Fractional-Order Derivatives and Integrals: Introductory Overview and Recent Developments

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참고문헌 (112)

1. A. Alsaedi, "On coupled systems of timefractional differential problems by using a new fractional derivative", J. Funct. Spaces, pp.8~ (2016)
2. M. A. Asiru, "Sumudu transform and the solution of integral equations of convolution type", Internat. J. Math. Ed. Sci. Tech, 32, pp.906~910 (2001)
3. A. Atangana, "On the new fractional derivative and application to nonlinear Fisher’s reaction diffusion equation", Appl. Math. Comput, 273, pp.948~956 (2016)
4. A. Atangana, "Extension of the RLC electrical circuit to fractional derivative without singular kernel", Adv. Mech. Engrg, 7(6), pp.1~6 (2015)
5. A. Atangana, "Analysis of the Keller-Segel model with a fractional derivative without singular kernel", Entropy, 17, pp.4439~4453 (2015)
6. Abdon Atangana, "New model of groundwater flowing within a confine aquifer: application of Caputo-Fabrizio derivative", Arabian Journal of Geosciences, 9(1) (2016), Springer Science and Business Media LLC[DOI]
7. A. Atangana, "New fractional derivatives with nonlocal and nonsingular kernel : theory and application to heat transfer model", Thermal Sci, 20, pp.763~769 (2016)
8. A. Atangana, "Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel", Adv. Mech. Engrg, 7(10), pp.1~7 (2015)
9. F. B. M. Belgacem, "Sumudu transform fundamental properties investigations and applications", J. Appl. Math. Stoch. Anal., pp.23~ (2006)
10. F. B. M. Belgacem, "Analytical investigations of the Sumudu transform and applications to integral production equations", Math. Probl. Eng, 3, pp.103~118 (2003)
11. R. G. Buschman, "The H function associated with a certain class of Feynman integrals", J. Phys. A Math. Gen, 23, pp.4707~4710 (1990)
12. M. Caputo, "A new definition of fractional derivative without singular kernel", Progr. Fract. Differ. Appl, 1, pp.73~85 (2015)
13. M. Caputo, "A new dissipation model based on memory mechanism", Pure Appl. Geophys, 91, pp.134~147 (1971)
14. C. Cattani, "Fractional dynamics" (2015), Emerging Science Publishers (De Gruyter Open)
15. V. B. L. Chaurasia, "On the solution of generalized fractional kinetic equations", Adv. Stud. Theoret. Phys, 4, pp.773~780 (2010)
16. J. Choi, "A note on fractional integral operator associated with multiindex Mittag-Leffler functions", Filomat, 30, pp.1931~1939 (2016)
17. J. Choi, "Solutions of generalized fractional kinetic equations involving Aleph functions", Math. Commun, 20, pp.113~123 (2015)
18. A. Erdelyi, "Higher transcendental functions, Vols. I and II" (1953), McGraw-Hill Book Company
19. A. Erdelyi, "Tables of integral transforms, Vol. II" (1954), McGraw-Hill Book Company
20. C. Fox, "The G and H functions as symmetrical Fourier kernels", Trans. Amer. Math. Soc, 98, pp.395~429 (1961)
21. M. Fukuhara, "Ordinary differential equations, Vol. II" (1941), Iwanami Shoten
22. L. Galue, "N-fractional calculus operator N ν method applied to some second order nonhomogeneous equations", J. Fract. Calc, 16, pp.85~97 (1999)
23. F. Gao, "A coupling method involving the Sumudu transform and the variational iteration method for a class of local fractional diffusion equations", J. Nonlinear Sci. Appl, 9, pp.5830~5835 (2016)
24. R. Gorenflo, "Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena", Proceedings of the Eighth International Colloquium on Differential Equations, pp.195~202 (1998), VSP Publishers
25. R. Gorenflo, "Abel integral equations: analysis and applications" (1991), Springer-Verlag
26. H. J. Haubold, "The fractional kinetic equation and thermonuclear functions", Astrophys. Space Sci., 273, pp.53~63 (2000)
27. R. Hilfer, "Applications of fractional calculus in physics" (2000), World Scientific Publishing Company
28. R. Hilfer, "Applications of Fractional Calculus in Physics", pp.87~130 (2000), World Scientific Publishing Company
29. R. Hilfer, "Fractional master equations and fractal time random walks", Phys. Rev. E, 51, pp.R848~R851 (1995)
30. E. L. Ince, "Ordinary differential equations" (1956), Dover Publications
31. A. A. Kilbas, "Theory and applications of fractional differential equations" (2006), Elsevier (NorthHolland) Science Publishers
32. V. Kiryakova, "Generalized fractional calculus and applications" (1993), Longman Scientific and Technical
33. V. Kiryakova, "Multiple(multiindex)Mittag-Leffler functions and relations to generalized fractional calculus", J. Comput. Appl. Math, 118, pp.214~259 (2000)
34. D. Kumar, "On the solution of generalized fractional kinetic equations", J. Global Res. Math. Arch, 1(4), pp.31~39 (2013)
35. Dinesh Kumar, "GENERALIZED FRACTIONAL KINETIC EQUATIONS ASSOCIATED WITH ALEPH FUNCTION", Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집), 19(1), pp.145~156 (2016), 장전수학회
36. Dinesh Kumar, "SOLUTION OF A GENERAL FAMILY OF FRACTIONAL KINETIC EQUATIONS ASSOCIATED WITH THE GENERALIZED MITTAG-LEFFLER FUNCTION", Nonlinear Functional Analysis and Applications, 23(3), pp.455~471 (2018), 경남대학교 수학교육과[DOI]
37. D. Kumar, "On generalized fractional kinetic equations involving generalized Bessel function of the first kind", Math. Probl. Engrg., pp.7~ (2015)
38. D. Kumar, "A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves", Math. Methods Appl. Sci., 40, pp.5642~5653 (2017)
39. S. -D. Lin, "A simple fractionalcalculus approach to the solutions of the Bessel differential equation of general order and some of its applications", Comput. Math. Appl, 49, pp.1487~1498 (2005)
40. S. -D. Lin, "N-Method to a generalized associated Legendre equation", J. Fract. Calc, 14, pp.95~111 (1998)
41. S. -D. Lin, "New finding of particular solutions for a generalized associated Legendre equation", J. Fract. Calc, 18, pp.9~37 (2000)
42. S. -D. Lin, "Some differintegral formulas for power, composite and rational functions", J. Fract. Calc, 32, pp.87~98 (2000)
43. S. -D. Lin, "Some families of linear ordinary and partial differential equations solvable by means of fractional calculus", Int. J. Differ. Equ. Appl, 4, pp.405~421 (2002)
44. S. -D. Lin, "Explicit solutions of a certain class of associated Legendre equations by means of fractional calculus", Appl. Math. Comput, 187, pp.280~289 (2007)
45. S. -D. Lin, "Explicit solutions of a certain family of fractional differintegral equations", Hyperion Sci. J. Ser. A Math. Phys. Electric. Engrg, 2, pp.85~90 (2001)
46. S. -D. Lin, "Explicit solutions of certain ordinary differential equations by means of fractional calculus", J. Fract. Calc, 20, pp.35~43 (2001)
47. S. -D. Lin, "Certain classes of ordinary and partial differential equations solvable by means of fractional calculus", Appl. Math. Comput, 131, pp.223~233 (2002)
48. S. -D. Lin, "Explicit solutions of some classes of non-Fuchsian differential equations by means of fractional calculus", J. Fract. Calc, 21, pp.49~60 (2002)
49. J. Liouville, "Memoire sur quelques de geometrie et de m´ecanique, et sur un nouveau genre de calcul pour resourdre ces wu´etions", J. Ecole Polytech., 13(21), pp.1~69 (1832)
50. A. C. McBride, "Fractional calculus and integral transforms of generalized functions" (1979), Pitman Publishing Limited
51. N. W. McLachlan, "Modern operational calculus with applications in technical mathematics" (1948), Macmillan
52. K. S. Miller, "An introduction to fractional calculus and fractional differential equations" (1993), A Wiley-Interscience Publication, John Wiley and Sons
53. G. M. Mittag-Leffler, "Sur la nouvelle fonction Eα(x)", C. R. Acad. Sci. Paris, 137, pp.554~558 (1903)
54. K. Nishimoto, "Fractional Calculus, Vols. I, II, III, IV, V" (1996), Descartes Press
55. K. Nishimoto, "An essence of Nishimoto’s fractional calculus (Calculus of the 21st century): integrations and differentiations of arbitrary order" (1991), Descartes Press
56. K. Nishimoto, "Operator Nν method to nonhomogeneous Gauss and Bessel equations", J. Fract. Calc, 9, pp.1~15 (1996)
57. K. Nishimoto, "N-Fractional calculus operator N νmethod to nonhomogeneous Fukuhara equations, I", J. Fract. Calc, 9, pp.23~31 (1996)
58. K. Nishimoto, "N-Fractional calculus operator N ν method to nonhomogeneous and homogeneous Whittaker equations, I", J. Fract. Calc, 9, pp.17~22 (1996)
59. K. Nishimoto, "N-Method to the homogeneous Whittaker equations", J. Fract. Calc, 15, pp.13~23 (1999)
60. K. Nishimoto, "N-Method to the homogeneous Whittaker equations(revise and supplement)", J. Fract. Calc, 16, pp.123~128 (1999)
61. K. Nishimoto, "Application of fractional calculus in solving certain classes of Fuchsian differential equations", J. College Engrg. Nihon Univ. Ser. B, 32, pp.119~126 (1991)
62. K. Nishimoto, "Solutions of some second-order linear differential equations by means of fractional calculus", J. College Engrg. Nihon Univ. Ser. B, 33, pp.15~25 (1992)
63. K. B. Oldham, "The fractional calculus: theory and applications of differentiation and integration to arbitrary order" (1974), Academic Press
64. I. Podlubny, "Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications" (1999), Academic Press
65. A. I. Prieto, "Some fractional calculus results involving the generalized Lommel-Wright and related functions", Appl. Math. Lett, 20, pp.17~22 (2007)
66. B. Rubin, "Fractional integrals and potentials, Pitman Monographs and Surveys in Pure and Applied Mathematics" (1996), Longman Scientific and Technical
67. K. M. Saad, "An application of homotopy analysis transform method for Riccati differential equation of fractional order", J. Fract. Calc. Appl, 7, pp.61~72 (2016)
68. A. Saichev, "Fractional kinetic equations : solutions and applications", Chaos, 7, pp.753~764 (1997)
69. S. Salinas de Romero, "N-Fractional calculus operator N ν method to nonhomogeneous and homogeneous Whittaker equations II, some illustrative examples", J. Fract. Calc, 12, pp.29~35 (1997)
70. S. Salinas de Romero, "An application of the N-fractional calculus operator method to a modified Whittaker equation", Appl. Math. Comput, 115, pp.11~21 (2000)
71. V. P. Saxena, "A trivial extension of Saxena’s I-function", Nat. Acad. Sci. Lett, 38, pp.243~245 (2015)
72. R. K. Saxena, "Further results on the generalized Mittag-Leffler function operator", J. Inequal. Appl., 75, pp.12~ (2015)
73. R. K. Saxena, "On the solutions of certain fractional kinetic equations", Appl. Math. Comput, 199, pp.504~511 (2008)
74. R. K. Saxena, "On fractional kinetic equations", Astrophys. Space Sci, 282, pp.281~287 (2002)
75. R. K. Saxena, "On generalized fractional kinetic equations", Phys. A, 344, pp.653~664 (2004)
76. R. K. Saxena, "Unified fractional kinetic equation and a fractional diffusion equation", Astrophys. Space Sci, 290, pp.299~310 (2004)
77. R. K. Saxena, "N-Fractional calculus of generalized Mittag-Leffler functions", J. Fract. Calc, 37, pp.43~52 (2010)
78. R. K. Saxena, "Alternative derivation of generalized kinetic equations", J. Fract. Calc. Appl, 4, pp.322~334 (2013)
79. R. K. Saxena, "Fractional differentiation and fractional integration of the generalized Mittag-Leffler function", J. Indian Acad. Math, 32, pp.153~162 (2010)
80. J. L. Schiff, "The Laplace transform: theory and applications" (1999), Springer-Verlag
81. A. K. Shukla, "On a generalization of Mittag-Leffler function and its properties", J. Math. Anal. Appl, 336, pp.797~811 (2007)
82. H. M. Srivastava, "On an extension of the Mittag-Leffler function", Yokohama Math. J, 16, pp.77~88 (1968)
83. H. M. Srivastava, "Convolution integral equations with special function kernels" (1977), Halsted Press, John Wiley and Sons
84. H. M. Srivastava, "Theory and applications of convolution integral equations" (1992), Kluwer Academic Publishers
85. H. M. Srivastava, "Zeta and q-zeta functions and associated series and integrals" (2012), Elsevier Science Publishers
86. H. M. Srivastava, "Local fractional Sumudu transform with application to IVPs on Cantor sets", Abstr. Appl. Anal., pp.7~ (2014)
87. H. M. Srivastava, "The H-functions of one and two variables with applications" (1982), South Asian Publishers
88. H. M. Srivastava, "Multiple Gaussian hypergeometric series" (1985), Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons
89. H. M. Srivastava, "Special functions in queuing theory and related stochastic processes" (1982), Academic Press
90. H. M. Srivastava, "Explicit solutions of a certain class differential equations by means of fractional calculus", Russian J. Math. Phys., 14, pp.357~365 (2007)
91. H. M. Srivastava, "A treatise on generating functions" (1984), Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons
92. H. M. Srivastava, "Univalent functions, fractional calculus, and their applications" (1989), Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons
93. H. M. Srivastava, "Current topics in analytic function theory" (1992), World Scientific Publishing Company
94. H. M. Srivastava, "Some fractional differintegral equations", J. Math. Anal. Appl, 106, pp.360~366 (1985)
95. H. M. Srivastava, "Some new models of the time-fractional gas dynamics equation", Adv. Math. Models Appl, 3(1), pp.5~17 (2018)
96. H. M. Srivastava, "Fractional calculus with an integral operator ˇcontaining a generalized Mittag-Leffler function in the kernel", Appl. Math. Comput, 211, pp.198~210 (2009)
97. Z. Tomovski, "Fractional and operational calculus with ˇgeneralized fractional derivative operators and Mittag-Leffler type functions", Integral Transforms Spec. Funct, 21, pp.797~814 (2010)
98. F. G. Tricomi, "Funzioni ipergeometriche confluenti" (1954), Edizioni Cremonese
99. S. -T. Tu, "Some families of ordinary and partial fractional differintegral equations", Integral Transforms Spec. Funct, 11, pp.291~302 (2001)
100. S. -T. Tu, "A certain family of fractional differintegral equations", Taiwanese J. Math, 4, pp.417~426 (2000)
101. S. -T. Tu, "Solutions of a certain class of fractional differintegral equations", Appl. Math. Lett, 14(2), pp.223~229 (2001)
102. S. -T. Tu, "Solutions of a class of ordinary and partial differential equations via fractional calculus", J. Fract. Calc, 18, pp.103~110 (2000)
103. J. -R. Wang, "Nonlocal Cauchy problems for semilinear differential inclusions with fractional order in Banach spaces", Commun. Nonlinear Sci. Numer. Simul, 27, pp.281~293 (2015)
104. J. -R. Wang, "On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives", Appl. Math. Lett, 39, pp.85~90 (2015)
105. J. -R. Wang, "On a new class of impulsive fractional differential equations", Appl. Math. Comput, 242, pp.649~657 (2014)
106. P. -Y. Wang, "Explicit solutions of Jacobi and Gauss differential equations by means of operators of fractional calculus", Appl. Math. Comput, 199, pp.760~769 (2008)
107. P. -Y. Wang, "A survey of fractional-calculus approaches the solutions of the Bessel differential equation of general order", Appl. Math. Comput, 187, pp.544~555 (2007)
108. G. N. Watson, "A treatise on the theory of Bessel functions" (1944), Cambridge University Press
109. G. K. Watugala, "Sumudu transform : A new integral transform to solve differential equations and control engineering problems", Math. Engrg. Industr, 6, pp.319~329 (1998)
110. E. T. Whittaker, "A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions" (1927), Cambridge University Press
111. A. Wiman, "Uber den fundamentalsatz in der theorie der funcktionen Eα(x)", Acta Math., 29, pp.191~201 (1905)
112. X.-J. Yang, "Local fractional integral transforms and their applications" (2016), Academic Press (Elsevier Science Publishers)

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