Fractional calculus (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Fractional calculus" in English language version.

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abelprize.no

Niels Henrik Abel (1823). "Oplösning af et Par Opgaver ved Hjelp af bestemte Integraler (Solution de quelques problèmes à l'aide d'intégrales définies, Solution of a couple of problems by means of definite integrals)" (PDF). Magazin for Naturvidenskaberne. Kristiania (Oslo): 55–68.

ams.org

mathscinet.ams.org

Erdélyi, Arthur (1950–1951). "On some functional transformations". Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino. 10: 217–234. MR 0047818.

arxiv.org

Katugampola, Udita N. (15 October 2014). "A New Approach To Generalized Fractional Derivatives" (PDF). Bulletin of Mathematical Analysis and Applications. 6 (4): 1–15. arXiv:1106.0965.
Podlubny, Igor; Magin, Richard L.; Trymorush, Irina (2017). "Niels Henrik Abel and the birth of fractional calculus". Fractional Calculus and Applied Analysis. 20 (5): 1068–1075. arXiv:1802.05441. doi:10.1515/fca-2017-0057. S2CID 119664694.
Atangana, Abdon; Baleanu, Dumitru (2016). "New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model". Thermal Science. 20 (2): 763–769. arXiv:1602.03408. doi:10.2298/TSCI160111018A. ISSN 0354-9836.
Bayın, Selçuk Ş. (5 December 2016). "Definition of the Riesz derivative and its application to space fractional quantum mechanics". Journal of Mathematical Physics. 57 (12): 123501. arXiv:1612.03046. Bibcode:2016JMP....57l3501B. doi:10.1063/1.4968819. S2CID 119099201.
Ferrari, Fausto (January 2018). "Weyl and Marchaud Derivatives: A Forgotten History". Mathematics. 6 (1): 6. arXiv:1711.08070. doi:10.3390/math6010006.
Mainardi, F.; Luchko, Y.; Pagnini, G. (2001). "The fundamental solution of the space-time fractional diffusion equation". Fractional Calculus and Applied Analysis. 4 (2): 153–192. arXiv:cond-mat/0702419. Bibcode:2007cond.mat..2419M.
Gorenflo, Rudolf; Mainardi, Francesco (2007). "Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk". In Rangarajan, G.; Ding, M. (eds.). Processes with Long-Range Correlations. Lecture Notes in Physics. Vol. 621. pp. 148–166. arXiv:0709.3990. Bibcode:2003LNP...621..148G. doi:10.1007/3-540-44832-2_8. ISBN 978-3-540-40129-2. S2CID 14946568.
Colbrook, Matthew J.; Ma, Xiangcheng; Hopkins, Philip F.; Squire, Jonathan (2017). "Scaling laws of passive-scalar diffusion in the interstellar medium". Monthly Notices of the Royal Astronomical Society. 467 (2): 2421–2429. arXiv:1610.06590. Bibcode:2017MNRAS.467.2421C. doi:10.1093/mnras/stx261. S2CID 20203131.
Näsholm, S. P.; Holm, S. (2012). "On a Fractional Zener Elastic Wave Equation". Fract. Calc. Appl. Anal. 16: 26–50. arXiv:1212.4024. doi:10.2478/s13540-013-0003-1. S2CID 120348311.
Holm, S.; Näsholm, S. P. (2013). "Comparison of fractional wave equations for power law attenuation in ultrasound and elastography". Ultrasound in Medicine & Biology. 40 (4): 695–703. arXiv:1306.6507. CiteSeerX 10.1.1.765.120. doi:10.1016/j.ultrasmedbio.2013.09.033. PMID 24433745. S2CID 11983716.
Pandey, Vikash; Holm, Sverre (2016-12-01). "Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations". The Journal of the Acoustical Society of America. 140 (6): 4225–4236. arXiv:1612.05557. Bibcode:2016ASAJ..140.4225P. doi:10.1121/1.4971289. ISSN 0001-4966. PMID 28039990. S2CID 29552742.
Laskin, N. (2002). "Fractional Schrodinger equation". Phys. Rev. E. 66 (5): 056108. arXiv:quant-ph/0206098. Bibcode:2002PhRvE..66e6108L. CiteSeerX 10.1.1.252.6732. doi:10.1103/PhysRevE.66.056108. PMID 12513557. S2CID 7520956.

bnf.fr

gallica.bnf.fr

Liouville, Joseph (1832), "Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions", Journal de l'École Polytechnique, 13, Paris: 1–69.
Liouville, Joseph (1832), "Mémoire sur le calcul des différentielles à indices quelconques", Journal de l'École Polytechnique, 13, Paris: 71–162.

books.google.com

Daniel Zwillinger (12 May 2014). Handbook of Differential Equations. Elsevier Science. ISBN 978-1-4832-2096-3.
Kilbas, A. Anatolii Aleksandrovich; Srivastava, Hari Mohan; Trujillo, Juan J. (2006). Theory And Applications of Fractional Differential Equations. Elsevier. p. 75 (Property 2.4). ISBN 978-0-444-51832-3.
Herrmann, Richard, ed. (2014). Fractional Calculus (2nd ed.). New Jersey: World Scientific Publishing Co. p. 54^{[verification needed]}. Bibcode:2014fcip.book.....H. doi:10.1142/8934. ISBN 978-981-4551-07-6. {{cite book}}: |journal= ignored (help)

doi.org

Podlubny, Igor; Magin, Richard L.; Trymorush, Irina (2017). "Niels Henrik Abel and the birth of fractional calculus". Fractional Calculus and Applied Analysis. 20 (5): 1068–1075. arXiv:1802.05441. doi:10.1515/fca-2017-0057. S2CID 119664694.
For a historical review of the subject up to the beginning of the 20th century, see: Bertram Ross (1977). "The development of fractional calculus 1695–1900". Historia Mathematica. 4: 75–89. doi:10.1016/0315-0860(77)90039-8. S2CID 122146887.
Valério, Duarte; Machado, José; Kiryakova, Virginia (2014-01-01). "Some pioneers of the applications of fractional calculus". Fractional Calculus and Applied Analysis. 17 (2): 552–578. doi:10.2478/s13540-014-0185-1. hdl:10400.22/5491. ISSN 1314-2224. S2CID 121482200.
Hermann, Richard (2014). Fractional Calculus: An Introduction for Physicists (2nd ed.). New Jersey: World Scientific Publishing. p. 46. Bibcode:2014fcip.book.....H. doi:10.1142/8934. ISBN 978-981-4551-07-6.
Mainardi, Francesco (May 2010). Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press. doi:10.1142/p614. ISBN 978-1-84816-329-4. S2CID 118719247.
Mostafanejad, Mohammad (2021). "Fractional paradigms in quantum chemistry". International Journal of Quantum Chemistry. 121 (20). doi:10.1002/qua.26762.
Al-Raeei, Marwan (2021). "Applying fractional quantum mechanics to systems with electrical screening effects". Chaos, Solitons & Fractals. 150 (September): 111209. Bibcode:2021CSF...15011209A. doi:10.1016/j.chaos.2021.111209.
Herrmann, Richard, ed. (2014). Fractional Calculus (2nd ed.). New Jersey: World Scientific Publishing Co. p. 54^{[verification needed]}. Bibcode:2014fcip.book.....H. doi:10.1142/8934. ISBN 978-981-4551-07-6. {{cite book}}: |journal= ignored (help)
Caputo, Michele (1967). "Linear model of dissipation whose Q is almost frequency independent. II". Geophysical Journal International. 13 (5): 529–539. Bibcode:1967GeoJ...13..529C. doi:10.1111/j.1365-246x.1967.tb02303.x..
Algahtani, Obaid Jefain Julaighim (2016-08-01). "Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model". Chaos, Solitons & Fractals. Nonlinear Dynamics and Complexity. 89: 552–559. Bibcode:2016CSF....89..552A. doi:10.1016/j.chaos.2016.03.026. ISSN 0960-0779.
Atangana, Abdon; Baleanu, Dumitru (2016). "New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model". Thermal Science. 20 (2): 763–769. arXiv:1602.03408. doi:10.2298/TSCI160111018A. ISSN 0354-9836.
Chen, YangQuan; Li, Changpin; Ding, Hengfei (22 May 2014). "High-Order Algorithms for Riesz Derivative and Their Applications". Abstract and Applied Analysis. 2014: 1–17. doi:10.1155/2014/653797.
Bayın, Selçuk Ş. (5 December 2016). "Definition of the Riesz derivative and its application to space fractional quantum mechanics". Journal of Mathematical Physics. 57 (12): 123501. arXiv:1612.03046. Bibcode:2016JMP....57l3501B. doi:10.1063/1.4968819. S2CID 119099201.
de Oliveira, Edmundo Capelas; Tenreiro Machado, José António (2014-06-10). "A Review of Definitions for Fractional Derivatives and Integral". Mathematical Problems in Engineering. 2014: 1–6. doi:10.1155/2014/238459. hdl:10400.22/5497.
Aslan, İsmail (2015-01-15). "An analytic approach to a class of fractional differential-difference equations of rational type via symbolic computation". Mathematical Methods in the Applied Sciences. 38 (1): 27–36. Bibcode:2015MMAS...38...27A. doi:10.1002/mma.3047. hdl:11147/5562. S2CID 120881978.
Ma, Li; Li, Changpin (2017-05-11). "On hadamard fractional calculus". Fractals. 25 (3): 1750033–2980. Bibcode:2017Fract..2550033M. doi:10.1142/S0218348X17500335. ISSN 0218-348X.
Miller, Kenneth S. (1975). "The Weyl fractional calculus". In Ross, Bertram (ed.). Fractional Calculus and Its Applications. Lecture Notes in Mathematics. Vol. 457. Springer. pp. 80–89. doi:10.1007/bfb0067098. ISBN 978-3-540-69975-0. {{cite book}}: |work= ignored (help)
Ferrari, Fausto (January 2018). "Weyl and Marchaud Derivatives: A Forgotten History". Mathematics. 6 (1): 6. arXiv:1711.08070. doi:10.3390/math6010006.
Khalili Golmankhaneh, Alireza (2022). Fractal Calculus and its Applications. Singapore: World Scientific Pub Co Inc. p. 328. doi:10.1142/12988. ISBN 978-981-126-110-7. S2CID 248575991.
Anderson, Douglas R.; Ulness, Darin J. (2015-06-01). "Properties of the Katugampola fractional derivative with potential application in quantum mechanics". Journal of Mathematical Physics. 56 (6): 063502. Bibcode:2015JMP....56f3502A. doi:10.1063/1.4922018. ISSN 0022-2488.
Caputo, Michele; Fabrizio, Mauro (2016-01-01). "Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels". Progress in Fractional Differentiation and Applications. 2 (1): 1–11. doi:10.18576/pfda/020101. ISSN 2356-9336.
Kober, Hermann (1940). "On fractional integrals and derivatives". The Quarterly Journal of Mathematics. os-11 (1): 193–211. Bibcode:1940QJMat..11..193K. doi:10.1093/qmath/os-11.1.193.
Wheatcraft, Stephen W.; Meerschaert, Mark M. (October 2008). "Fractional conservation of mass" (PDF). Advances in Water Resources. 31 (10): 1377–1381. Bibcode:2008AdWR...31.1377W. doi:10.1016/j.advwatres.2008.07.004. ISSN 0309-1708.
Atangana, Abdon; Bildik, Necdet (2013). "The Use of Fractional Order Derivative to Predict the Groundwater Flow". Mathematical Problems in Engineering. 2013: 1–9. doi:10.1155/2013/543026.
Atangana, Abdon; Vermeulen, P. D. (2014). "Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation". Abstract and Applied Analysis. 2014: 1–11. doi:10.1155/2014/381753.
Benson, D.; Wheatcraft, S.; Meerschaert, M. (2000). "Application of a fractional advection-dispersion equation". Water Resources Research. 36 (6): 1403–1412. Bibcode:2000WRR....36.1403B. CiteSeerX 10.1.1.1.4838. doi:10.1029/2000wr900031. S2CID 7669161.
Benson, D.; Wheatcraft, S.; Meerschaert, M. (2000). "The fractional-order governing equation of Lévy motion". Water Resources Research. 36 (6): 1413–1423. Bibcode:2000WRR....36.1413B. doi:10.1029/2000wr900032. S2CID 16579630.
Wheatcraft, Stephen W.; Meerschaert, Mark M.; Schumer, Rina; Benson, David A. (2001-01-01). "Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests". Transport in Porous Media. 42 (1–2): 211–240. CiteSeerX 10.1.1.58.2062. doi:10.1023/A:1006733002131. ISSN 1573-1634. S2CID 189899853.
Atangana, Abdon; Kilicman, Adem (2014). "On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative". Mathematical Problems in Engineering. 2014: 9. doi:10.1155/2014/542809.
Metzler, R.; Klafter, J. (2000). "The random walk's guide to anomalous diffusion: a fractional dynamics approach". Phys. Rep. 339 (1): 1–77. Bibcode:2000PhR...339....1M. doi:10.1016/s0370-1573(00)00070-3.
Gorenflo, Rudolf; Mainardi, Francesco (2007). "Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk". In Rangarajan, G.; Ding, M. (eds.). Processes with Long-Range Correlations. Lecture Notes in Physics. Vol. 621. pp. 148–166. arXiv:0709.3990. Bibcode:2003LNP...621..148G. doi:10.1007/3-540-44832-2_8. ISBN 978-3-540-40129-2. S2CID 14946568.
Colbrook, Matthew J.; Ma, Xiangcheng; Hopkins, Philip F.; Squire, Jonathan (2017). "Scaling laws of passive-scalar diffusion in the interstellar medium". Monthly Notices of the Royal Astronomical Society. 467 (2): 2421–2429. arXiv:1610.06590. Bibcode:2017MNRAS.467.2421C. doi:10.1093/mnras/stx261. S2CID 20203131.
Tenreiro Machado, J. A.; Silva, Manuel F.; Barbosa, Ramiro S.; Jesus, Isabel S.; Reis, Cecília M.; Marcos, Maria G.; Galhano, Alexandra F. (2010). "Some Applications of Fractional Calculus in Engineering". Mathematical Problems in Engineering. 2010: 1–34. doi:10.1155/2010/639801. hdl:10400.22/13143.
Holm, S.; Näsholm, S. P. (2011). "A causal and fractional all-frequency wave equation for lossy media". Journal of the Acoustical Society of America. 130 (4): 2195–2201. Bibcode:2011ASAJ..130.2195H. doi:10.1121/1.3631626. hdl:10852/103311. PMID 21973374. S2CID 7804006.
Näsholm, S. P.; Holm, S. (2011). "Linking multiple relaxation, power-law attenuation, and fractional wave equations". Journal of the Acoustical Society of America. 130 (5): 3038–3045. Bibcode:2011ASAJ..130.3038N. doi:10.1121/1.3641457. hdl:10852/103312. PMID 22087931. S2CID 10376751.
Näsholm, S. P.; Holm, S. (2012). "On a Fractional Zener Elastic Wave Equation". Fract. Calc. Appl. Anal. 16: 26–50. arXiv:1212.4024. doi:10.2478/s13540-013-0003-1. S2CID 120348311.
Holm, S.; Näsholm, S. P. (2013). "Comparison of fractional wave equations for power law attenuation in ultrasound and elastography". Ultrasound in Medicine & Biology. 40 (4): 695–703. arXiv:1306.6507. CiteSeerX 10.1.1.765.120. doi:10.1016/j.ultrasmedbio.2013.09.033. PMID 24433745. S2CID 11983716.
Holm, S. (2019). Waves with Power-Law Attenuation. Springer and Acoustical Society of America Press. doi:10.1007/978-3-030-14927-7. ISBN 978-3-030-14926-0. S2CID 145880744.
Pandey, Vikash; Holm, Sverre (2016-12-01). "Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations". The Journal of the Acoustical Society of America. 140 (6): 4225–4236. arXiv:1612.05557. Bibcode:2016ASAJ..140.4225P. doi:10.1121/1.4971289. ISSN 0001-4966. PMID 28039990. S2CID 29552742.
Pandey, Vikash; Holm, Sverre (2016-09-23). "Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity". Physical Review E. 94 (3): 032606. Bibcode:2016PhRvE..94c2606P. doi:10.1103/PhysRevE.94.032606. hdl:10852/53091. PMID 27739858.
Laskin, N. (2002). "Fractional Schrodinger equation". Phys. Rev. E. 66 (5): 056108. arXiv:quant-ph/0206098. Bibcode:2002PhRvE..66e6108L. CiteSeerX 10.1.1.252.6732. doi:10.1103/PhysRevE.66.056108. PMID 12513557. S2CID 7520956.
Laskin, Nick (2018). Fractional Quantum Mechanics. CiteSeerX 10.1.1.247.5449. doi:10.1142/10541. ISBN 978-981-322-379-0.
Bhrawy, A.H.; Zaky, M.A. (2017). "An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations". Applied Numerical Mathematics. 111: 197–218. doi:10.1016/j.apnum.2016.09.009.

emis.de

Katugampola, Udita N. (15 October 2014). "A New Approach To Generalized Fractional Derivatives" (PDF). Bulletin of Mathematical Analysis and Applications. 6 (4): 1–15. arXiv:1106.0965.

free.fr

s.dugowson.free.fr

For the history of the subject, see the thesis (in French): Stéphane Dugowson, Les différentielles métaphysiques (histoire et philosophie de la généralisation de l'ordre de dérivation), Thèse, Université Paris Nord (1994)

handle.net

hdl.handle.net

Valério, Duarte; Machado, José; Kiryakova, Virginia (2014-01-01). "Some pioneers of the applications of fractional calculus". Fractional Calculus and Applied Analysis. 17 (2): 552–578. doi:10.2478/s13540-014-0185-1. hdl:10400.22/5491. ISSN 1314-2224. S2CID 121482200.
de Oliveira, Edmundo Capelas; Tenreiro Machado, José António (2014-06-10). "A Review of Definitions for Fractional Derivatives and Integral". Mathematical Problems in Engineering. 2014: 1–6. doi:10.1155/2014/238459. hdl:10400.22/5497.
Aslan, İsmail (2015-01-15). "An analytic approach to a class of fractional differential-difference equations of rational type via symbolic computation". Mathematical Methods in the Applied Sciences. 38 (1): 27–36. Bibcode:2015MMAS...38...27A. doi:10.1002/mma.3047. hdl:11147/5562. S2CID 120881978.
Tenreiro Machado, J. A.; Silva, Manuel F.; Barbosa, Ramiro S.; Jesus, Isabel S.; Reis, Cecília M.; Marcos, Maria G.; Galhano, Alexandra F. (2010). "Some Applications of Fractional Calculus in Engineering". Mathematical Problems in Engineering. 2010: 1–34. doi:10.1155/2010/639801. hdl:10400.22/13143.
Holm, S.; Näsholm, S. P. (2011). "A causal and fractional all-frequency wave equation for lossy media". Journal of the Acoustical Society of America. 130 (4): 2195–2201. Bibcode:2011ASAJ..130.2195H. doi:10.1121/1.3631626. hdl:10852/103311. PMID 21973374. S2CID 7804006.
Näsholm, S. P.; Holm, S. (2011). "Linking multiple relaxation, power-law attenuation, and fractional wave equations". Journal of the Acoustical Society of America. 130 (5): 3038–3045. Bibcode:2011ASAJ..130.3038N. doi:10.1121/1.3641457. hdl:10852/103312. PMID 22087931. S2CID 10376751.
Pandey, Vikash; Holm, Sverre (2016-09-23). "Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity". Physical Review E. 94 (3): 032606. Bibcode:2016PhRvE..94c2606P. doi:10.1103/PhysRevE.94.032606. hdl:10852/53091. PMID 27739858.

harvard.edu

ui.adsabs.harvard.edu

Hermann, Richard (2014). Fractional Calculus: An Introduction for Physicists (2nd ed.). New Jersey: World Scientific Publishing. p. 46. Bibcode:2014fcip.book.....H. doi:10.1142/8934. ISBN 978-981-4551-07-6.
Al-Raeei, Marwan (2021). "Applying fractional quantum mechanics to systems with electrical screening effects". Chaos, Solitons & Fractals. 150 (September): 111209. Bibcode:2021CSF...15011209A. doi:10.1016/j.chaos.2021.111209.
Herrmann, Richard, ed. (2014). Fractional Calculus (2nd ed.). New Jersey: World Scientific Publishing Co. p. 54^{[verification needed]}. Bibcode:2014fcip.book.....H. doi:10.1142/8934. ISBN 978-981-4551-07-6. {{cite book}}: |journal= ignored (help)
Caputo, Michele (1967). "Linear model of dissipation whose Q is almost frequency independent. II". Geophysical Journal International. 13 (5): 529–539. Bibcode:1967GeoJ...13..529C. doi:10.1111/j.1365-246x.1967.tb02303.x..
Algahtani, Obaid Jefain Julaighim (2016-08-01). "Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model". Chaos, Solitons & Fractals. Nonlinear Dynamics and Complexity. 89: 552–559. Bibcode:2016CSF....89..552A. doi:10.1016/j.chaos.2016.03.026. ISSN 0960-0779.
Bayın, Selçuk Ş. (5 December 2016). "Definition of the Riesz derivative and its application to space fractional quantum mechanics". Journal of Mathematical Physics. 57 (12): 123501. arXiv:1612.03046. Bibcode:2016JMP....57l3501B. doi:10.1063/1.4968819. S2CID 119099201.
Aslan, İsmail (2015-01-15). "An analytic approach to a class of fractional differential-difference equations of rational type via symbolic computation". Mathematical Methods in the Applied Sciences. 38 (1): 27–36. Bibcode:2015MMAS...38...27A. doi:10.1002/mma.3047. hdl:11147/5562. S2CID 120881978.
Ma, Li; Li, Changpin (2017-05-11). "On hadamard fractional calculus". Fractals. 25 (3): 1750033–2980. Bibcode:2017Fract..2550033M. doi:10.1142/S0218348X17500335. ISSN 0218-348X.
Anderson, Douglas R.; Ulness, Darin J. (2015-06-01). "Properties of the Katugampola fractional derivative with potential application in quantum mechanics". Journal of Mathematical Physics. 56 (6): 063502. Bibcode:2015JMP....56f3502A. doi:10.1063/1.4922018. ISSN 0022-2488.
Kober, Hermann (1940). "On fractional integrals and derivatives". The Quarterly Journal of Mathematics. os-11 (1): 193–211. Bibcode:1940QJMat..11..193K. doi:10.1093/qmath/os-11.1.193.
Wheatcraft, Stephen W.; Meerschaert, Mark M. (October 2008). "Fractional conservation of mass" (PDF). Advances in Water Resources. 31 (10): 1377–1381. Bibcode:2008AdWR...31.1377W. doi:10.1016/j.advwatres.2008.07.004. ISSN 0309-1708.
Benson, D.; Wheatcraft, S.; Meerschaert, M. (2000). "Application of a fractional advection-dispersion equation". Water Resources Research. 36 (6): 1403–1412. Bibcode:2000WRR....36.1403B. CiteSeerX 10.1.1.1.4838. doi:10.1029/2000wr900031. S2CID 7669161.
Benson, D.; Wheatcraft, S.; Meerschaert, M. (2000). "The fractional-order governing equation of Lévy motion". Water Resources Research. 36 (6): 1413–1423. Bibcode:2000WRR....36.1413B. doi:10.1029/2000wr900032. S2CID 16579630.
Metzler, R.; Klafter, J. (2000). "The random walk's guide to anomalous diffusion: a fractional dynamics approach". Phys. Rep. 339 (1): 1–77. Bibcode:2000PhR...339....1M. doi:10.1016/s0370-1573(00)00070-3.
Mainardi, F.; Luchko, Y.; Pagnini, G. (2001). "The fundamental solution of the space-time fractional diffusion equation". Fractional Calculus and Applied Analysis. 4 (2): 153–192. arXiv:cond-mat/0702419. Bibcode:2007cond.mat..2419M.
Gorenflo, Rudolf; Mainardi, Francesco (2007). "Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk". In Rangarajan, G.; Ding, M. (eds.). Processes with Long-Range Correlations. Lecture Notes in Physics. Vol. 621. pp. 148–166. arXiv:0709.3990. Bibcode:2003LNP...621..148G. doi:10.1007/3-540-44832-2_8. ISBN 978-3-540-40129-2. S2CID 14946568.
Colbrook, Matthew J.; Ma, Xiangcheng; Hopkins, Philip F.; Squire, Jonathan (2017). "Scaling laws of passive-scalar diffusion in the interstellar medium". Monthly Notices of the Royal Astronomical Society. 467 (2): 2421–2429. arXiv:1610.06590. Bibcode:2017MNRAS.467.2421C. doi:10.1093/mnras/stx261. S2CID 20203131.
Holm, S.; Näsholm, S. P. (2011). "A causal and fractional all-frequency wave equation for lossy media". Journal of the Acoustical Society of America. 130 (4): 2195–2201. Bibcode:2011ASAJ..130.2195H. doi:10.1121/1.3631626. hdl:10852/103311. PMID 21973374. S2CID 7804006.
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msu.edu

stt.msu.edu

Wheatcraft, Stephen W.; Meerschaert, Mark M. (October 2008). "Fractional conservation of mass" (PDF). Advances in Water Resources. 31 (10): 1377–1381. Bibcode:2008AdWR...31.1377W. doi:10.1016/j.advwatres.2008.07.004. ISSN 0309-1708.

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Atangana, Abdon; Baleanu, Dumitru (2016). "New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model". Thermal Science. 20 (2): 763–769. arXiv:1602.03408. doi:10.2298/TSCI160111018A. ISSN 0354-9836.

nih.gov

pubmed.ncbi.nlm.nih.gov

Holm, S.; Näsholm, S. P. (2011). "A causal and fractional all-frequency wave equation for lossy media". Journal of the Acoustical Society of America. 130 (4): 2195–2201. Bibcode:2011ASAJ..130.2195H. doi:10.1121/1.3631626. hdl:10852/103311. PMID 21973374. S2CID 7804006.
Näsholm, S. P.; Holm, S. (2011). "Linking multiple relaxation, power-law attenuation, and fractional wave equations". Journal of the Acoustical Society of America. 130 (5): 3038–3045. Bibcode:2011ASAJ..130.3038N. doi:10.1121/1.3641457. hdl:10852/103312. PMID 22087931. S2CID 10376751.
Holm, S.; Näsholm, S. P. (2013). "Comparison of fractional wave equations for power law attenuation in ultrasound and elastography". Ultrasound in Medicine & Biology. 40 (4): 695–703. arXiv:1306.6507. CiteSeerX 10.1.1.765.120. doi:10.1016/j.ultrasmedbio.2013.09.033. PMID 24433745. S2CID 11983716.
Pandey, Vikash; Holm, Sverre (2016-12-01). "Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations". The Journal of the Acoustical Society of America. 140 (6): 4225–4236. arXiv:1612.05557. Bibcode:2016ASAJ..140.4225P. doi:10.1121/1.4971289. ISSN 0001-4966. PMID 28039990. S2CID 29552742.
Pandey, Vikash; Holm, Sverre (2016-09-23). "Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity". Physical Review E. 94 (3): 032606. Bibcode:2016PhRvE..94c2606P. doi:10.1103/PhysRevE.94.032606. hdl:10852/53091. PMID 27739858.
Laskin, N. (2002). "Fractional Schrodinger equation". Phys. Rev. E. 66 (5): 056108. arXiv:quant-ph/0206098. Bibcode:2002PhRvE..66e6108L. CiteSeerX 10.1.1.252.6732. doi:10.1103/PhysRevE.66.056108. PMID 12513557. S2CID 7520956.

psu.edu

citeseerx.ist.psu.edu

Benson, D.; Wheatcraft, S.; Meerschaert, M. (2000). "Application of a fractional advection-dispersion equation". Water Resources Research. 36 (6): 1403–1412. Bibcode:2000WRR....36.1403B. CiteSeerX 10.1.1.1.4838. doi:10.1029/2000wr900031. S2CID 7669161.
Wheatcraft, Stephen W.; Meerschaert, Mark M.; Schumer, Rina; Benson, David A. (2001-01-01). "Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests". Transport in Porous Media. 42 (1–2): 211–240. CiteSeerX 10.1.1.58.2062. doi:10.1023/A:1006733002131. ISSN 1573-1634. S2CID 189899853.
Holm, S.; Näsholm, S. P. (2013). "Comparison of fractional wave equations for power law attenuation in ultrasound and elastography". Ultrasound in Medicine & Biology. 40 (4): 695–703. arXiv:1306.6507. CiteSeerX 10.1.1.765.120. doi:10.1016/j.ultrasmedbio.2013.09.033. PMID 24433745. S2CID 11983716.
Laskin, N. (2002). "Fractional Schrodinger equation". Phys. Rev. E. 66 (5): 056108. arXiv:quant-ph/0206098. Bibcode:2002PhRvE..66e6108L. CiteSeerX 10.1.1.252.6732. doi:10.1103/PhysRevE.66.056108. PMID 12513557. S2CID 7520956.
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sciencedirect.com

Al-Raeei, Marwan (2021). "Applying fractional quantum mechanics to systems with electrical screening effects". Chaos, Solitons & Fractals. 150 (September): 111209. Bibcode:2021CSF...15011209A. doi:10.1016/j.chaos.2021.111209.
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semanticscholar.org

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Podlubny, Igor; Magin, Richard L.; Trymorush, Irina (2017). "Niels Henrik Abel and the birth of fractional calculus". Fractional Calculus and Applied Analysis. 20 (5): 1068–1075. arXiv:1802.05441. doi:10.1515/fca-2017-0057. S2CID 119664694.
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Holm, S.; Näsholm, S. P. (2011). "A causal and fractional all-frequency wave equation for lossy media". Journal of the Acoustical Society of America. 130 (4): 2195–2201. Bibcode:2011ASAJ..130.2195H. doi:10.1121/1.3631626. hdl:10852/103311. PMID 21973374. S2CID 7804006.
Näsholm, S. P.; Holm, S. (2011). "Linking multiple relaxation, power-law attenuation, and fractional wave equations". Journal of the Acoustical Society of America. 130 (5): 3038–3045. Bibcode:2011ASAJ..130.3038N. doi:10.1121/1.3641457. hdl:10852/103312. PMID 22087931. S2CID 10376751.
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Holm, S.; Näsholm, S. P. (2013). "Comparison of fractional wave equations for power law attenuation in ultrasound and elastography". Ultrasound in Medicine & Biology. 40 (4): 695–703. arXiv:1306.6507. CiteSeerX 10.1.1.765.120. doi:10.1016/j.ultrasmedbio.2013.09.033. PMID 24433745. S2CID 11983716.
Holm, S. (2019). Waves with Power-Law Attenuation. Springer and Acoustical Society of America Press. doi:10.1007/978-3-030-14927-7. ISBN 978-3-030-14926-0. S2CID 145880744.
Pandey, Vikash; Holm, Sverre (2016-12-01). "Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations". The Journal of the Acoustical Society of America. 140 (6): 4225–4236. arXiv:1612.05557. Bibcode:2016ASAJ..140.4225P. doi:10.1121/1.4971289. ISSN 0001-4966. PMID 28039990. S2CID 29552742.
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springer.com

link.springer.com

Holm, S. (2019). Waves with Power-Law Attenuation. Springer and Acoustical Society of America Press. doi:10.1007/978-3-030-14927-7. ISBN 978-3-030-14926-0. S2CID 145880744.

worldcat.org

Valério, Duarte; Machado, José; Kiryakova, Virginia (2014-01-01). "Some pioneers of the applications of fractional calculus". Fractional Calculus and Applied Analysis. 17 (2): 552–578. doi:10.2478/s13540-014-0185-1. hdl:10400.22/5491. ISSN 1314-2224. S2CID 121482200.
Algahtani, Obaid Jefain Julaighim (2016-08-01). "Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model". Chaos, Solitons & Fractals. Nonlinear Dynamics and Complexity. 89: 552–559. Bibcode:2016CSF....89..552A. doi:10.1016/j.chaos.2016.03.026. ISSN 0960-0779.
Atangana, Abdon; Baleanu, Dumitru (2016). "New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model". Thermal Science. 20 (2): 763–769. arXiv:1602.03408. doi:10.2298/TSCI160111018A. ISSN 0354-9836.
Ma, Li; Li, Changpin (2017-05-11). "On hadamard fractional calculus". Fractals. 25 (3): 1750033–2980. Bibcode:2017Fract..2550033M. doi:10.1142/S0218348X17500335. ISSN 0218-348X.
Anderson, Douglas R.; Ulness, Darin J. (2015-06-01). "Properties of the Katugampola fractional derivative with potential application in quantum mechanics". Journal of Mathematical Physics. 56 (6): 063502. Bibcode:2015JMP....56f3502A. doi:10.1063/1.4922018. ISSN 0022-2488.
Caputo, Michele; Fabrizio, Mauro (2016-01-01). "Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels". Progress in Fractional Differentiation and Applications. 2 (1): 1–11. doi:10.18576/pfda/020101. ISSN 2356-9336.
Wheatcraft, Stephen W.; Meerschaert, Mark M. (October 2008). "Fractional conservation of mass" (PDF). Advances in Water Resources. 31 (10): 1377–1381. Bibcode:2008AdWR...31.1377W. doi:10.1016/j.advwatres.2008.07.004. ISSN 0309-1708.
Wheatcraft, Stephen W.; Meerschaert, Mark M.; Schumer, Rina; Benson, David A. (2001-01-01). "Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests". Transport in Porous Media. 42 (1–2): 211–240. CiteSeerX 10.1.1.58.2062. doi:10.1023/A:1006733002131. ISSN 1573-1634. S2CID 189899853.
Pandey, Vikash; Holm, Sverre (2016-12-01). "Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations". The Journal of the Acoustical Society of America. 140 (6): 4225–4236. arXiv:1612.05557. Bibcode:2016ASAJ..140.4225P. doi:10.1121/1.4971289. ISSN 0001-4966. PMID 28039990. S2CID 29552742.

worldscientific.com

Khalili Golmankhaneh, Alireza (2022). Fractal Calculus and its Applications. Singapore: World Scientific Pub Co Inc. p. 328. doi:10.1142/12988. ISBN 978-981-126-110-7. S2CID 248575991.