Fractional partial differential equations and their numerical solutions:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2015
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 323 - 336 |
| Physical Description: | IX, 336 S. graph. Darst. |
| ISBN: | 9789814667043 |
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Record in the Search Index
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| adam_text | Titel: Fractional partial differential equations and their numerical solutions
Autor: Guo, Boling
Jahr: 2015
Contents
Preface v
1 Physics Background 1
1.1 Origin of the fractional derivative 1
1.2 Anomalous diffusion and fractional advection-diffusion 5
1.2.1 The random walk and fractional equations 6
1.2.2 Fractional advection-diffusion equation 10
1.2.3 Fractional Fokker-Planck equation 11
1.2.4 Fractional Klein-Framers equation 15
1.3 Fractional quasi-geostrophic equation 16
1.4 Fractional nonlinear Schrodinger equation 20
1.5 Fractional Ginzburg-Landau equation 23
1.6 Fractional Landau-Lifshitz equation 27
1.7 Some applications of fractional differential equations 29
2 Fractional Calculus and Fractional Differential Equations 34
2.1 Fractional integrals and derivatives 34
2.1.1 Riemann-Liouville fractional integrals 34
2.1.2 R-L fractional derivatives 42
2.1.3 Laplace transforms of R-L fractional derivatives 48
2.1.4 Caputo s definitions of fractional derivatives 50
2.1.5 Weyl s definition for fractional derivatives 52
2.2 Fractional Laplacian 56
2.2.1 Definition and properties 56
2.2.2 Pseudo-differential operator 63
2.2.3 Riesz potential and Bessel potential 69
2.2.4 Fractional Sobolev space 71
2.2.5 Commutator estimates 76
2.3 An existence theorem 82
2.4 Distributed order differential equations 89
2.4.1 Distributed order diffusion-wave equation 91
2.4.2 Initial boundary value problem of distributed order 94
vii
viii Contents
2.5 Appendix A: the Fourier transform 96
2.6 Appendix B: Laplace transform 104
2.7 Appendix C: Mittag-Leffler function 106
2.7.1 Gamma function and Beta function 106
2.7.2 Mittag-Leffler function 107
3 Fractional Partial Differential Equations 109
3.1 Fractional diffusion equation 109
3.2 Fractional nonlinear Schrodinger equation 113
3.2.1 Space fractional nonlinear Schrodinger equation 113
3.2.2 Time fractional nonlinear Schrodinger equation 125
3.2.3 Global well-posedness of the one-dimensional fractional
nonlinear Schrodinger equation 129
3.3 Fractional Ginzburg-Landau equation 138
3.3.1 Existence of weak solutions 138
3.3.2 Global existence of strong solutions 143
3.3.3 Existence of attractors 150
3.4 Fractional Landau-Lifshitz equation 155
3.4.1 Vanishing viscosity method 155
3.4.2 Ginzburg-Landau approximation and asymptotic limit 162
3.4.3 Higher dimensional case—Galerkin approximation 170
3.4.4 Local well-posedness 185
3.5 Fractional QG equations 199
3.5.1 Existence and uniqueness of solutions 200
3.5.2 Inviscid limit 209
3.5.3 Decay and approximation 213
3.5.4 Existence of attractors 221
3.6 Fractional Boussinesq approximation 229
3.7 Boundary value problems 247
4 Numerical Approximations in Fractional Calculus 257
4.1 Fundamentals of fractional calculus 258
4.2 G-Algorithms for Riemann-Liouville fractional derivative 261
4.3 D-Algorithm for Riemann-Liouville fractional derivative 266
4.4 R-Algorithms for Riemann-Liouville fractional integral 269
4.5 L-Algorithms for fractional derivative 272
Contents ix
4.6 General form of fractional difference quotient
approximations 274
4.7 Extensions of integer-Order numerical differentiation and
integration 276
4.7.1 Extensions of backward and central difference
quotient schemes 276
4.7.2 Extension of interpolation-type integration
quadrature formulas 279
4.7.3 Extension of linear multi-step method: Lubich fractional
linear multi-step method 280
4.8 Applications of other approximation techniques 283
4.8.1 Approximations of fractional integral and derivative of
periodic function using fourier expansion 283
4.8.2 Short memory principle 284
5 Numerical Methods for the Fractional Ordinary
Differential Equations 286
5.1 Solution of fractional linear differential equation 286
5.2 Solution of the general fractional differential equations 287
5.2.1 Direct method 289
5.2.2 Indirect method 292
6 Numerical Methods for Fractional Partial
Differential Equations 299
6.1 Space fractional advection-diffusion equation 301
6.2 Time fractional partial differential equation 305
6.2.1 Finite difference schemes 306
6.2.2 Stability analysis: Fourier-von Neumann method 307
6.2.3 Error analysis 308
6.3 Time-space fractional partial differential equation 310
6.3.1 Finite difference schemes 310
6.3.2 Stability and convergence analysis 312
6.4 Numerical methods for non-linear fractional partial
differential equations 318
6.4.1 Adomina decomposition method 318
6.4.2 Variational iteration method 320
Bibliography 323
|
| any_adam_object | 1 |
| author | Guo, Boling 1936- Pu, Xueke Huang, Fenghui 1974- |
| author_GND | (DE-588)1073302334 (DE-588)1073303098 |
| author_facet | Guo, Boling 1936- Pu, Xueke Huang, Fenghui 1974- |
| author_role | aut aut aut |
| author_sort | Guo, Boling 1936- |
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| building | Verbundindex |
| bvnumber | BV042668185 |
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| dewey-ones | 515 - Analysis |
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| illustrated | Illustrated |
| indexdate | 2024-07-10T07:07:03Z |
| institution | BVB |
| isbn | 9789814667043 |
| language | English |
| lccn | 2015002791 |
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| physical | IX, 336 S. graph. Darst. |
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| spelling | Guo, Boling 1936- Verfasser (DE-588)1073302334 aut Fractional partial differential equations and their numerical solutions Boling Guo ; Xueke Pu ; Fenghui Huang Singapore [u.a.] World Scientific 2015 IX, 336 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 323 - 336 Fractional differential equations Partial Differential equations Differential equations Numerical solutions Gebrochene Analysis (DE-588)4722475-7 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Numerisches Verfahren (DE-588)4128130-5 s Gebrochene Analysis (DE-588)4722475-7 s DE-604 Pu, Xueke Verfasser aut Huang, Fenghui 1974- Verfasser (DE-588)1073303098 aut DE-601 pdf/application http://www.gbv.de/dms/tib-ub-hannover/817227784.pdf Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028100289&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Guo, Boling 1936- Pu, Xueke Huang, Fenghui 1974- Fractional partial differential equations and their numerical solutions Fractional differential equations Partial Differential equations Differential equations Numerical solutions Gebrochene Analysis (DE-588)4722475-7 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4722475-7 (DE-588)4128130-5 (DE-588)4044779-0 |
| title | Fractional partial differential equations and their numerical solutions |
| title_auth | Fractional partial differential equations and their numerical solutions |
| title_exact_search | Fractional partial differential equations and their numerical solutions |
| title_full | Fractional partial differential equations and their numerical solutions Boling Guo ; Xueke Pu ; Fenghui Huang |
| title_fullStr | Fractional partial differential equations and their numerical solutions Boling Guo ; Xueke Pu ; Fenghui Huang |
| title_full_unstemmed | Fractional partial differential equations and their numerical solutions Boling Guo ; Xueke Pu ; Fenghui Huang |
| title_short | Fractional partial differential equations and their numerical solutions |
| title_sort | fractional partial differential equations and their numerical solutions |
| topic | Fractional differential equations Partial Differential equations Differential equations Numerical solutions Gebrochene Analysis (DE-588)4722475-7 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Fractional differential equations Partial Differential equations Differential equations Numerical solutions Gebrochene Analysis Numerisches Verfahren Partielle Differentialgleichung |
| url | http://www.gbv.de/dms/tib-ub-hannover/817227784.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028100289&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT guoboling fractionalpartialdifferentialequationsandtheirnumericalsolutions AT puxueke fractionalpartialdifferentialequationsandtheirnumericalsolutions AT huangfenghui fractionalpartialdifferentialequationsandtheirnumericalsolutions |
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