Chapter 1. Non-Newtonian Real Numbers
1.1 Basic α-Operations
1.2 α-Real Field and Related Properties
1.3 Order Axioms
1.4 α-Modulus
1.5 α-Powers
1.6 Least Upper and Greatest Lower Bounds
1.7 Some Classes of α-Real Numbers
1.8 Advanced Practical Problems
Chapter 2. Non-Newtonian Sequences
2.1 Definitions. Examples
2.2 α-Bounded and α-Unbounded α-Sequences
2.3 α-Convergent α-Sequences
2.4 Properties of α-Convergent α-Sequences
2.5 α-Arithmetic of α-Sequences
2.6 Advanced Practical Problems
Chapter 3. Non-Newtonian Elementary Functions
3.1 The α-Exponential Functions
3.2 The α-Logarithm Function
3.3 α-Trigonometric Functions
3.4 Inverse α-Trigonometric Functions
3.5 α-Hyperbolic Functions
3.6 Advanced Practical Problems
Chapter 4. Non-Newtonian Functions
4.1 Definition. Some Classes of α-Functions
4.2 α-Limits of α-Functions
4.3 α-Continuous α-Functions
4.4 Advanced Practical Problems
Chapter 5. Non-Newtonian Differentiation
5.1 Definition for α-Derivatives. Examples
5.2 α-Derivatives of α-Elementary Functions
5.3 Properties of α-Derivatives
5.4 Mean Value Theorems
5.5 The α-L’Hôpital Rule
5.6 Advanced Practical Problems
Chapter 6. Higher Order Non-Newtonian Derivatives
6.1 Definition. Examples
6.2 The α-Taylor Formula
6.3 Expansion of Some Elementary α-Functions
6.4 Local α-Extremum
6.5 α-Convex and α-Concave α-Functions
6.6 Advanced Practical Problems
Chapter 7. Non-Newtonian Integration
7.1 Definition. Examples
7.2 Basic α-Integrals
7.3 Change of Variables
7.4 α-Integration by Parts
7.5 α-Inequalities for α-Integrals
7.6 Mean Value Theorems for α-Integrals
7.7 The α-Taylor Formula in α-Integral Form
7.8 Advanced Practical Problems
Chapter 8. Improper Non-Newtonian Integrals
8.1 Definition for Improper α-Integrals Over Finite Intervals
8.2 Improper α-Integrals Over Infinite Intervals
8.3 Properties of the Improper α-Integrals
8.4 Criteria for Comparison of Improper α-Integrals
8.5 Non-Newtonian Abel-Dirichlet Criterion
8.6 Advanced Practical Problems
Chapter 9. Applications: Non-Newtonian Differential Equations
9.1 Solutions to α-Differential Equations
9.2 Gronwall Type α-Integral Inequalities
9.3 Picard’s Method of Successive Approximations and Existence Theorems
9.4 Uniqueness
9.5 α-Continuous Dependence on Initial Data
9.6 Advanced Practical Problems
Biography
Svetlin G. Georgiev is a mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, functional analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, integral equations, and dynamic calculus on time scales.
Muhittin Evren Aydin is a mathematician who works on various aspects of mathematics. Currently, he focuses on differential geometry, Riemannian geometry, fractional calculus, microeconomics, and applications of differential geometry.
