Anastassiou Fractional Differentiation Inequalities

1. Auflage 2009
ISBN: 978-0-387-98128-4
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 686 Seiten

In this book the author presents the Opial, Poincaré, Sobolev, Hilbert, and Ostrowski fractional differentiation inequalities. Results for the above are derived using three different types of fractional derivatives, namely by Canavati, Riemann-Liouville and Caputo. The univariate and multivariate cases are both examined. Each chapter is self-contained. The theory is presented systematically along with the applications. The application to information theory is also examined. This monograph is suitable for researchers and graduate students in pure mathematics. Applied mathematicians, engineers, and other applied scientists will also find this book useful.

Autoren/Hrsg.

Anastassiou, George A.

Inhaltsverzeichnis

1;Contents;6
2;Preface;11
3;Introduction;13
4;Opial-Type Inequalities for Functions and Their Ordinary and Canavati Fractional Derivatives;18
4.1;Preliminaries;18
4.2;Main Results;22
4.3;Applications;29
5;Canavati Fractional Opial-Type Inequalities and Fractional Differential Equations;34
5.1;Introduction;34
5.2;Preliminaries;35
5.3;Main Results;37
5.4;Applications;45
5.5;Other Fractional Differential Equations;49
6;Riemann--Liouville Opial-Type Inequalities for Fractional Derivatives ;51
6.1;Introduction and Preliminaries;51
6.2;Main Results;54
6.3;Applications;58
7;Opial-Type Lp-Inequalities for Riemann--LiouvilleFractional Derivatives;63
7.1;Introduction and Preliminaries;63
7.2;Main Results;66
8;Opial-Type Inequalities Involving Canavati FractionalDerivatives of Two Functions and Applications;76
8.1;Introduction;76
8.2;Preliminaries;78
8.3;Main Results;81
8.4;Applications;105
9;Opial-Type Inequalities for Riemann--Liouville Fractional Derivatives of Two Functions with Applications;116
9.1;Introduction;116
9.2;Background;117
9.3;Main Results;118
9.4;Applications;147
10;Canavati Fractional Opial-Type Inequalities for SeveralFunctions and Applications;157
10.1;Introduction;157
10.2;Preliminaries;158
10.3;Main Results;160
10.4;Applications;177
11;Riemann--Liouville Fractional Opial-Type Inequalitiesfor Several Functions and Applications;186
11.1;Introduction;186
11.2;Background;187
11.3;Main Results;188
11.4;Applications;202
12;Converse Canavati Fractional Opial-Type Inequalitiesfor Several Functions;211
12.1;Introduction;211
12.2;Preliminaries;212
12.3;Main Results;215
12.3.1;Results Involving Two Functions;215
12.3.2;Results Involving Several Functions;226
13;Converse Riemann--Liouville Fractional Opial-TypeInequalities for Several Functions;234
13.1;Introduction;234
13.2;Background;235
13.3;Main Results;236
13.3.1;Results Involving Two Functions;236
13.3.2;Results Involving Several Functions;249
13.3.3;Results with Respect to Generalized Riemann --Liouville Fractional Derivative;256
14;Multivariate Canavati Fractional Taylor Formula;261
14.1;Introduction;261
14.2;Results;262
15;Multivariate Caputo Fractional Taylor Formula;273
15.1;Background;273
15.2;Results;274
16;Canavati Fractional Multivariate Opial-TypeInequalities on Spherical Shells;283
16.1;Introduction;283
16.2;Results;284
17;Riemann--Liouville Fractional Multivariate Opial-TypeInequalities over a Spherical Shell;322
17.1;Introduction;322
17.2;Background---I;323
17.3;Background---II;326
17.4;Background---III;332
17.5;Main Results;337
17.5.1;Riemann--Liouville Fractional Opial-TypeInequalities Involving One Function;337
17.5.2;Riemann--Liouville Fractional Opial-TypeInequalities Involving Two Functions;353
17.5.3;Riemann--Liouville Fractional Opial-TypeInequalities Involving Several Functions;372
18;Caputo Fractional Multivariate Opial-Type Inequalitiesover a Spherical Shell;394
18.1;Introduction;394
18.2;Background---I;395
18.3;Main Results;400
18.3.1;Results Involving One Function;400
18.3.2;Results Involving Two Functions;405
18.3.3;Results Involving Several Functions;414
18.4;Background---II;422
18.5;Main Results on a Spherical Shell;427
18.5.1;Results Involving One Function;427
18.5.2;Results Involving Two Functions;430
18.5.3;Results Involving Several Functions;434
18.6;Applications;439
19;Poincaré-Type Fractional Inequalities;448
19.1;Introduction;448
19.2;Fractional Poincaré Inequalities Results;449
19.3;Applications of Fractional Poincaré Inequalities;460
19.4;Fractional Mean Poincaré Inequalities;476
19.5;Applications of Fractional Mean Poincaré Inequalities;482
20;Various Sobolev-Type Fractional Inequalities;486
20.1;Introduction;486
20.2;Various Univariate Sobolev-Type Fractional Inequalities;487
20.3;Applications;506
21;General Hilbert--Pachpatte-Type Integral Inequalities;508
21.1;Introduction;508
21.2;Main Results;509
22;General Multivariate Hilbert--Pachpatte-TypeIntegral Inequalities;526
22.1;Introduction;526
22.2;Symbols and Basics;527
22.3;Main Results;530
23;Other Hilbert--Pachpatte-Type Fractional IntegralInequalities;548
23.1;Background;548
23.2;Univariate Results;553
23.3;Multivariate Results;556
24;Canavati Fractional and Other Approximationof Csiszar's f-Divergence;565
24.1;Preliminaries;565
24.2;Main Results;570
25;Caputo and Riemann--Liouville FractionalApproximation of Csiszar's f-Divergence;579
25.1;Preliminaries;579
25.2;Results;583
26;Canavati Fractional Ostrowski-Type Inequalities;590
26.1;Background;590
26.2;Results;592
27;Multivariate Canavati Fractional Ostrowski-TypeInequalities;596
27.1;Background;596
27.2;Results;599
28;Caputo Fractional Ostrowski-Type Inequalities;615
28.1;Background;615
28.2;Univariate Results;618
28.3;Multivariate Results;622
29;Appendix;634
29.1;Conversion Formulae for Different Kinds of FractionalDerivatives;634
29.2;Some Basic Fractional Derivatives;637
30;References;639
31;List of Symbols;668
32;Index;670