Non-Newtonian Sequence Spaces with Applications
Series: Chapman & Hall/CRC Monographs and Research Notes in Mathematics;
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Product details:
- Edition number 1
- Publisher Chapman and Hall
- Date of Publication 13 May 2025
- ISBN 9781032988900
- Binding Hardback
- No. of pages202 pages
- Size 254x178 mm
- Language English
- Illustrations 9 Illustrations, black & white; 9 Line drawings, black & white; 11 Tables, black & white 700
Categories
Short description:
Non-Newtonian Sequence Spaces with Applications presents an alternative to the usual calculus based on multiplication instead of addition.
MoreLong description:
Non-Newtonian Sequence Spaces with Applications presents an alternative to the usual calculus based on multiplication instead of addition. This book is intended for graduate students and researchers with a special interest in non-Newtonian calculus, its applications, and related topics.
Key features:
- Valuable material for postgraduate researchers studying non-Newtonian calculus
- Suitable as supplementary reading to a Computational Physics course
Table of Contents:
Preface vii
Acknowledgements ix
List of Abbreviations and Symbols x
1 Sequence and Function Spaces over the Non-newtonian ... 1
1.1 Some Basic Results on the Spaces of Sequences ... . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Preliminaries, background and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Geometric complex field and related properties . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Geometric metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Convergence and completeness in (GC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.5 Sequence spaces over C(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Some Results on Sequence Spaces with ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.1 Preliminaries, backround and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Non-newtonian real field and related properties . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Non-newtonian metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.4 Convergence and completeness in (NC) . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Sequence Spaces Over the Non-newtonian ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Certain Non-newtonian Complex Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Preliminaries, background and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6 Some Sequence Spaces and Matrix Transformations in ... . . . . . . . . . . . . . . . . . . . . . . 29
1.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.6.2 Preliminaries, background and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.6.3 Characterizations of some matrix classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6.4 Multiplicative dual summability methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 Application of Geometric Calculus in Numerical Analysis and Difference Sequence Spaces 39
2.1 Introduction and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 ?-generator and Geometric Real Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.1 Some useful relations between geometric operations and ordinary arithmetic operations . 40
2.3 Geometric Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Dual Spaces of ?G
?(?G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.1 Geometric form of Abel?s partial summation formula . . . . . . . . . . . . . . . . . . . . 46
2.5 ?-, ?- and ?-duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Some Applications of Geometric Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6.1 Geometric Newton-Gregory backward interpolation formula . . . . . . . . . . . . . . . . 53
2.6.2 Advantages of geometric interpolation formulae over ordinary interpolation formulae . . 55
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Bigeometric Integral Calculus 56
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Geometric Arithmetic and Geometric Real Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.1 G-derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.2 Some standard G-derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
iv
3.4 G-Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.1 Some standard G-integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.2 Integration by transforming the function to the form ex f?(x)
f(x) . . . . . . . . . . . . . . . . 58
3.4.3 Integration by the relation between G-integral and ordinary integral . . . . . . . . . . . 58
3.4.4 Properties of G-integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Definite Bigeometric Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.1 Properties of definite G-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.2 Definite bigeometric integral as a limit of geometric sum . . . . . . . . . . . . . . . . . . 63
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Bigeometric Calculus and Its Applications 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Some useful relations between geometric operations and ordinary arithmetic operations . 67
4.2 Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.1 Geometric binomial formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.2 Geometric real number line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.3 Geometric coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.4 Geometric factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.5 Generalized geometric forward difference operator ?n
G . . . . . . . . . . . . . . . . . . . . 69
4.2.6 Generalized Geometric Backward Difference Operator ?n
G . . . . . . . . . . . . . . . . . 69
4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.1 Geometric Pythagorean triplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.2 Geometric trigonometric ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.3 Relation between geometric trigonometry and ordinary trigonometry . . . . . . . . . . . 71
4.3.4 Geometric trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.5 G-limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.6 G-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 Basic Properties of G-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.1 G-derivative and its interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.2 Relation between G-derivative and ordinary derivative . . . . . . . . . . . . . . . . . . . 77
4.4.3 G-derivatives of some standard functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.4 Geometric Taylor?s series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Some Applications of G-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5.1 Expansion of some useful functions in Taylor?s product . . . . . . . . . . . . . . . . . . . 83
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Solution of Bigeometric-Differential Equations by Numerical Methods 87
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Basic Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.1 Geometric factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.2 Geometric Newton-Gregory formula for forward interpolation . . . . . . . . . . . . . . . 88
5.2.3 Geometric Newton-Gregory formula for backward interpolation . . . . . . . . . . . . . . 88
5.2.4 G-derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.5 Some standard G-derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.6 Geometric Taylor?s series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Numerical Methods and Solution of G-Differential Equations . . . . . . . . . . . . . . . . . . . . 89
5.3.1 G-Euler?s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3.2 Taylor?s G-series method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3.3 G-Runge-Kutta method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3.4 G-Runge-Kutta method of order four . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Certain Spaces of Functions over the Set of Non-Newtonian Complex Numbers 100
6.1 Preliminaries, Backround and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 The Set of ?-Complex Numbers and ?-Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3 Continuous Function Space over the Field C? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Multiplicative Type Complex Calculus 110
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.2 Definitions, Methods, and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2.1 A multiplicative group, an additive group, and an isomorphism . . . . . . . . . . . . . . 111
7.2.2 Remoteness of two values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.2.3 Change rate of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.2.4 Derivative and integral operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.2.5 Euler?s simple method in differential equation solving . . . . . . . . . . . . . . . . . . . . 117
7.2.6 Some fundamental theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8 Function Sequences and Series ... 124
8.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.2 ?-Function Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.2.1 ?-function sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.2.2 ?-function series and consequences of ?-uniform convergence . . . . . . . . . . . . . . . . 129
8.2.3 ?-uniform convergence and ?-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8.2.4 ?-uniform convergence and ?-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.2.5 ?-Uniform Convergence and ?-Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9 On Non-newtonian Power Series and its Applications 139
9.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.2.1 ?-Dirichlet?s and ?-Abel?s tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.2.2 ?-power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Bibliography 150
Index 153
