Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena

. To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamental differences between the power law, exponential decay, Mittag-Leffler law and their possible applications in nature....

Full description

Saved in:

Bibliographic Details
Published in:	European physical journal plus Vol. 133; no. 4; p. 166
Main Authors:	Atangana, Abdon, Gómez-Aguilar, J. F.
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2018 Springer Nature B.V
Subjects:	Applied and Technical Physics Associativity Atomic Calculus Commutativity Complex Systems Condensed Matter Physics Derivatives Differential equations Focus Point on Modelling Complex Real-World Problems with Fractal and New Trends of Fractional Differentiation Fractional calculus Group theory Mathematical analysis Mathematical and Computational Physics Mathematical models Molecular Operators (mathematics) Optical and Plasma Physics Physics Physics and Astronomy Power law Principles Regular Article Statistical analysis Steady state Theoretical
ISSN:	2190-5444, 2190-5444
Online Access:	Get full text
Tags:	Add Tag No Tags, Be the first to tag this record!

Abstract	. To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamental differences between the power law, exponential decay, Mittag-Leffler law and their possible applications in nature. We demonstrate the failure of the semi-group principle in modeling real-world problems. We use natural phenomena to illustrate the importance of non-commutative and non-associative operators under which the Caputo-Fabrizio and Atangana-Baleanu fractional operators fall. We present statistical properties of generator for each fractional derivative, including Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu ones. The Atangana-Baleanu and Caputo-Fabrizio fractional derivatives show crossover properties for the mean-square displacement, while the Riemann-Liouville is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the Caputo Fabrizio kernel has a steady state between the transition. Only the Atangana-Baleanu kernel is a crossover for the waiting time distribution from stretched exponential to power law. A new criterion was suggested, namely the Atangana-Gómez fractional bracket, that helps describe the energy needed by a fractional derivative to characterize a 2-pletic manifold. Based on these properties, we classified fractional derivatives in three categories: weak, mild and strong fractional differential and integral operators. We presented some applications of fractional differential operators to describe real-world problems and we proved, with numerical simulations, that the Riemann-Liouville power-law derivative provides a description of real-world problems with much additional information, that can be seen as noise or error due to specific memory properties of its power-law kernel. The Caputo-Fabrizio derivative is less noisy while the Atangana-Baleanu fractional derivative provides an excellent description, due to its Mittag-Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state. The study suggests that the properties of associativity and commutativity or the semi-group principle are just irrelevant in fractional calculus. Properties of classical derivatives were established for the ordinary calculus with no memory effect and it is a failure of mathematical investigation to attempt to describe more complex natural phenomena using the same notions.
AbstractList	. To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamental differences between the power law, exponential decay, Mittag-Leffler law and their possible applications in nature. We demonstrate the failure of the semi-group principle in modeling real-world problems. We use natural phenomena to illustrate the importance of non-commutative and non-associative operators under which the Caputo-Fabrizio and Atangana-Baleanu fractional operators fall. We present statistical properties of generator for each fractional derivative, including Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu ones. The Atangana-Baleanu and Caputo-Fabrizio fractional derivatives show crossover properties for the mean-square displacement, while the Riemann-Liouville is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the Caputo Fabrizio kernel has a steady state between the transition. Only the Atangana-Baleanu kernel is a crossover for the waiting time distribution from stretched exponential to power law. A new criterion was suggested, namely the Atangana-Gómez fractional bracket, that helps describe the energy needed by a fractional derivative to characterize a 2-pletic manifold. Based on these properties, we classified fractional derivatives in three categories: weak, mild and strong fractional differential and integral operators. We presented some applications of fractional differential operators to describe real-world problems and we proved, with numerical simulations, that the Riemann-Liouville power-law derivative provides a description of real-world problems with much additional information, that can be seen as noise or error due to specific memory properties of its power-law kernel. The Caputo-Fabrizio derivative is less noisy while the Atangana-Baleanu fractional derivative provides an excellent description, due to its Mittag-Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state. The study suggests that the properties of associativity and commutativity or the semi-group principle are just irrelevant in fractional calculus. Properties of classical derivatives were established for the ordinary calculus with no memory effect and it is a failure of mathematical investigation to attempt to describe more complex natural phenomena using the same notions. To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamental differences between the power law, exponential decay, Mittag-Leffler law and their possible applications in nature. We demonstrate the failure of the semi-group principle in modeling real-world problems. We use natural phenomena to illustrate the importance of non-commutative and non-associative operators under which the Caputo-Fabrizio and Atangana-Baleanu fractional operators fall. We present statistical properties of generator for each fractional derivative, including Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu ones. The Atangana-Baleanu and Caputo-Fabrizio fractional derivatives show crossover properties for the mean-square displacement, while the Riemann-Liouville is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the Caputo Fabrizio kernel has a steady state between the transition. Only the Atangana-Baleanu kernel is a crossover for the waiting time distribution from stretched exponential to power law. A new criterion was suggested, namely the Atangana-Gómez fractional bracket, that helps describe the energy needed by a fractional derivative to characterize a 2-pletic manifold. Based on these properties, we classified fractional derivatives in three categories: weak, mild and strong fractional differential and integral operators. We presented some applications of fractional differential operators to describe real-world problems and we proved, with numerical simulations, that the Riemann-Liouville power-law derivative provides a description of real-world problems with much additional information, that can be seen as noise or error due to specific memory properties of its power-law kernel. The Caputo-Fabrizio derivative is less noisy while the Atangana-Baleanu fractional derivative provides an excellent description, due to its Mittag-Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state. The study suggests that the properties of associativity and commutativity or the semi-group principle are just irrelevant in fractional calculus. Properties of classical derivatives were established for the ordinary calculus with no memory effect and it is a failure of mathematical investigation to attempt to describe more complex natural phenomena using the same notions.
ArticleNumber	166
Author	Atangana, Abdon Gómez-Aguilar, J. F.
Author_xml	– sequence: 1 givenname: Abdon surname: Atangana fullname: Atangana, Abdon email: AtanganaA@ufs.ac.za organization: Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State – sequence: 2 givenname: J. F. surname: Gómez-Aguilar fullname: Gómez-Aguilar, J. F. organization: CONACyT-Tecnológico Nacional de México/CENIDET
BookMark	eNp9kEtPxCAYRYkZE8fHH3BF4roKlLHUnY7PZBI3uiZfKShjCxWoxn8vM6PRuJAF3JB7vsDZRRPnnUbokJJjSjk50cNyOLGMUFFQRhgtyi00ZbQmxYxzPvmVd9BBjEuSF68pr_kUvV9q5TvvbIRkvcPeYBNArTJ0WEGnxm6MOIydjmf4Imh4se4JK9_3Y8rIm00fGFyLIUav7PdN8pkd0hg07n3eHOScBw7P2vleO9hH2wa6qA--zj30eH31ML8tFvc3d_PzRaFKWqeiIS0VipmWNQpK1oiqqbhq60YRLojJmYvSMCaAAJCKUQFtTasZzKgyRtNyDx1t5g7Bv446Jrn0Y8h_i5LVjJyyU1GVuSU2LRV8jEEbqWxaC0kBbCcpkSvTcmVark3LtWm5QtkfdAi2h_DxP1RuoJjL7kmHn1f9Q30C6peYzw
CitedBy_id	crossref_primary_10_1186_s13662_018_1696_6 crossref_primary_10_1155_2018_6162043 crossref_primary_10_1016_j_nahs_2018_12_001 crossref_primary_10_3390_fractalfract8070399 crossref_primary_10_1140_epjp_i2018_12072_4 crossref_primary_10_1186_s13662_021_03494_7 crossref_primary_10_1016_j_chaos_2018_08_003 crossref_primary_10_1155_2020_4867927 crossref_primary_10_1002_mma_5421 crossref_primary_10_1016_j_physa_2018_05_137 crossref_primary_10_1155_2019_4178073 crossref_primary_10_1016_j_chaos_2019_03_022 crossref_primary_10_1016_j_chaos_2019_109588 crossref_primary_10_1371_journal_pone_0223476 crossref_primary_10_1186_s13662_020_03140_8 crossref_primary_10_1016_j_physa_2019_01_094 crossref_primary_10_1140_epjs_s11734_022_00460_6 crossref_primary_10_1140_epjp_s13360_020_00558_7 crossref_primary_10_1016_j_chaos_2020_110573 crossref_primary_10_1016_j_jfranklin_2018_11_060 crossref_primary_10_1016_j_isatra_2020_11_021 crossref_primary_10_1002_mma_7164 crossref_primary_10_1002_mma_11111 crossref_primary_10_1515_geo_2022_0446 crossref_primary_10_1016_j_chaos_2019_03_038 crossref_primary_10_1140_epjp_i2019_12730_y crossref_primary_10_1142_S0218348X25401188 crossref_primary_10_1155_2020_1274251 crossref_primary_10_1016_j_chaos_2019_07_057 crossref_primary_10_1155_2021_6662808 crossref_primary_10_1016_j_chaos_2019_07_019 crossref_primary_10_1016_j_cjph_2020_02_003 crossref_primary_10_1016_j_chaos_2019_06_014 crossref_primary_10_1016_j_chaos_2019_06_009 crossref_primary_10_1186_s13662_020_02993_3 crossref_primary_10_3390_math7020200 crossref_primary_10_1140_epjp_i2019_12765_0 crossref_primary_10_1007_s00034_020_01447_1 crossref_primary_10_1002_asjc_3838 crossref_primary_10_1016_j_chaos_2019_07_023 crossref_primary_10_1016_j_amc_2019_124735 crossref_primary_10_1007_s41478_020_00292_4 crossref_primary_10_1186_s13662_020_02709_7 crossref_primary_10_1016_j_chaos_2019_06_005 crossref_primary_10_1371_journal_pone_0218425 crossref_primary_10_1080_02286203_2024_2377898 crossref_primary_10_1140_epjp_i2019_12507_4 crossref_primary_10_1007_s12190_025_02519_8 crossref_primary_10_1016_j_chaos_2019_109491 crossref_primary_10_1016_j_cnsns_2021_106076 crossref_primary_10_1038_s41598_025_88310_y crossref_primary_10_1002_mma_5754 crossref_primary_10_3390_math8060923 crossref_primary_10_1002_mma_7816 crossref_primary_10_1007_s00366_020_01185_7 crossref_primary_10_1186_s13662_019_2138_9 crossref_primary_10_1007_s10973_020_09700_0 crossref_primary_10_1016_j_chaos_2020_109695 crossref_primary_10_3390_math10173071 crossref_primary_10_1186_s13662_020_2514_5 crossref_primary_10_1140_epjp_i2018_12120_1 crossref_primary_10_1002_num_22577 crossref_primary_10_1016_j_bspc_2019_101584 crossref_primary_10_1016_j_chaos_2018_09_009 crossref_primary_10_1016_j_chaos_2018_10_010 crossref_primary_10_1016_j_chaos_2019_06_037 crossref_primary_10_1016_j_chaos_2022_112285 crossref_primary_10_1155_2020_8890820 crossref_primary_10_1155_2021_2643572 crossref_primary_10_1007_s41478_022_00419_9 crossref_primary_10_1186_s13661_019_1194_0 crossref_primary_10_1002_mma_5981 crossref_primary_10_1016_j_cam_2020_112811 crossref_primary_10_1016_j_amc_2019_05_029 crossref_primary_10_3934_math_2025131 crossref_primary_10_1007_s00034_019_01211_0 crossref_primary_10_1007_s00366_019_00843_9 crossref_primary_10_1016_j_chaos_2018_09_013 crossref_primary_10_1016_j_chaos_2019_109541 crossref_primary_10_33889_IJMEMS_2025_10_4_043 crossref_primary_10_1016_j_physa_2019_04_024 crossref_primary_10_1080_02286203_2021_2015818 crossref_primary_10_1016_j_chaos_2020_110095 crossref_primary_10_1016_j_chaos_2018_10_007 crossref_primary_10_1002_mma_8325 crossref_primary_10_1016_j_matcom_2020_11_017 crossref_primary_10_3390_fractalfract2040024 crossref_primary_10_1016_j_chaos_2019_109557 crossref_primary_10_1016_j_chaos_2018_09_021 crossref_primary_10_32604_cmes_2022_023694 crossref_primary_10_1002_cta_3230 crossref_primary_10_1016_j_chaos_2020_110418 crossref_primary_10_1016_j_chaos_2018_09_026 crossref_primary_10_1186_s13662_018_1903_5 crossref_primary_10_1140_epjs_s11734_022_00456_2 crossref_primary_10_1007_s10973_019_08992_1 crossref_primary_10_1038_s41598_025_92195_2 crossref_primary_10_1016_j_chaos_2018_08_021 crossref_primary_10_1140_epjp_i2019_12861_1 crossref_primary_10_1016_j_chaos_2019_05_013 crossref_primary_10_1186_s13662_018_1855_9 crossref_primary_10_1002_mma_7228 crossref_primary_10_1186_s13662_019_2336_5 crossref_primary_10_1016_j_chaos_2018_09_033 crossref_primary_10_2298_TSCI23S1287P crossref_primary_10_1016_j_chaos_2018_09_038 crossref_primary_10_1016_j_chaos_2018_09_039 crossref_primary_10_1177_1077546320908412 crossref_primary_10_1016_j_chaos_2018_10_025 crossref_primary_10_1186_s13662_020_03042_9 crossref_primary_10_1016_j_chaos_2018_10_020 crossref_primary_10_1016_j_matcom_2021_02_002 crossref_primary_10_3233_ASY_221811 crossref_primary_10_1007_s00034_022_02213_1 crossref_primary_10_1016_j_chaos_2019_05_008 crossref_primary_10_1140_epjp_i2018_12315_4 crossref_primary_10_1140_epjp_i2018_12160_5 crossref_primary_10_1016_j_chaos_2020_109774 crossref_primary_10_1016_j_chaos_2018_09_043 crossref_primary_10_1155_2021_6636882 crossref_primary_10_1016_j_chaos_2018_09_047 crossref_primary_10_1016_j_heliyon_2024_e34160 crossref_primary_10_1038_s41598_022_21372_4 crossref_primary_10_1016_j_chaos_2023_114266 crossref_primary_10_1088_1402_4896_ac0c58 crossref_primary_10_1002_mma_6483 crossref_primary_10_1016_j_chaos_2020_110062 crossref_primary_10_3233_JCM_190029 crossref_primary_10_1007_s00366_021_01367_x crossref_primary_10_1016_j_chaos_2018_11_025 crossref_primary_10_1002_mma_6477 crossref_primary_10_1016_j_cjph_2020_08_019 crossref_primary_10_1016_j_chaos_2020_109867 crossref_primary_10_1155_2020_8709393 crossref_primary_10_3390_e22080824 crossref_primary_10_1088_1402_4896_abf67c crossref_primary_10_1016_j_physa_2019_03_085 crossref_primary_10_1002_asjc_2231 crossref_primary_10_1002_asjc_3680 crossref_primary_10_1186_s13662_020_02729_3 crossref_primary_10_3390_fractalfract4020015 crossref_primary_10_1140_epjp_i2018_12186_7 crossref_primary_10_1016_j_jksus_2020_101246 crossref_primary_10_1016_j_chaos_2020_110171 crossref_primary_10_1016_j_chaos_2018_11_035 crossref_primary_10_1007_s00366_019_00852_8 crossref_primary_10_1016_j_chaos_2018_07_033 crossref_primary_10_2298_TSCI2406169A crossref_primary_10_1016_j_camwa_2019_03_043 crossref_primary_10_1016_j_chaos_2020_109991 crossref_primary_10_1016_j_chaos_2018_07_036 crossref_primary_10_1016_j_chaos_2022_112269 crossref_primary_10_1016_j_cjph_2019_08_014 crossref_primary_10_1016_j_chaos_2022_112822 crossref_primary_10_3390_e23050610 crossref_primary_10_1016_j_physa_2019_01_102 crossref_primary_10_1155_cmm4_9946126 crossref_primary_10_1016_j_egyr_2018_09_009 crossref_primary_10_1016_j_physa_2019_01_105 crossref_primary_10_1002_mma_7430 crossref_primary_10_1186_s13662_018_1785_6 crossref_primary_10_1002_mma_6459 crossref_primary_10_1002_mma_7305 crossref_primary_10_1016_j_chaos_2018_11_009 crossref_primary_10_1016_j_physa_2019_03_069 crossref_primary_10_1016_j_chaos_2018_07_022 crossref_primary_10_1016_j_chaos_2019_109401 crossref_primary_10_1016_j_ijheatmasstransfer_2020_119592 crossref_primary_10_3389_fphy_2023_1178154 crossref_primary_10_1002_asjc_3661 crossref_primary_10_1016_j_chaos_2019_109405 crossref_primary_10_1140_epjp_i2019_12878_4 crossref_primary_10_1140_epjp_i2019_13009_1 crossref_primary_10_1016_j_comcom_2020_11_014 crossref_primary_10_1016_j_dsp_2021_103305 crossref_primary_10_1140_epjp_i2019_12939_8 crossref_primary_10_1186_s13662_019_2024_5 crossref_primary_10_1155_2023_5322092 crossref_primary_10_3390_fractalfract6070348 crossref_primary_10_1088_1402_4896_ad20ba crossref_primary_10_1002_mma_6568 crossref_primary_10_1088_1402_4896_acb591 crossref_primary_10_1016_j_physa_2019_02_018 crossref_primary_10_1016_j_chaos_2020_109970 crossref_primary_10_1016_j_physa_2019_121085 crossref_primary_10_1016_j_chaos_2018_09_001 crossref_primary_10_1016_j_chaos_2018_11_017 crossref_primary_10_1016_j_chaos_2022_112812 crossref_primary_10_1016_j_chaos_2022_112204 crossref_primary_10_3390_axioms9030095 crossref_primary_10_1007_s00419_024_02676_5 crossref_primary_10_1186_s13662_021_03262_7 crossref_primary_10_1109_ACCESS_2021_3077561 crossref_primary_10_1016_j_chaos_2019_04_029 crossref_primary_10_1016_j_ijmecsci_2020_105902 crossref_primary_10_3390_math13030536 crossref_primary_10_1016_j_chaos_2018_06_032 crossref_primary_10_1016_j_physa_2018_08_033 crossref_primary_10_3390_axioms12040321 crossref_primary_10_1002_mma_7766 crossref_primary_10_1515_fca_2019_0017 crossref_primary_10_1016_j_chaos_2019_07_003 crossref_primary_10_1016_j_chaos_2022_111915 crossref_primary_10_1186_s13662_020_2527_0 crossref_primary_10_1016_j_ijmecsci_2019_105204 crossref_primary_10_1002_num_22983 crossref_primary_10_1016_j_chaos_2019_07_002 crossref_primary_10_1038_s41598_024_56944_z crossref_primary_10_3390_en13215768 crossref_primary_10_1016_j_chaos_2020_110373 crossref_primary_10_1038_s41598_023_45773_1 crossref_primary_10_3390_axioms13020133 crossref_primary_10_1002_mma_9264 crossref_primary_10_1016_j_apnum_2020_11_020 crossref_primary_10_1140_epjp_i2019_12777_8 crossref_primary_10_1002_mma_6423 crossref_primary_10_3390_fractalfract7060473 crossref_primary_10_1002_num_22518 crossref_primary_10_1016_j_chaos_2019_07_015 crossref_primary_10_1016_j_chaos_2019_07_014 crossref_primary_10_1016_j_chaos_2019_07_017 crossref_primary_10_1016_j_chaos_2019_07_010 crossref_primary_10_1016_j_chaos_2019_07_013 crossref_primary_10_1155_2022_1899130 crossref_primary_10_1088_1402_4896_ad7f98 crossref_primary_10_1016_j_chaos_2023_113399 crossref_primary_10_1002_mma_7192 crossref_primary_10_1007_s12043_021_02224_8 crossref_primary_10_1016_j_chaos_2019_08_018 crossref_primary_10_1177_1687814019866540 crossref_primary_10_1177_10775463221135912 crossref_primary_10_1016_j_chaos_2019_03_006 crossref_primary_10_1007_s11868_025_00708_4 crossref_primary_10_1016_j_fss_2021_04_012 crossref_primary_10_1007_s11044_020_09752_y crossref_primary_10_1140_epjp_i2019_12499_y crossref_primary_10_1016_j_chaos_2021_110891 crossref_primary_10_1016_j_jafrearsci_2022_104753 crossref_primary_10_1016_j_physa_2019_122896 crossref_primary_10_1016_j_physa_2019_123864 crossref_primary_10_3390_e20090649 crossref_primary_10_1016_j_chaos_2018_12_015 crossref_primary_10_1088_1402_4896_acd27d crossref_primary_10_1140_epjp_i2019_13003_7 crossref_primary_10_1007_s00034_021_01942_z crossref_primary_10_1016_j_chaos_2018_12_019 crossref_primary_10_1016_j_cjph_2019_09_005 crossref_primary_10_1002_cta_2640 crossref_primary_10_1088_1402_4896_ac1218
Cites_doi	10.1515/fca-2015-0026 10.1007/BF00642162 10.1016/j.jcp.2014.07.019 10.1007/s00022-005-0009-x 10.1142/S0129055X03001886 10.1017/S1446788700036636 10.1016/j.chaos.2005.02.023 10.1101/lm.033118.113 10.1103/PhysRevLett.30.1343 10.1016/j.aml.2006.08.013 10.1080/04353684.1986.11879523 10.1103/PhysRevLett.94.151602 10.1007/978-3-319-63949-9_16 10.1002/num.22195 10.1080/0022250X.2017.1356828 10.1016/j.camwa.2009.07.050 10.1023/B:NUMA.0000027736.85078.be 10.1103/RevModPhys.73.977 10.1007/s00220-009-0951-9 10.1007/BF01397280 10.3389/fphy.2017.00052 10.2298/TSCI160111018A 10.1017/CBO9780511524479 10.1007/BF02104515 10.1002/num.22219 10.2298/TSCI160229115H 10.2174/9781681085999118010013 10.18576/pfda/020101 10.1016/j.chaos.2016.03.020 10.1016/j.gmod.2004.01.003 10.1007/s12597-013-0169-7 10.1142/1407 10.1016/j.cnsns.2013.04.001 10.1007/BF00050786 10.1016/B978-1-4832-5663-4.50009-0 10.12785/jsap/010102 10.1103/PhysRevLett.109.100404 10.2298/TSCI15S1S31A 10.1016/0168-9274(92)90017-8 10.1037/11183-000 10.1016/j.chaos.2016.02.012 10.1140/epjp/i2017-11798-7 10.1007/s004410051245 10.1111/j.1365-246X.1967.tb02303.x 10.1016/j.chaos.2016.12.025
ContentType	Journal Article
Copyright	Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018 Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018.
Copyright_xml	– notice: Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018 – notice: Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018.
DBID	AAYXX CITATION 8FE 8FG AEUYN AFKRA ARAPS BENPR BGLVJ BHPHI BKSAR CCPQU DWQXO HCIFZ P5Z P62 PCBAR PHGZM PHGZT PKEHL PQEST PQGLB PQQKQ PQUKI
DOI	10.1140/epjp/i2018-12021-3
DatabaseName	CrossRef ProQuest SciTech Collection ProQuest Technology Collection ProQuest One Sustainability (subscription) ProQuest Central UK/Ireland Advanced Technologies & Computer Science Collection ProQuest Central Technology collection Natural Science Collection ProQuest SciTech Premium Collection‎ Natural Science Collection Earth, Atmospheric & Aquatic Science Collection ProQuest One Community College ProQuest Central SciTech Premium Collection ProQuest advanced technologies & aerospace journals ProQuest Advanced Technologies & Aerospace Collection Earth, Atmospheric & Aquatic Science Database ProQuest Central Premium ProQuest One Academic ProQuest One Academic Middle East (New) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic (retired) ProQuest One Academic UKI Edition
DatabaseTitle	CrossRef Advanced Technologies & Aerospace Collection Technology Collection ProQuest One Academic Middle East (New) ProQuest Advanced Technologies & Aerospace Collection ProQuest One Academic Eastern Edition Earth, Atmospheric & Aquatic Science Database SciTech Premium Collection ProQuest One Community College ProQuest Technology Collection ProQuest SciTech Collection Earth, Atmospheric & Aquatic Science Collection ProQuest Central Advanced Technologies & Aerospace Database ProQuest One Applied & Life Sciences ProQuest One Sustainability ProQuest One Academic UKI Edition Natural Science Collection ProQuest Central Korea ProQuest Central (New) ProQuest One Academic ProQuest One Academic (New)
DatabaseTitleList	Advanced Technologies & Aerospace Collection
Database_xml	– sequence: 1 dbid: P5Z name: Advanced Technologies & Aerospace Database url: https://search.proquest.com/hightechjournals sourceTypes: Aggregation Database
DeliveryMethod	fulltext_linktorsrc
Discipline	Physics
EISSN	2190-5444
ExternalDocumentID	10_1140_epjp_i2018_12021_3
GroupedDBID	-5F -5G -BR -EM -~C 06D 0R~ 203 29~ 2JN 2KG 30V 4.4 406 408 8UJ 95. 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH AAZMS ABAKF ABDZT ABECU ABFTV ABHLI ABJNI ABJOX ABKCH ABMQK ABQBU ABSXP ABTEG ABTHY ABTKH ABTMW ABXPI ACAOD ACDTI ACGFS ACHSB ACKNC ACMDZ ACMLO ACOKC ACPIV ACREN ACZOJ ADHHG ADINQ ADKNI ADKPE ADURQ ADYFF ADZKW AEFQL AEGNC AEJHL AEJRE AEKMD AEMSY AENEX AEOHA AEPYU AESKC AETCA AEUYN AEVLU AEXYK AFBBN AFKRA AFQWF AFWTZ AFZKB AGAYW AGDGC AGMZJ AGQEE AGQMX AGRTI AGWZB AGYKE AHAVH AHBYD AHYZX AIAKS AIGIU AIIXL AILAN AITGF AJRNO AJZVZ ALFXC ALMA_UNASSIGNED_HOLDINGS AMKLP AMXSW AMYLF AMYQR ANMIH AOCGG ARAPS ARMRJ AXYYD AYJHY BENPR BGLVJ BGNMA BHPHI BKSAR CCPQU CSCUP DDRTE DNIVK DPUIP EBLON EBS EIOEI EJD ESBYG FERAY FFXSO FIGPU FINBP FNLPD FRRFC FSGXE GGCAI GGRSB GJIRD GNWQR GQ6 GQ7 HCIFZ HMJXF HRMNR HZ~ I0C IKXTQ IWAJR IXD J-C JBSCW JZLTJ KOV LLZTM M4Y NPVJJ NQJWS NU0 O93 O9J P9T PCBAR PT4 RID RLLFE ROL RSV S27 S3B SHX SISQX SJYHP SNE SNPRN SNX SOHCF SOJ SPH SPISZ SRMVM SSLCW STPWE SZN T13 TSG U2A UG4 UOJIU UTJUX UZXMN VC2 VFIZW W48 WK8 Z7S Z7Y ZMTXR ~A9 2VQ AAPKM AAYXX ABBRH ABDBE ABFSG ABRTQ ACSTC ADQRH AEBTG AEZWR AFDZB AFFHD AFHIU AFOHR AGJBK AHPBZ AHSBF AHWEU AIXLP AJBLW ATHPR AYFIA CITATION O9- PHGZM PHGZT PQGLB S1Z 8FE 8FG DWQXO P62 PKEHL PQEST PQQKQ PQUKI
ID	FETCH-LOGICAL-c319t-b0d18c2fd2bca32b87b74cd9bc0480f74c483f228a0aa07218ad9175a51cffe13
IEDL.DBID	BENPR
ISICitedReferencesCount	347
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000431462500004&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	2190-5444
IngestDate	Wed Nov 05 09:05:34 EST 2025 Tue Nov 18 22:26:13 EST 2025 Sat Nov 29 03:58:10 EST 2025 Fri Feb 21 02:31:39 EST 2025
IsPeerReviewed	true
IsScholarly	true
Issue	4
Language	English
LinkModel	DirectLink
MergedId	FETCHMERGED-LOGICAL-c319t-b0d18c2fd2bca32b87b74cd9bc0480f74c483f228a0aa07218ad9175a51cffe13
Notes	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
PQID	2920626873
PQPubID	2044220
ParticipantIDs	proquest_journals_2920626873 crossref_citationtrail_10_1140_epjp_i2018_12021_3 crossref_primary_10_1140_epjp_i2018_12021_3 springer_journals_10_1140_epjp_i2018_12021_3
PublicationCentury	2000
PublicationDate	2018-04-01
PublicationDateYYYYMMDD	2018-04-01
PublicationDate_xml	– month: 04 year: 2018 text: 2018-04-01 day: 01
PublicationDecade	2010
PublicationPlace	Berlin/Heidelberg
PublicationPlace_xml	– name: Berlin/Heidelberg – name: Heidelberg
PublicationTitle	European physical journal plus
PublicationTitleAbbrev	Eur. Phys. J. Plus
PublicationYear	2018
Publisher	Springer Berlin Heidelberg Springer Nature B.V
Publisher_xml	– name: Springer Berlin Heidelberg – name: Springer Nature B.V
References	J. Hristov, Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, in Frontiers in Fractional Calculus (Bentham Science Publishers, 2017) pp. 235--295 BrouckeR.Astrophys. Space Sci.198072331980Ap&SS..72...33B58866210.1007/BF00642162 PillaiR.N.Ann. Inst. Stat. Math.19904215710.1007/BF00050786 CantrijnF.IbortA.De-LeonM.J. Aust. Math. Soc. A19996630310.1017/S1446788700036636 CaputoM.FabrizioM.Progr. Fract. Differ. Appl.20162110.18576/pfda/020101 El-SayedA.M.A.El-MesiryA.E.M.El-SakaH.A.A.Appl. Math. Lett.200720817231471510.1016/j.aml.2006.08.013 J. Filonik, Athenian Impiety Trials: A Reappraisal (Dike, 2013) Morales-DelgadoV.F.Gómez-AguilarJ.F.Taneco-HernándezM.A.Eur. Phys. J. Plus201713252710.1140/epjp/i2017-11798-7 Gómez-AguilarJ.F.J. Math. Sociol.201741172370496510.1080/0022250X.2017.1356828 TarasovV.E.Nonlinear Sci. Numer. Simul.201318294510.1016/j.cnsns.2013.04.001 BiswasA.ShapiroV.Graph. Models20046613310.1016/j.gmod.2004.01.003 DouglasM.R.NekrasovN.A.Rev. Mod. Phys.2001739772001RvMP...73..977D10.1103/RevModPhys.73.977 TateishiA.A.RibeiroH.V.LenziE.K.Front. Phys.201755210.3389/fphy.2017.00052 CaputoM.Geophys. J. Int.1967135291967GeoJI..13..529C10.1111/j.1365-246X.1967.tb02303.x DiethelmK.FordN.J.FreedA.D.Numer. Algorithms200436312004NuAlg..36...31D206357210.1023/B:NUMA.0000027736.85078.be J.R. Kantor, The Scientific Evolution of Psychology (Principia Press, Chicago, 1963) KnightR.D.J. Geom.200583137219323210.1007/s00022-005-0009-x G.B. Folland, Advanced Calculus (Prentice Hall, 2002) PitkänenM.Prespacetime J.2016766 KumarC.S.NairB.U.OPSEARCH20155286332037610.1007/s12597-013-0169-7 AtanganaA.DumitruB.Therm. Sci.20162076310.2298/TSCI160111018A KumarC.S.NairB.U.J. Stat. Appl.2011623 S. Okubo, Introduction to Octonion and Other Non-associative Algebras in Physics (Cambridge University Press, 1995) AlkahtaniB.S.T.Chaos, Solitons Fractals2016895472016CSF....89..547A350706310.1016/j.chaos.2016.03.020 Gómez-AguilarJ.F.Chaos, Solitons Fractals2017951792017CSF....95..179G359751010.1016/j.chaos.2016.12.025 AtanganaA.KocaI.Chaos, Solitons Fractals2016894472016CSF....89..447A350705210.1016/j.chaos.2016.02.012 CaputoM.FabricioM.Progr. Fract. Differ. Appl.2015173 T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore/New Jersey, 1992) R. Sorabji (Editor), Aristotle Transformed (Bloomsburg Academic, London, 1990) W. Blaschke, Differentialgeometrie der Kreise und Kugeln, in Vorlesungen über Differentialgeometrie, Grundlehren der Mathematischen Wissenschaften (Springer, Berlin, 1929) E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies, 4th ed. (Dover Publications, New York, 1937) A. Atangana, J.F. Gómez-Aguilar, Numer. Methods Part. Differ. Equ. (2017) https://doi.org/10.1002/num.22195 P. Green, Alexander of Macedon (University of California Press Ltd., Oxford, 1991) R.D. Schafer, An Introduction to Nonassociative Algebras, Vol. 22 (Academic Press, 1966) H. Takayasu, Fractals in the Physical Sciences (Manchester University Press, Manchester, 1990) CouclelisH.GaleN.Geogr. Ann. Ser. B Human Geogr.198668110.1080/04353684.1986.11879523 WilléD.R.BakerC.T.Appl. Numer. Math.1992922310.1016/0168-9274(92)90017-8 I. Yaglom, Complex Numbers in Geometry (Academic Press, N.Y., 1968) pp. 195-219, translated by E. Primrose from 1963 Russian original, appendix: Non-Euclidean geometries in the plane and complex numbers AtanganaA.NietoJ.J.Adv. Mech. Eng.201571 LuJ.G.Chaos, Solitons Fractals20052611252005CSF....26.1125L10.1016/j.chaos.2005.02.023 HeisenbergW.Z. Phys.1927431721927ZPhy...43..172H10.1007/BF01397280 GrossD.J.WilczekF.Phys. Rev. Lett.19733013431973PhRvL..30.1343G10.1103/PhysRevLett.30.1343 GaliziaC.G.McIlwrathS.L.MenzelR.Cell Tissue Res.199929538310.1007/s004410051245 GorenfloR.LoutchkoJ.LuchkoY.Fract. Calc. Appl. Anal.200254911967847 I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Academic Press, New York, 1998) Daftardar-GejjiV.SukaleY.BhalekarS.Fract. Calc. Appl. Anal.201518400332390910.1515/fca-2015-0026 K.C. Louden, Compiler Construction: Principles and Practice (1997) J. Briggs, Fractals: The Patterns of Chaos (Thames and Hudson, London, 1992) DoplicherS.FredenhagenK.RobertsJ.E.Commun. Math. Phys.19951721871995CMaPh.172..187D10.1007/BF02104515 ChangpinL.ChunxingT.Comput. Math. Appl.2009581573256240510.1016/j.camwa.2009.07.050 HristovJ.Therm. Sci.20162182710.2298/TSCI160229115H KuroshA.G.Mat. Sbornik.19472023720986 J. Hristov, Electrical Circuits of Non-integer Order: Introduction to an Emerging Interdisciplinary Area with Examples, in Analysis and Simulation of Electrical and Computer Systems (Springer, 2018) pp. 251--273 OrtigueiraM.D.MachadoJ.A.T.J. Comput. Phys.201529342015JCoPh.293....4O334245210.1016/j.jcp.2014.07.019 RozemaL.A.DarabiA.MahlerD.H.HayatA.SoudagarY.SteinbergA.M.Phys. Rev. Lett.20121091004042012PhRvL.109j0404R10.1103/PhysRevLett.109.100404 AtanganaA.Doungmo-GoufoE.F.Therm. Sci.20151923110.2298/TSCI15S1S31A EisenhardtD.Learn Mem.20142153410.1101/lm.033118.113 J.F. Gómez-Aguilar, Numer. Methods Part. Differ. Equ. (2017) https://doi.org/10.1002/num.22219 BaezJ.HoffnungA.RogersC.Commun. Math. Phys.20102937012010CMaPh.293..701B10.1007/s00220-009-0951-9 ChaichianM.PresnajderP.TureanuA.Phys. Rev. Lett.2005941516022005PhRvL..94o1602C10.1103/PhysRevLett.94.151602 MorettiV.Rev. Math. Phys.2003151171203806810.1142/S0129055X03001886 MathaiA.M.MoschopoulosP.J. Stat. Appl. Prob.201211510.12785/jsap/010102 J. Baez (12021_CR26) 2010; 293 D. Eisenhardt (12021_CR6) 2014; 21 M. Caputo (12021_CR18) 2016; 2 12021_CR42 12021_CR43 K. Diethelm (12021_CR53) 2004; 36 A. Biswas (12021_CR24) 2004; 66 R. Broucke (12021_CR23) 1980; 72 M. Chaichian (12021_CR30) 2005; 94 H. Couclelis (12021_CR29) 1986; 68 M.D. Ortigueira (12021_CR13) 2015; 293 L. Changpin (12021_CR52) 2009; 58 D.R. Willé (12021_CR59) 1992; 9 R.D. Knight (12021_CR11) 2005; 83 V.E. Tarasov (12021_CR14) 2013; 18 W. Heisenberg (12021_CR34) 1927; 43 12021_CR51 C.G. Galizia (12021_CR19) 1999; 295 A.M. Mathai (12021_CR41) 2012; 1 12021_CR10 R.N. Pillai (12021_CR38) 1990; 42 R. Gorenflo (12021_CR37) 2002; 5 J.F. Gómez-Aguilar (12021_CR48) 2017; 41 M. Caputo (12021_CR54) 2015; 1 A. Atangana (12021_CR55) 2015; 7 J.F. Gómez-Aguilar (12021_CR49) 2017; 95 F. Cantrijn (12021_CR25) 1999; 66 12021_CR20 12021_CR21 12021_CR61 12021_CR15 12021_CR12 L.A. Rozema (12021_CR35) 2012; 109 D.J. Gross (12021_CR36) 1973; 30 12021_CR56 V. Moretti (12021_CR33) 2003; 15 A. Atangana (12021_CR62) 2015; 19 C.S. Kumar (12021_CR39) 2011; 6 A. Atangana (12021_CR44) 2016; 89 A.A. Tateishi (12021_CR50) 2017; 5 12021_CR1 12021_CR2 12021_CR3 V.F. Morales-Delgado (12021_CR47) 2017; 132 12021_CR4 A.G. Kurosh (12021_CR5) 1947; 20 12021_CR7 12021_CR8 C.S. Kumar (12021_CR40) 2015; 52 12021_CR9 A. Atangana (12021_CR17) 2016; 20 M. Pitkänen (12021_CR28) 2016; 7 12021_CR27 12021_CR22 S. Doplicher (12021_CR31) 1995; 172 V. Daftardar-Gejji (12021_CR60) 2015; 18 M. Caputo (12021_CR16) 1967; 13 M.R. Douglas (12021_CR32) 2001; 73 B.S.T. Alkahtani (12021_CR46) 2016; 89 A.M.A. El-Sayed (12021_CR57) 2007; 20 J. Hristov (12021_CR45) 2016; 21 J.G. Lu (12021_CR58) 2005; 26
References_xml	– reference: R.D. Schafer, An Introduction to Nonassociative Algebras, Vol. 22 (Academic Press, 1966) – reference: E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies, 4th ed. (Dover Publications, New York, 1937) – reference: GrossD.J.WilczekF.Phys. Rev. Lett.19733013431973PhRvL..30.1343G10.1103/PhysRevLett.30.1343 – reference: I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Academic Press, New York, 1998) – reference: AtanganaA.Doungmo-GoufoE.F.Therm. Sci.20151923110.2298/TSCI15S1S31A – reference: J.R. Kantor, The Scientific Evolution of Psychology (Principia Press, Chicago, 1963) – reference: CaputoM.Geophys. J. Int.1967135291967GeoJI..13..529C10.1111/j.1365-246X.1967.tb02303.x – reference: ChangpinL.ChunxingT.Comput. Math. Appl.2009581573256240510.1016/j.camwa.2009.07.050 – reference: W. Blaschke, Differentialgeometrie der Kreise und Kugeln, in Vorlesungen über Differentialgeometrie, Grundlehren der Mathematischen Wissenschaften (Springer, Berlin, 1929) – reference: El-SayedA.M.A.El-MesiryA.E.M.El-SakaH.A.A.Appl. Math. Lett.200720817231471510.1016/j.aml.2006.08.013 – reference: BrouckeR.Astrophys. Space Sci.198072331980Ap&SS..72...33B58866210.1007/BF00642162 – reference: KumarC.S.NairB.U.OPSEARCH20155286332037610.1007/s12597-013-0169-7 – reference: R. Sorabji (Editor), Aristotle Transformed (Bloomsburg Academic, London, 1990) – reference: PitkänenM.Prespacetime J.2016766 – reference: Morales-DelgadoV.F.Gómez-AguilarJ.F.Taneco-HernándezM.A.Eur. Phys. J. Plus201713252710.1140/epjp/i2017-11798-7 – reference: RozemaL.A.DarabiA.MahlerD.H.HayatA.SoudagarY.SteinbergA.M.Phys. Rev. Lett.20121091004042012PhRvL.109j0404R10.1103/PhysRevLett.109.100404 – reference: G.B. Folland, Advanced Calculus (Prentice Hall, 2002) – reference: CaputoM.FabricioM.Progr. Fract. Differ. Appl.2015173 – reference: TarasovV.E.Nonlinear Sci. Numer. Simul.201318294510.1016/j.cnsns.2013.04.001 – reference: GorenfloR.LoutchkoJ.LuchkoY.Fract. Calc. Appl. Anal.200254911967847 – reference: DoplicherS.FredenhagenK.RobertsJ.E.Commun. Math. Phys.19951721871995CMaPh.172..187D10.1007/BF02104515 – reference: J. Hristov, Electrical Circuits of Non-integer Order: Introduction to an Emerging Interdisciplinary Area with Examples, in Analysis and Simulation of Electrical and Computer Systems (Springer, 2018) pp. 251--273 – reference: CouclelisH.GaleN.Geogr. Ann. Ser. B Human Geogr.198668110.1080/04353684.1986.11879523 – reference: AtanganaA.KocaI.Chaos, Solitons Fractals2016894472016CSF....89..447A350705210.1016/j.chaos.2016.02.012 – reference: ChaichianM.PresnajderP.TureanuA.Phys. Rev. Lett.2005941516022005PhRvL..94o1602C10.1103/PhysRevLett.94.151602 – reference: KnightR.D.J. Geom.200583137219323210.1007/s00022-005-0009-x – reference: EisenhardtD.Learn Mem.20142153410.1101/lm.033118.113 – reference: Gómez-AguilarJ.F.Chaos, Solitons Fractals2017951792017CSF....95..179G359751010.1016/j.chaos.2016.12.025 – reference: LuJ.G.Chaos, Solitons Fractals20052611252005CSF....26.1125L10.1016/j.chaos.2005.02.023 – reference: CantrijnF.IbortA.De-LeonM.J. Aust. Math. Soc. A19996630310.1017/S1446788700036636 – reference: HeisenbergW.Z. Phys.1927431721927ZPhy...43..172H10.1007/BF01397280 – reference: BiswasA.ShapiroV.Graph. Models20046613310.1016/j.gmod.2004.01.003 – reference: TateishiA.A.RibeiroH.V.LenziE.K.Front. Phys.201755210.3389/fphy.2017.00052 – reference: H. Takayasu, Fractals in the Physical Sciences (Manchester University Press, Manchester, 1990) – reference: J. Briggs, Fractals: The Patterns of Chaos (Thames and Hudson, London, 1992) – reference: WilléD.R.BakerC.T.Appl. Numer. Math.1992922310.1016/0168-9274(92)90017-8 – reference: S. Okubo, Introduction to Octonion and Other Non-associative Algebras in Physics (Cambridge University Press, 1995) – reference: BaezJ.HoffnungA.RogersC.Commun. Math. Phys.20102937012010CMaPh.293..701B10.1007/s00220-009-0951-9 – reference: Gómez-AguilarJ.F.J. Math. Sociol.201741172370496510.1080/0022250X.2017.1356828 – reference: KuroshA.G.Mat. Sbornik.19472023720986 – reference: GaliziaC.G.McIlwrathS.L.MenzelR.Cell Tissue Res.199929538310.1007/s004410051245 – reference: AlkahtaniB.S.T.Chaos, Solitons Fractals2016895472016CSF....89..547A350706310.1016/j.chaos.2016.03.020 – reference: MorettiV.Rev. Math. Phys.2003151171203806810.1142/S0129055X03001886 – reference: AtanganaA.DumitruB.Therm. Sci.20162076310.2298/TSCI160111018A – reference: Daftardar-GejjiV.SukaleY.BhalekarS.Fract. Calc. Appl. Anal.201518400332390910.1515/fca-2015-0026 – reference: CaputoM.FabrizioM.Progr. Fract. Differ. Appl.20162110.18576/pfda/020101 – reference: AtanganaA.NietoJ.J.Adv. Mech. Eng.201571 – reference: HristovJ.Therm. Sci.20162182710.2298/TSCI160229115H – reference: J. Hristov, Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, in Frontiers in Fractional Calculus (Bentham Science Publishers, 2017) pp. 235--295 – reference: K.C. Louden, Compiler Construction: Principles and Practice (1997) – reference: DouglasM.R.NekrasovN.A.Rev. Mod. Phys.2001739772001RvMP...73..977D10.1103/RevModPhys.73.977 – reference: PillaiR.N.Ann. Inst. Stat. Math.19904215710.1007/BF00050786 – reference: T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore/New Jersey, 1992) – reference: A. Atangana, J.F. Gómez-Aguilar, Numer. Methods Part. Differ. Equ. (2017) https://doi.org/10.1002/num.22195 – reference: J.F. Gómez-Aguilar, Numer. Methods Part. Differ. Equ. (2017) https://doi.org/10.1002/num.22219 – reference: KumarC.S.NairB.U.J. Stat. Appl.2011623 – reference: OrtigueiraM.D.MachadoJ.A.T.J. Comput. Phys.201529342015JCoPh.293....4O334245210.1016/j.jcp.2014.07.019 – reference: P. Green, Alexander of Macedon (University of California Press Ltd., Oxford, 1991) – reference: MathaiA.M.MoschopoulosP.J. Stat. Appl. Prob.201211510.12785/jsap/010102 – reference: DiethelmK.FordN.J.FreedA.D.Numer. Algorithms200436312004NuAlg..36...31D206357210.1023/B:NUMA.0000027736.85078.be – reference: J. Filonik, Athenian Impiety Trials: A Reappraisal (Dike, 2013) – reference: I. Yaglom, Complex Numbers in Geometry (Academic Press, N.Y., 1968) pp. 195-219, translated by E. Primrose from 1963 Russian original, appendix: Non-Euclidean geometries in the plane and complex numbers – ident: 12021_CR22 – volume: 18 start-page: 400 year: 2015 ident: 12021_CR60 publication-title: Fract. Calc. Appl. Anal. doi: 10.1515/fca-2015-0026 – volume: 72 start-page: 33 year: 1980 ident: 12021_CR23 publication-title: Astrophys. Space Sci. doi: 10.1007/BF00642162 – ident: 12021_CR51 – volume: 293 start-page: 4 year: 2015 ident: 12021_CR13 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2014.07.019 – ident: 12021_CR8 – volume: 83 start-page: 137 year: 2005 ident: 12021_CR11 publication-title: J. Geom. doi: 10.1007/s00022-005-0009-x – volume: 15 start-page: 1171 year: 2003 ident: 12021_CR33 publication-title: Rev. Math. Phys. doi: 10.1142/S0129055X03001886 – volume: 66 start-page: 303 year: 1999 ident: 12021_CR25 publication-title: J. Aust. Math. Soc. A doi: 10.1017/S1446788700036636 – volume: 26 start-page: 1125 year: 2005 ident: 12021_CR58 publication-title: Chaos, Solitons Fractals doi: 10.1016/j.chaos.2005.02.023 – volume: 7 start-page: 1 year: 2015 ident: 12021_CR55 publication-title: Adv. Mech. Eng. – volume: 21 start-page: 534 year: 2014 ident: 12021_CR6 publication-title: Learn Mem. doi: 10.1101/lm.033118.113 – volume: 30 start-page: 1343 year: 1973 ident: 12021_CR36 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.30.1343 – volume: 20 start-page: 817 year: 2007 ident: 12021_CR57 publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2006.08.013 – ident: 12021_CR15 – ident: 12021_CR4 – volume: 68 start-page: 1 year: 1986 ident: 12021_CR29 publication-title: Geogr. Ann. Ser. B Human Geogr. doi: 10.1080/04353684.1986.11879523 – volume: 94 start-page: 151602 year: 2005 ident: 12021_CR30 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.94.151602 – ident: 12021_CR43 doi: 10.1007/978-3-319-63949-9_16 – volume: 20 start-page: 237 year: 1947 ident: 12021_CR5 publication-title: Mat. Sbornik. – ident: 12021_CR10 – ident: 12021_CR61 doi: 10.1002/num.22195 – volume: 6 start-page: 23 year: 2011 ident: 12021_CR39 publication-title: J. Stat. Appl. – volume: 41 start-page: 172 year: 2017 ident: 12021_CR48 publication-title: J. Math. Sociol. doi: 10.1080/0022250X.2017.1356828 – volume: 58 start-page: 1573 year: 2009 ident: 12021_CR52 publication-title: Comput. Math. Appl. doi: 10.1016/j.camwa.2009.07.050 – volume: 36 start-page: 31 year: 2004 ident: 12021_CR53 publication-title: Numer. Algorithms doi: 10.1023/B:NUMA.0000027736.85078.be – volume: 73 start-page: 977 year: 2001 ident: 12021_CR32 publication-title: Rev. Mod. Phys. doi: 10.1103/RevModPhys.73.977 – volume: 293 start-page: 701 year: 2010 ident: 12021_CR26 publication-title: Commun. Math. Phys. doi: 10.1007/s00220-009-0951-9 – ident: 12021_CR20 – volume: 43 start-page: 172 year: 1927 ident: 12021_CR34 publication-title: Z. Phys. doi: 10.1007/BF01397280 – volume: 5 start-page: 52 year: 2017 ident: 12021_CR50 publication-title: Front. Phys. doi: 10.3389/fphy.2017.00052 – volume: 20 start-page: 763 year: 2016 ident: 12021_CR17 publication-title: Therm. Sci. doi: 10.2298/TSCI160111018A – ident: 12021_CR21 doi: 10.1017/CBO9780511524479 – ident: 12021_CR3 – volume: 172 start-page: 187 year: 1995 ident: 12021_CR31 publication-title: Commun. Math. Phys. doi: 10.1007/BF02104515 – ident: 12021_CR56 doi: 10.1002/num.22219 – volume: 21 start-page: 827 year: 2016 ident: 12021_CR45 publication-title: Therm. Sci. doi: 10.2298/TSCI160229115H – ident: 12021_CR42 doi: 10.2174/9781681085999118010013 – volume: 2 start-page: 1 year: 2016 ident: 12021_CR18 publication-title: Progr. Fract. Differ. Appl. doi: 10.18576/pfda/020101 – volume: 89 start-page: 547 year: 2016 ident: 12021_CR46 publication-title: Chaos, Solitons Fractals doi: 10.1016/j.chaos.2016.03.020 – volume: 66 start-page: 133 year: 2004 ident: 12021_CR24 publication-title: Graph. Models doi: 10.1016/j.gmod.2004.01.003 – volume: 52 start-page: 86 year: 2015 ident: 12021_CR40 publication-title: OPSEARCH doi: 10.1007/s12597-013-0169-7 – volume: 1 start-page: 73 year: 2015 ident: 12021_CR54 publication-title: Progr. Fract. Differ. Appl. – ident: 12021_CR9 doi: 10.1142/1407 – volume: 18 start-page: 2945 year: 2013 ident: 12021_CR14 publication-title: Nonlinear Sci. Numer. Simul. doi: 10.1016/j.cnsns.2013.04.001 – volume: 42 start-page: 157 year: 1990 ident: 12021_CR38 publication-title: Ann. Inst. Stat. Math. doi: 10.1007/BF00050786 – ident: 12021_CR7 doi: 10.1016/B978-1-4832-5663-4.50009-0 – ident: 12021_CR27 – volume: 1 start-page: 15 year: 2012 ident: 12021_CR41 publication-title: J. Stat. Appl. Prob. doi: 10.12785/jsap/010102 – volume: 7 start-page: 66 year: 2016 ident: 12021_CR28 publication-title: Prespacetime J. – volume: 109 start-page: 100404 year: 2012 ident: 12021_CR35 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.109.100404 – volume: 19 start-page: 231 year: 2015 ident: 12021_CR62 publication-title: Therm. Sci. doi: 10.2298/TSCI15S1S31A – volume: 9 start-page: 223 year: 1992 ident: 12021_CR59 publication-title: Appl. Numer. Math. doi: 10.1016/0168-9274(92)90017-8 – ident: 12021_CR2 – ident: 12021_CR1 doi: 10.1037/11183-000 – volume: 89 start-page: 447 year: 2016 ident: 12021_CR44 publication-title: Chaos, Solitons Fractals doi: 10.1016/j.chaos.2016.02.012 – volume: 132 start-page: 527 year: 2017 ident: 12021_CR47 publication-title: Eur. Phys. J. Plus doi: 10.1140/epjp/i2017-11798-7 – ident: 12021_CR12 – volume: 295 start-page: 383 year: 1999 ident: 12021_CR19 publication-title: Cell Tissue Res. doi: 10.1007/s004410051245 – volume: 5 start-page: 491 year: 2002 ident: 12021_CR37 publication-title: Fract. Calc. Appl. Anal. – volume: 13 start-page: 529 year: 1967 ident: 12021_CR16 publication-title: Geophys. J. Int. doi: 10.1111/j.1365-246X.1967.tb02303.x – volume: 95 start-page: 179 year: 2017 ident: 12021_CR49 publication-title: Chaos, Solitons Fractals doi: 10.1016/j.chaos.2016.12.025
SSID	ssj0000491494
Score	2.583676
Snippet	. To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in... To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in...
SourceID	proquest crossref springer
SourceType	Aggregation Database Enrichment Source Index Database Publisher
StartPage	166
SubjectTerms	Applied and Technical Physics Associativity Atomic Calculus Commutativity Complex Systems Condensed Matter Physics Derivatives Differential equations Focus Point on Modelling Complex Real-World Problems with Fractal and New Trends of Fractional Differentiation Fractional calculus Group theory Mathematical analysis Mathematical and Computational Physics Mathematical models Molecular Operators (mathematics) Optical and Plasma Physics Physics Physics and Astronomy Power law Principles Regular Article Statistical analysis Steady state Theoretical
SummonAdditionalLinks	– databaseName: Springer Nature - Connect here FIRST to enable access dbid: RSV link: http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3dS8MwEA86FXzxW5xOyYNvWtYl6Zr65tfwQYbgB3srSZrAZHSl7fTf95K2G4oK-laaXCi5pHeX_O53CJ0SKQhNYANyiDU8RhnxoqQvvcjIiAthNKmqltyHwyEfjaKHOimsaNDuzZWk-1NXfLZ-V2evWXcM9gqiHmKBBXQZrQSWbcbG6I8v85MV8HnB7WdNhsy3op-t0MK1_HIb6ozMYPN_n7eFNmqnEl9Wq2AbLel0B605cKcqdtH7jbbc1GmN3MFTg01eJTSAFCjJHgEWOJ9NdHGBr8CNtOfnWNnckbIuL4FFmmDR6NK9Kacgm9kbCGzRuthxhMKAFjVmeR3EHnoe3D5d33l1wQVPwU4sPeknPa6ISYhUghLJQxkylURS2cxzA8-MU0MIF74QlliNiwTCvUAEPWWM7tF91EqnqT5AmIeCyZAYHUYRAxdTGh4oKfqGGgH2kLVRr1FArGo2clsUYxJXmdJ-bCc0dhMauwmNaRudzWWyiovj196dRq9xvS-L2NbmghCOh9B83uhx0fzzaId_636E1t1ScBCfDmqV-Uwfo1X1Vo6L_MSt1w8QWO1p priority: 102 providerName: Springer Nature
Title	Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena
URI	https://link.springer.com/article/10.1140/epjp/i2018-12021-3 https://www.proquest.com/docview/2920626873
Volume	133
WOSCitedRecordID	wos000431462500004&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVPQU databaseName: Advanced Technologies & Aerospace Database customDbUrl: eissn: 2190-5444 dateEnd: 20241213 omitProxy: false ssIdentifier: ssj0000491494 issn: 2190-5444 databaseCode: P5Z dateStart: 20110101 isFulltext: true titleUrlDefault: https://search.proquest.com/hightechjournals providerName: ProQuest – providerCode: PRVPQU databaseName: AUTh Library subscriptions: ProQuest Central customDbUrl: eissn: 2190-5444 dateEnd: 20241213 omitProxy: false ssIdentifier: ssj0000491494 issn: 2190-5444 databaseCode: BENPR dateStart: 20110101 isFulltext: true titleUrlDefault: https://www.proquest.com/central providerName: ProQuest – providerCode: PRVPQU databaseName: Earth, Atmospheric & Aquatic Science Database customDbUrl: eissn: 2190-5444 dateEnd: 20241213 omitProxy: false ssIdentifier: ssj0000491494 issn: 2190-5444 databaseCode: PCBAR dateStart: 20110101 isFulltext: true titleUrlDefault: https://search.proquest.com/eaasdb providerName: ProQuest – providerCode: PRVAVX databaseName: SpringerLink customDbUrl: eissn: 2190-5444 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000491494 issn: 2190-5444 databaseCode: RSV dateStart: 20110101 isFulltext: true titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22 providerName: Springer Nature
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1La9wwEB7yaCGXJH2RzQsdekvN7kraWM4lZPMgh7AseZTQi5FkCVKC11077d_vjCxnaaC55GbZljB8I89DM98AfOVGc1HgBlToayRSSJ5kxaFJMm8ypbV3vO1acpVOJur-PpvGgFsd0yq7f2L4URczSzHyPnVVQuNbpeK4-pVQ1yg6XY0tNJZhlZjKUM5Xx-eT6fVzlAXtX3QBZFctIwd9V_2s-g-o9tB54pSfIP7VSAsz88XJaFA4Fxtv_dRNWI-mJjtpZeMDLLnyI7wPKZ-2_gR_zhwxVpcxn4fNPPPztswBZyF0FBis2fzp0dVHbIzGJUXVmaWKkiY2nWC6LJjuEA53mhnOrehcglEOLwvMobgg5ZIR24P-DHcX57enl0lsw5BY3J9NYgbFUFnuC26sFtyo1KTSFpmxVI_u8Voq4TlXeqA10a0pXaATONKjofXeDcUXWClnpdsCplItTcq9S7NMouFpvBpZow-98Bq1pOzBsIMit5GjnFplPOZt_fQgJ_jyAF8e4MtFDw6e51QtQ8erb-92mOVxt9b5ArAefOtQXzz-_2rbr6-2A2tB0EKizy6sNPMntwfv7O_moZ7vR1ndh-Xp6AeOpqfjExpd33z_C1YZ-gY
linkProvider	ProQuest
linkToHtml	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V1Lb9QwEB6VLahceFddKOADnCDaXdu7cSohRGmrVl1WFSpSb8Z2bKlVlQ2blIo_xW_sjJN0BRK99cAtSuKRbH-2ZzyPD-ANt4aLHBegQlsjkULyJMsnNsmCzZQxwfOGtWSazmbq5CQ7WoHfXS4MhVV2e2LcqPO5ozvyAbEqofKtUvGx_JEQaxR5VzsKjQYWh_7XJZps1YeDHZzft5zv7R5_3k9aVoHEIdzqxA7zkXI85Nw6I7hVqU2lyzPrKL064LNUInCuzNAYqh6mTI42zdiMRy4EPxIo9w6sYrcm4x6sbu_Ojr5e3-qgvo0mh-yyc-Rw4MuzcnCKxywaa5ziIcSfJ-BSrf3LExsPuL2H_9vQPIIHrSrNPjXYfwwrvngC92JIq6uewuWOp4rcRRuvxOaBhUWTxoGtEJp08VmxxcW5r7bYNirP5DVgjjJm6pZUg5kiZ6ZDcHxTz7FtSX4XRjHKLFZGRYEUK0fVLMwz-HYrvV6HXjEv_AYwlRppUx58mmUSFWsb1NhZMwkiGNQCZB9G3dRr19ZgJyqQc93khw81wUVHuOgIFy368O66TdlUILnx780OI7rdjSq9BEgf3ncoW37-t7TnN0t7DWv7x1-menowO3wB9yPIY1DTJvTqxYV_CXfdz_q0Wrxq1wmD77eNvysSpVRy
linkToPdf	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1ZS8QwEA7e-OItrmcefNOybZq1qW9ei6Is4oVvIUkTWFm6pe3q33eStl6oIL6VJBNKJmFmkm--QWiXSEHCBA4gg1jDoyElXpwcSC82MmZCGE2qqiVXUa_HHh_j6w9Z_A7t3jxJVjkNlqUpLdtZYmpuW7-ts6es3QfbBREQsSCDcBxNUohkLKjr5vbh7ZYF_F8IAWiTLfOt6GeL9O5mfnkZdQanO___X11Ac7WziY-q3bGIxnS6hKYd6FMVy-jlVFvO6rRG9OChwSavEh1ACpRnrwYLnI8GujjEx-Be2nt1rGxOSVmXncAiTbBodOxayiHIZvZlAlsUL3bcoTChRZNZvgexgu67Z3cn515diMFTcEJLT_pJwBQxCZFKhESySEZUJbFUNiPdwDdloSGECV8IS7jGRAJhYEd0AmWMDsJVNJEOU72GMIsElRExOopjCq6nNKyjpDgwoRFgJ2kLBY0yuKpZym2xjAGvMqh9bheUuwXlbkF52EJ7bzJZxdHx6-jNRse8Pq8FtzW7ILRjEXTvNzp97_55tvW_Dd9BM9enXX510bvcQLNuVzgU0CaaKPOR3kJT6rnsF_m228avgEL5MQ
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Decolonisation+of+fractional+calculus+rules%3A+Breaking+commutativity+and+associativity+to+capture+more+natural+phenomena&rft.jtitle=European+physical+journal+plus&rft.au=Atangana%2C+Abdon&rft.au=G%C3%B3mez-Aguilar%2C+J.+F.&rft.date=2018-04-01&rft.issn=2190-5444&rft.eissn=2190-5444&rft.volume=133&rft.issue=4&rft_id=info:doi/10.1140%2Fepjp%2Fi2018-12021-3&rft.externalDBID=n%2Fa&rft.externalDocID=10_1140_epjp_i2018_12021_3
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2190-5444&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2190-5444&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2190-5444&client=summon