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E-grāmata: Intelligent Numerical Methods: Applications to Fractional Calculus

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Intelligent Numerical Methods: Applications to Fractional CalculusPalielināt
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In this monograph the authors present Newton-type, Newton-like and other numerical methods, which involve fractional derivatives and fractional integral operators, for the first time studied in the literature. All for the purpose to solve numerically equations whose associated functions can be also non-differentiable in the ordinary sense. That is among others extending the classical Newton method theory which requires usual differentiability of function.

Chapters are self-contained and can be read independently and several advanced courses can be taught out of this book. An extensive list of references is given per chapter. The books results are expected to find applications in many areas of applied mathematics, stochastics, computer science and engineering.As such this monograph is suitable for researchers, graduate students, and seminars of the above subjects, also to be in all science and engineering libraries.
1 Newton-Like Methods on Generalized Banach Spaces and Fractional Calculus
1(22)
1.1 Introduction
1(1)
1.2 Generalized Banach Spaces
2(3)
1.3 Semilocal Convergence
5(3)
1.4 Special Cases and Applications
8(6)
1.5 Applications to Fractional Calculus
14(9)
References
20(3)
2 Semilocal Convegence of Newton-Like Methods and Fractional Calculus
23(16)
2.1 Introduction
23(1)
2.2 Generalized Banach Spaces
24(2)
2.3 Semilocal Convergence
26(4)
2.4 Special Cases and Applications
30(1)
2.5 Applications to Fractional Calculus
30(9)
References
36(3)
3 Convergence of Iterative Methods and Generalized Fractional Calculus
39(18)
3.1 Introduction
39(1)
3.2 Generalized Banach Spaces
40(2)
3.3 Semilocal Convergence
42(4)
3.4 Special Cases and Applications
46(1)
3.5 Applications to Generalized Fractional Calculus
47(10)
References
56(1)
4 Fixed Point Techniques and Generalized Right Fractional Calculus
57(18)
4.1 Introduction
57(1)
4.2 Generalized Banach Spaces
58(2)
4.3 Semilocal Convergence
60(4)
4.4 Special Cases and Applications
64(1)
4.5 Applications to Generalized Right Fractional Calculus
65(10)
References
74(1)
5 Approximating Fixed Points and k-Fractional Calculus
75(20)
5.1 Introduction
75(1)
5.2 Generalized Banach Spaces
76(2)
5.3 Semilocal Convergence
78(4)
5.4 Special Cases and Applications
82(1)
5.5 Applications to k-Fractional Calculus
83(12)
References
93(2)
6 Iterative Methods and Generalized g-Fractional Calculus
95(12)
6.1 Introduction
95(1)
6.2 Generalized Banach Spaces
96(2)
6.3 Applications to g-Fractional Calculus
98(9)
References
106(1)
7 Unified Convergence Analysis for Iterative Algorithms and Fractional Calculus
107(20)
7.1 Introduction
107(1)
7.2 Convergence Analysis
108(6)
7.3 Applications to Fractional Calculus
114(13)
References
124(3)
8 Convergence Analysis for Extended Iterative Algorithms and Fractional and Vector Calculus
127(22)
8.1 Introduction
127(1)
8.2 Convergence Analysis
128(6)
8.3 Applications to Fractional and Vector Calculus
134(15)
References
147(2)
9 Convergence Analysis for Extended Iterative Algorithms and Fractional Calculus
149(14)
9.1 Introduction
149(1)
9.2 Convergence Analysis
150(6)
9.3 Applications to Fractional Calculus
156(7)
References
161(2)
10 Secant-Like Methods and Fractional Calculus
163(14)
10.1 Introduction
163(1)
10.2 Convergence Analysis
164(5)
10.3 Applications to Right Fractional Calculus
169(8)
References
174(3)
11 Secant-Like Methods and Modified g-Fractional Calculus
177(20)
11.1 Introduction
177(1)
11.2 Convergence Analysis
178(5)
11.3 Applications to Modified g-Fractional Calculus
183(14)
References
196(1)
12 Secant-Like Algorithms and Generalized Fractional Calculus
197(18)
12.1 Introduction
197(1)
12.2 Convergence Analysis
198(5)
12.3 Applications to g-Fractional Calculus
203(12)
References
214(1)
13 Secant-Like Methods and Generalized g-Fractional Calculus of Canavati-Type
215(16)
13.1 Introduction
215(1)
13.2 Convergence Analysis
216(5)
13.3 Applications to g-Fractional Calculus of Canavati Type
221(10)
References
230(1)
14 Iterative Algorithms and Left-Right Caputo Fractional Derivatives
231(14)
14.1 Introduction
231(1)
14.2 Convergence Analysis
232(7)
14.3 Applications to Fractional Calculus
239(6)
References
243(2)
15 Iterative Methods on Banach Spaces with a Convergence Structure and Fractional Calculus
245(18)
15.1 Introduction
245(1)
15.2 Banach Spaces with Convergence Structure
246(2)
15.3 Semilocal Convergence
248(4)
15.4 Special Cases and Examples
252(2)
15.5 Applications to Fractional Calculus
254(9)
References
262(1)
16 Inexact Gauss-Newton Method for Singular Equations
263(20)
16.1 Introduction
263(3)
16.2 Preliminaries
266(4)
16.3 Local Convergence
270(3)
16.4 Proofs
273(5)
16.4.1 Proof of Theorem 16.9
274(2)
16.4.2 Proof of Theorem 16.12
276(1)
16.4.3 Proof of Theorem 16.14
277(1)
16.5 Numerical Examples
278(1)
16.6 Conclusion
279(4)
References
279(4)
17 The Asymptotic Mesh Independence Principle
283(14)
17.1 Introduction
283(1)
17.2 The Mesh Independence Principle
284(10)
17.3 Numerical Examples
294(3)
References
295(2)
18 Ball Convergence of a Sixth Order Iterative Method
297(12)
18.1 Introduction
297(2)
18.2 Local Convergence Analysis
299(7)
18.3 Numerical Examples
306(3)
References
306(3)
19 Broyden's Method with Regularly Continuous Divided Differences
309(8)
19.1 Introduction
309(1)
19.2 Semilocal Convergence Analysis of Broyden's Method
310(7)
References
316(1)
20 Left General Fractional Monotone Approximation
317(20)
20.1 Introduction and Preparation
317(10)
20.2 Main Result
327(7)
20.3 Applications (to Theorem 20.15)
334(3)
References
335(2)
21 Right General Fractional Monotone Approximation Theory
337(16)
21.1 Introduction and Preparation
337(7)
21.2 Main Result
344(7)
21.3 Applications (to Theorem 21.14)
351(2)
References
351(2)
22 Left Generalized High Order Fractional Monotone Approximation
353(20)
22.1 Introduction
353(9)
22.2 Main Result
362(9)
22.3 Applications (to Theorem 22.16)
371(2)
References
371(2)
23 Right Generalized High Order Fractional Monotone Approximation
373(18)
23.1 Introduction
373(7)
23.2 Main Result
380(9)
23.3 Applications (to Theorem 23.15)
389(2)
References
389(2)
24 Advanced Fractional Taylor's Formulae
391(22)
24.1 Introduction
391(1)
24.2 Results
392(21)
References
412(1)
25 Generalized Canavati Type Fractional Taylor's Formulae
413(8)
25.1 Results
413(8)
References
420(1)
Index 421
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