but not limited to generalized quantum integration, generalized quantum chain rules, and generalized quantum Taylor formula.

Since generalized quantum calculus includes the q-quantum and time-scale calculus, this book can be utilized by a wide audience of researchers and students. This text is one of few foundational books on generalized quantum calculus and can be used for future discoveries in the area of integral transforms, variational calculus, integral equations, and inequalities in the language of generalized quantum calculus. This book also offers detailed proofs, exercises, and examples to aid instructors, researchers, and users in their studies.

Key Features

Explores cutting-edge research trends in quantum calculus
Provides practical information and techniques for building fundamental knowledge and applying contemporary quantum calculus in upper-undergraduate and graduate-level studies
Serves as a front-line book for budding researchers and experts of mathematics, along with students from several interdisciplinary fields
Offers additional resources such as detailed proofs, exercises, and examples to aid instructors and students in their work

About the author

By Svetlin G. Georgiev, Professor on the Faculty of Mathematics and Informatics, Sorbonne University, Paris, France and Sanket Tikare, Assistant Professor, Department of Mathematics at Ramniranjan Jhunjhunwala College, Mumbai, India

1. Generalized Quantum Differentiation
1.1 The ß-Operator
1.2 Definition for ß-Derivative. Examples
1.3 Properties of the ß-Derivative
1.4 Rules for ß-Differentiation
1.5 Properties of ß-Differentiable Functions
1.6 Chain Rules
1.7 A Mean Value Theorem
1.8 Higher Order ß-Derivatives
1.9 Advanced Practical Problems
1.10 Notes and References

2. Generalized Quantum Integration
2.1 ß-Antiderivatives
2.2 Definition for ß-Integral. Examples
2.3 Properties of ß-Integrals
2.4 Inequalities and ß-Integrals
2.5 Generalized Quantum Monomials
2.6 The Taylor Formula
2.7 Improper Integrals of the First Kind
2.8 Improper Integrals of the Second Kind
2.9 Advanced Practical Problems
2.10 Notes and References

3. ß-Elementary Functions
3.1 ß- Regressive Functions
3.2 ß-Exponential Functions
3.3 ß-Trigonometric Functions
3.4 ß-Hyperbolic Functions
3.5 Advanced Practical Problems
3.6 Notes and References

4. The ß-Laplace Transform
4.1 Functions of Exponential Orders
4.2 Definition for the ß-Laplace Transform. Properties
4.3 The ß-Laplace Transform of ß-Derivative
4.4 The ß-Laplace Transform of ß-Integrals
4.5 Advanced Practical Problems
4.6 Notes and References

5. First Order Linear ß-Differential Equations
5.1 Linear ß-Differential operators
5.2 Homogeneous Initial Value Problems
5.3 Nonhomogeneous Initial Value Problems
5.4 The Laplace Transform Method
5.5 Notes and References

6. Second Order Linear ß-Differential Equations
6.1 ß-Wronskians
6.2 The Abel Theorem
6.3 Homogeneous Second Order Linear ß-Differential Equations with Constant Coefficients
6.4 Reduction of Order
6.5 Method of Factoring
6.6 Nonconstant Coefficients
6.7 ß-Euler-Cauchy Equations
6.8 Variation of Parameters
6.9 The Anihilator Method
6.10 The ß-Laplace Transform Method
6.11 Advanced Practical Problems
6.12 Notes and References

7. ß-Differential Systems
7.1 Structure of ß-Differential Systems
7.2 ß-Matrix Exponential Function
7.3 The ß-Liouville Theorem
7.4 Constant Coefficients
7.5 Advanced Practical Problems
7.6 Notes and References

8. Linear ß- Integral Inequalities
8.1 Gronwall-Type Inequalities
8.2 Bellman-Type Inequalities
8.3 Volterra-Type ß-Integral Inequalities
8.4 Notes and References

ISBN: 9780443328046

Page Count: 288

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Upper-level undergraduate students, post-graduate students, researchers, and professors in mathematics