Local Fractional Integral Transforms and Their Applications

Local Fractional Integral Transforms and Their Applications

Volumn , Issue , 2015, Pages 1-249

  Yang, Xiao Jun ^a   Baleanu, Dumitru ^b,c   Srivastava, H M ^d  

^a CHINA UNIVERSITY OF MINING AND TECHNOLOGY   (China)

^b CANKAYA UNIVERSITY   (Turkey)

^c INSTITUTE OF SPACE SCIENCES   (Romania)

^d UNIVERSITY OF VICTORIA   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85088982362     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Book

References (430)

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20. He J.-H., Elagan S.K., Li Z.-B. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 2012, 376(4):257-259.
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22. Yang X.-J., Srivastava H.M., He J.-H., Baleanu D. Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives. Phys. Lett. A 2013, 377(28):1696-1700.
23. Yang X.-J., Baleanu D., Srivastava H.M. Local fractional similarity solution for the diffusion equation defined on Cantor sets. Appl. Math. Lett. 2015, 47:54-60.
24. Oliveira E.C.D., Machado J.A.T. A review of definitions for fractional derivatives and integral. Math. Probl. Eng. 2014, 6 pages.
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26. Chen W., Zhang X.-D., Korovsak D. Investigation on fractional and fractal derivative relaxation-oscillation models. Int. J. Nonlinear Sci. Numer. Simul. 2010, 11(1):3-10.
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30. Ortigueira M.D., Trujillo J.J. Generalized Grünwald-Letnikov fractional derivative and its Laplace and Fourier transforms. J. Comput. Nonlinear Dyn. 2011, 6(3).
31. Machado J.A.T. Fractional derivatives: probability interpretation and frequency response of rational approximations. Commun. Nonlinear Sci. Numer. Simul. 2009, 14(9):3492-3497.
32. Machado J.A.T. Fractional coins and fractional derivatives. Abstr. Appl. Anal. 2013, 5 pages.
33. Ortigueira M.D., Coito F. From differences to derivatives. Fract. Calc. Appl. Anal. 2004, 7(4):459.
34. Atangana A., Secer A. A note on fractional order derivatives and table of fractional derivatives of some special functions. Abstr. Appl. Anal. 2013, 8 pages.
35. Jumarie G. On the representation of fractional Brownian motion as an integral with respect to dt?. Appl. Math. Lett. 2005, 18(7):739-748.
36. Jumarie G. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 2006, 51(9):1367-1376.
37. Jumarie G. Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions. Appl. Math. Model. 2008, 32(5):836-859.
38. Jumarie G. Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for nondifferentiable functions. Appl. Math. Lett. 2009, 22(3):378-385.
39. Li C.-P., Zhao Z.-G. Introduction to fractional integrability and differentiability. Eur. Phys. J. 2011, 193(1):5-26.
40. Khalil R., Al-Horani M., Yousef A., Sababheh M. A new definition of fractional derivative. J. Comput. Appl. Math. 2014, 264:65-70.
41. Abdeljawad T. On conformable fractional calculus. J. Comput. Appl. Math. 2015, 279:57-66.
42. Sabzikar F., Meerschaert M.M., Chen J. Tempered fractional calculus. J. Comput. Phys. 2014, http://dx.doi.org/10.1016/j.jcp.2014.04.024.
43. Katugampola U.N. New approach to a generalized fractional integral. Appl. Math. Comput. 2011, 218(3):860-865.
44. Caputo M., Fabrizio M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1(2):73-85.
45. Losada J., Nieto J.J. Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1(2):87-92.
46. Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations to Methods of Their Solution and Some of Their Applications 1998, vol. 198. Academic Press, New York.
47. Machado J.A.T., Kiryakova V., Mainardi F. A poster about the recent history of fractional calculus. Fract. Calc. Appl. Anal. 2010, 13(3):329-334.
48. Machado J.A.T., Kiryakova V., Mainardi F. Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(3):1140-1153.
49. Magin R.L. Fractional Calculus in Bioengineering 2006, 149. Begell House Publishers, Redding.
50. Kiryakova V.S. Generalized Fractional Calculus and Applications 1993, 301. Longman Scientific and Technical, Harlow (Essex).
51. Debnath L. A brief historical introduction to fractional calculus. Int. J. Math. Educ. Sci. Technol. 2004, 35(4):487-501.
52. Tarasov V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media 2011, Springer Science and Business Media, New York.
53. Ortigueira M.D. Fractional Calculus for Scientists and Engineers 2011, 84. Springer Science and Business Media, New York.
54. Meerschaert M.M., Sikorskii A. Stochastic Models for Fractional Calculus 2011, 43. Walter de Gruyter, New York.
55. Machado J.A.T., Galhano A.M., Trujillo J.J. On development of fractional calculus during the last fifty years. Scientometrics 2014, 98(1):577-582.
56. Monje C.A., Chen Y., Vinagre B.M., Xue D., Feliu-Batlle V. Fractional-Order Systems and Controls: Fundamentals and Applications 2010, Springer Science and Business Media, New York.
57. Baleanu D., Diethelm K., Scalas E., Trujillo J.J. Models and Numerical Methods 2012, 3:10-16. World Scientific, Singapore.
58. Caponetto R. Fractional Order Systems: Modeling and Control Applications 2010, 72. World Scientific, Singapore.
59. Anastassiou G.A. Fractional Differentiation Inequalities 2009, Springer, New York.
60. West B., Bologna M., Grigolini P. Physics of Fractal Operators 2003, Springer Science and Business Media, New York.
61. Zaslavsky G.M. Hamiltonian Chaos and Fractional Dynamics 2008, Oxford University (Clarendon) Press, Oxford, London, New York.
62. Baleanu D., Machado J.A.T., Luo A.-C. Fractional Dynamics and Control 2011, Springer Science and Business Media, New York.
63. Klafter J., Lim S.-C., Metzler R. Fractional Dynamics: Recent Advances 2012, World Scientific, Singapore.
64. Diethelm K. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type 2004, Springer Science and Business Media, New York.
65. Mainardi F. Fractional Calculus Waves in Linear Viscoelasticity: An Introduction to Mathematical Models 2010, World Scientific, Singapore.
66. Malinowska A.B., Torres D.F.M. Fractional Calculus of Variations 2012, Imperial College Press, Singapore.
67. Das S., Pan I. Fractional Order Signal Processing: Introductory Concepts and Applications 2011, Springer Science and Business Media, New York.
68. Uchaikin V.V. Fractional Derivatives for Physicists and Engineers 2013, Springer, Berlin.
69. Herrmann R. Fractional Calculus: An Introduction for Physicists 2014, World Scientific, Singapore.
70. Chen G.-S. Generalizations of Hölder's and some related integral inequalities on fractal space. J. Funct. Spaces Appl. 2013, 9 pages.
71. Wei W., Srivastava H.M., Zhang Y., Wang L., Shen P., Zhang J. A local fractional integral inequality on fractal space analogous to Anderson's inequality. Abstr. Appl. Anal. 2014, 7 pages.
72. Chen G.-S., Srivastava H.M., Wang P., Wie W. Some further generalizations of Hölder's inequality and related results on fractal space. Abstr. Appl. Anal. 2014, 7 pages.
73. Zhang Y., Baleanu D., Yang X.-J. On a local fractional wave equation under fixed entropy arising in fractal hydrodynamics. Entropy 2014, 16(12):6254-6262.
74. Li Y.-Y., Zhao L.Y., Xie G.-N., Baleanu D., Yang X.-J., Zhao K. Local fractional Poisson and Laplace equations with applications to electrostatics in fractal domain. Adv. Math. Phys. 2014, 5 pages.
75. Yang X.-J., Baleanu D. Local fractional variational iteration method for Fokker-Planck equation on a Cantor set. Acta Univ. 2013, 23(2):3-8.
76. Yang X.-J., Baleanu D., Machado J.A.T. Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis. Bound. Value Probl. 2013, 1:1-16.
77. Su W.-H., Baleanu D., Yang X.-J., Jafari H. Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method. Fixed Point Theory Appl. 2013, 1:1-11.
78. Yang X.-J., Baleanu D., Machado J.A.T. Systems of Navier-Stokes equations on Cantor sets. Math. Probl. Eng. 2013, 8 pages.
79. Liu H.-Y., He J.-H., Li Z.B. Fractional calculus for nanoscale flow and heat transfer. Int. J. Numer. Methods Heat Fluid Flow 2014, 24(6):1227-1250.
80. Zhao Y., Baleanu D., Cattani C., Cheng D.-F., Yang X.-J. Maxwell's equations on Cantor sets: a local fractional approach. Adv. High Energy Phys. 2013, 6 pages.
81. Wang L.-F., Yang X.-J., Baleanu D., Cattani C., Zhao Y. Fractal dynamical model of vehicular traffic flow within the local fractional conservation laws. Abstr. Appl. Anal. 2014, 5 pages.
82. Yang A.-M., Zhang Y.-Z., Cattani C., Xie G.-N., Rashidi M.M., Zhou Y.-J., ang X.-J. Application of local fractional series expansion method to solve Klein-Gordon equations on Cantor sets. Abstr. Appl. Anal. 2014, 6 pages.
83. Yang X.-J., Baleanu D., He J.-H. Transport equations in fractal porous media within fractional complex transform method. Proc. Rom. Acad. Series A 2013, 14(4):287-292.
84. Hao Y.-J., Srivastava H.M., Jafari H., Yang X.-J. Helmholtz and diffusion equations associated with local fractional derivative operators involving the Cantorian and Cantor-type cylindrical coordinates. Adv. Math. Phys. 2013, 5 pages.
85. Yang X.-J., Hristov J., Srivastava H.M., Ahmad B. Modelling fractal waves on shallow water surfaces via local fractional Korteweg-de Vries equation. Abstr. Appl. Anal. 2014, 10 pages.
86. Yang X.-J., Machado J.A.T., Hristov J. Nonlinear dynamics for local fractional Burgers' equation arising in fractal flow. Nonlinear Dyn. 2015, 10.1007/s11071-015-2085-2.
87. Niu X.-F., Zhang C.-L., Li Z.-B., Zhao Y. Local fractional derivative boundary value problems for Tricomi equation arising in fractal transonic flow. Abstr. Appl. Anal. 2014, 5 pages.
88. Cattani C., Srivastava H.M., Yang X.-J. Fractional Dynamics 2015, Emerging Science Publishers, Berlin.
89. Yang Y.-J., Baleanu D., Baleanu M.C. Observing diffusion problems defined on Cantor sets in different coordinate systems. Thermal Sci. 2015, 10.2298/TSCI141126065Y.
90. Yang X.-J. Advanced Local Fractional Calculus and Its Applications 2012, World Science, New York.
91. Yang X.-J. Local fractional integral transforms. Prog. Nonlinear Sci. 2011, 4(1):1-225.
92. Chen W. Time-space fabric underlying anomalous diffusion. Chaos Soliton. Fract. 2006, 28(4):923-929.
93. Chen W., Sun H., Zhang X., Korovsak D. Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl. 2010, 59(5):1754-1758.
94. He J.-H. A new fractal derivation. Therm. Sci. 2011, 15(Suppl. 1):145-147.
95. He J.-H., Elagan S.K., Li Z.-B. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 2012, 376(4):257-259.
96. Yang X.-J. Local Fractional Functional Analysis and Its Applications 2011, Asian Academic Publisher Limited, Hong Kong.
97. Yang X.-J., Srivastava H.M., He J.-H., Baleanu D. Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives. Phys. Lett. A 2013, 377(28):1696-1700.
98. Yang X.-J., Baleanu D., Srivastava H.M. Local fractional similarity solution for the diffusion equation defined on Cantor sets. Appl. Math. Lett. 2015, 47:54-60.
99. Oliveira E.C.D., Machado J.A.T. A review of definitions for fractional derivatives and integral. Math. Probl. Eng. 2014, 6 pages.
100. Zhang Y. Solving initial-boundary value problems for local fractional differential equation by local fractional Fourier series method. Abstr. Appl. Anal. 2014, 5 pages.
101. Chen W., Zhang X.-D., Korovsak D. Investigation on fractional and fractal derivative relaxation-oscillation models. Int. J. Nonlinear Sci. Numer. Simul. 2010, 11(1):3-10.
102. Yang A.-M., Zhang Y.-Z., Long Y. The Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar. Therm. Sci. 2013, 17(3):707-713.
103. Samko S.G., Kilbas A.A., Marichev O.I. Fractional Integrals and Derivatives: Theory and Applications 1993, Gordon and Breach, Yverdon.
104. Ortigueira M.D. Fractional central differences and derivatives. J. Vib. Control. 2008, 14(9-10):1255-1266.
105. Ortigueira M.D., Trujillo J.J. Generalized Grünwald-Letnikov fractional derivative and its Laplace and Fourier transforms. J. Comput. Nonlinear Dyn. 2011, 6(3).
106. Machado J.A.T. Fractional derivatives: probability interpretation and frequency response of rational approximations. Commun. Nonlinear Sci. Numer. Simul. 2009, 14(9):3492-3497.
107. Machado J.A.T. Fractional coins and fractional derivatives. Abstr. Appl. Anal. 2013, 5 pages.
108. Ortigueira M.D., Coito F. From differences to derivatives. Fract. Calc. Appl. Anal. 2004, 7(4):459.
109. Atangana A., Secer A. A note on fractional order derivatives and table of fractional derivatives of some special functions. Abstr. Appl. Anal. 2013, 8 pages.
110. Jumarie G. On the representation of fractional Brownian motion as an integral with respect to dt?. Appl. Math. Lett. 2005, 18(7):739-748.
111. Jumarie G. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 2006, 51(9):1367-1376.
112. Jumarie G. Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions. Appl. Math. Model. 2008, 32(5):836-859.
113. Jumarie G. Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for nondifferentiable functions. Appl. Math. Lett. 2009, 22(3):378-385.
114. Li C.-P., Zhao Z.-G. Introduction to fractional integrability and differentiability. Eur. Phys. J. 2011, 193(1):5-26.
115. Khalil R., Al-Horani M., Yousef A., Sababheh M. A new definition of fractional derivative. J. Comput. Appl. Math. 2014, 264:65-70.
116. Abdeljawad T. On conformable fractional calculus. J. Comput. Appl. Math. 2015, 279:57-66.
117. Sabzikar F., Meerschaert M.M., Chen J. Tempered fractional calculus. J. Comput. Phys. 2014, http://dx.doi.org/10.1016/j.jcp.2014.04.024.
118. Katugampola U.N. New approach to a generalized fractional integral. Appl. Math. Comput. 2011, 218(3):860-865.
119. Caputo M., Fabrizio M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1(2):73-85.
120. Losada J., Nieto J.J. Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1(2):87-92.
121. Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations to Methods of Their Solution and Some of Their Applications 1998, vol. 198. Academic Press, New York.
122. Machado J.A.T., Kiryakova V., Mainardi F. A poster about the recent history of fractional calculus. Fract. Calc. Appl. Anal. 2010, 13(3):329-334.
123. Machado J.A.T., Kiryakova V., Mainardi F. Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(3):1140-1153.
124. Magin R.L. Fractional Calculus in Bioengineering 2006, 149. Begell House Publishers, Redding.
125. Kiryakova V.S. Generalized Fractional Calculus and Applications 1993, 301. Longman Scientific and Technical, Harlow (Essex).
126. Debnath L. A brief historical introduction to fractional calculus. Int. J. Math. Educ. Sci. Technol. 2004, 35(4):487-501.
127. Tarasov V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media 2011, Springer Science and Business Media, New York.
128. Ortigueira M.D. Fractional Calculus for Scientists and Engineers 2011, 84. Springer Science and Business Media, New York.
129. Meerschaert M.M., Sikorskii A. Stochastic Models for Fractional Calculus 2011, 43. Walter de Gruyter, New York.
130. Machado J.A.T., Galhano A.M., Trujillo J.J. On development of fractional calculus during the last fifty years. Scientometrics 2014, 98(1):577-582.
131. Monje C.A., Chen Y., Vinagre B.M., Xue D., Feliu-Batlle V. Fractional-Order Systems and Controls: Fundamentals and Applications 2010, Springer Science and Business Media, New York.
132. Baleanu D., Diethelm K., Scalas E., Trujillo J.J. Models and Numerical Methods 2012, 3:10-16. World Scientific, Singapore.
133. Caponetto R. Fractional Order Systems: Modeling and Control Applications 2010, 72. World Scientific, Singapore.
134. Anastassiou G.A. Fractional Differentiation Inequalities 2009, Springer, New York.
135. West B., Bologna M., Grigolini P. Physics of Fractal Operators 2003, Springer Science and Business Media, New York.
136. Zaslavsky G.M. Hamiltonian Chaos and Fractional Dynamics 2008, Oxford University (Clarendon) Press, Oxford, London, New York.
137. Baleanu D., Machado J.A.T., Luo A.-C. Fractional Dynamics and Control 2011, Springer Science and Business Media, New York.
138. Klafter J., Lim S.-C., Metzler R. Fractional Dynamics: Recent Advances 2012, World Scientific, Singapore.
139. Diethelm K. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type 2004, Springer Science and Business Media, New York.
140. Mainardi F. Fractional Calculus Waves in Linear Viscoelasticity: An Introduction to Mathematical Models 2010, World Scientific, Singapore.
141. Malinowska A.B., Torres D.F.M. Fractional Calculus of Variations 2012, Imperial College Press, Singapore.
142. Das S., Pan I. Fractional Order Signal Processing: Introductory Concepts and Applications 2011, Springer Science and Business Media, New York.
143. Uchaikin V.V. Fractional Derivatives for Physicists and Engineers 2013, Springer, Berlin.
144. Herrmann R. Fractional Calculus: An Introduction for Physicists 2014, World Scientific, Singapore.
145. Chen G.-S. Generalizations of Hölder's and some related integral inequalities on fractal space. J. Funct. Spaces Appl. 2013, 9 pages.
146. Wei W., Srivastava H.M., Zhang Y., Wang L., Shen P., Zhang J. A local fractional integral inequality on fractal space analogous to Anderson's inequality. Abstr. Appl. Anal. 2014, 7 pages.
147. Chen G.-S., Srivastava H.M., Wang P., Wie W. Some further generalizations of Hölder's inequality and related results on fractal space. Abstr. Appl. Anal. 2014, 7 pages.
148. Zhang Y., Baleanu D., Yang X.-J. On a local fractional wave equation under fixed entropy arising in fractal hydrodynamics. Entropy 2014, 16(12):6254-6262.
149. Li Y.-Y., Zhao L.Y., Xie G.-N., Baleanu D., Yang X.-J., Zhao K. Local fractional Poisson and Laplace equations with applications to electrostatics in fractal domain. Adv. Math. Phys. 2014, 5 pages.
150. Yang X.-J., Baleanu D. Local fractional variational iteration method for Fokker-Planck equation on a Cantor set. Acta Univ. 2013, 23(2):3-8.
151. Yang X.-J., Baleanu D., Machado J.A.T. Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis. Bound. Value Probl. 2013, 1:1-16.
152. Su W.-H., Baleanu D., Yang X.-J., Jafari H. Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method. Fixed Point Theory Appl. 2013, 1:1-11.
153. Yang X.-J., Baleanu D., Machado J.A.T. Systems of Navier-Stokes equations on Cantor sets. Math. Probl. Eng. 2013, 8 pages.
154. Liu H.-Y., He J.-H., Li Z.B. Fractional calculus for nanoscale flow and heat transfer. Int. J. Numer. Methods Heat Fluid Flow 2014, 24(6):1227-1250.
155. Zhao Y., Baleanu D., Cattani C., Cheng D.-F., Yang X.-J. Maxwell's equations on Cantor sets: a local fractional approach. Adv. High Energy Phys. 2013, 6 pages.
156. Wang L.-F., Yang X.-J., Baleanu D., Cattani C., Zhao Y. Fractal dynamical model of vehicular traffic flow within the local fractional conservation laws. Abstr. Appl. Anal. 2014, 5 pages.
157. Yang A.-M., Zhang Y.-Z., Cattani C., Xie G.-N., Rashidi M.M., Zhou Y.-J., ang X.-J. Application of local fractional series expansion method to solve Klein-Gordon equations on Cantor sets. Abstr. Appl. Anal. 2014, 6 pages.
158. Yang X.-J., Baleanu D., He J.-H. Transport equations in fractal porous media within fractional complex transform method. Proc. Rom. Acad. Series A 2013, 14(4):287-292.
159. Hao Y.-J., Srivastava H.M., Jafari H., Yang X.-J. Helmholtz and diffusion equations associated with local fractional derivative operators involving the Cantorian and Cantor-type cylindrical coordinates. Adv. Math. Phys. 2013, 5 pages.
160. Yang X.-J., Hristov J., Srivastava H.M., Ahmad B. Modelling fractal waves on shallow water surfaces via local fractional Korteweg-de Vries equation. Abstr. Appl. Anal. 2014, 10 pages.
161. Yang X.-J., Machado J.A.T., Hristov J. Nonlinear dynamics for local fractional Burgers' equation arising in fractal flow. Nonlinear Dyn. 2015, 10.1007/s11071-015-2085-2.
162. Niu X.-F., Zhang C.-L., Li Z.-B., Zhao Y. Local fractional derivative boundary value problems for Tricomi equation arising in fractal transonic flow. Abstr. Appl. Anal. 2014, 5 pages.
163. Cattani C., Srivastava H.M., Yang X.-J. Fractional Dynamics 2015, Emerging Science Publishers, Berlin.
164. Yang Y.-J., Baleanu D., Baleanu M.C. Observing diffusion problems defined on Cantor sets in different coordinate systems. Thermal Sci. 2015, 10.2298/TSCI141126065Y.
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180. Yang X.-J. Advanced Local Fractional Calculus and Its Applications 2012, World Science, New York.
181. Kolwankar K.M., Gangal A.D. Fractional differentiability of nowhere differentiable functions and dimensions. Chaos 1996, 6(4):505-513.
182. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations North-Holland Mathematical Studies 2006, 204. Elsevier (North-Holland) Science Publishers, Amsterdam, London, New York.
183. Sabatier J., Agrawal O.P., Machado J.A.T. Advances in Fractional Calculus 2007, Springer, Berlin, Heidelberg, New York.
184. Carpinteri A., Mainardi F. Fractals and Fractional Calculus in Continuum Mechanics 1997, Springer, New York.
185. Oldham K.B., Spanier J. The Fractional Calculus 1974, Academic Press, New York.
186. Kolwankar K.M., Gangal A.D. Hölder exponents of irregular signals and local fractional derivatives. Pramana 1997, 48(1):49-68.
187. Kolwankar K.M., Gangal A.D. Local fractional Fokker-Planck equation. Phys. Rev. Lett. 1998, 80(2):214.
188. Carpinteri A., Cornetti P. A fractional calculus approach to the description of stress and strain localization in fractal media. Chaos Soliton. Fract. 2002, 13(1):85-94.
189. Carpinteri A., Chiaia B., Cornetti P. Static-kinematic duality and the principle of virtual work in the mechanics of fractal media. Comput. Methods Appl. Mech. Eng. 2001, 191(1):3-19.
190. Carpinteri A., Chiaia B., Cornetti P. The elastic problem for fractal media: basic theory and finite element formulation. Comput. Struct. 2004, 82(6):499-508.
191. Carpinteri A., Cornetti P., Kolwankar K.M. Calculation of the tensile and flexural strength of disordered materials using fractional calculus. Chaos Soliton. Fract. 2004, 21(3):623-632.
192. Chen Y., Yan Y., Zhang K. On the local fractional derivative. J. Math. Anal. Appl. 2010, 362(1):17-33.
193. Adda F.B., Cresson J. About non-differentiable functions. J. Math. Anal. Appl. 2001, 263(2):721-737.
194. Babakhani A., Daftardar-Gejji V. On calculus of local fractional derivatives. J. Math. Anal. Appl. 2002, 270(1):66-79.
195. Yang X.-J. Local fractional integral transforms. Prog. Nonlinear Sci. 2011, 4(1):1-225.
196. Chen W. Time-space fabric underlying anomalous diffusion. Chaos Soliton. Fract. 2006, 28(4):923-929.
197. Chen W., Sun H., Zhang X., Korovsak D. Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl. 2010, 59(5):1754-1758.
198. He J.-H. A new fractal derivation. Therm. Sci. 2011, 15(Suppl. 1):145-147.
199. He J.-H., Elagan S.K., Li Z.-B. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 2012, 376(4):257-259.
200. Yang X.-J. Local Fractional Functional Analysis and Its Applications 2011, Asian Academic Publisher Limited, Hong Kong.
201. Yang X.-J., Srivastava H.M., He J.-H., Baleanu D. Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives. Phys. Lett. A 2013, 377(28):1696-1700.
202. Yang X.-J., Baleanu D., Srivastava H.M. Local fractional similarity solution for the diffusion equation defined on Cantor sets. Appl. Math. Lett. 2015, 47:54-60.
203. Oliveira E.C.D., Machado J.A.T. A review of definitions for fractional derivatives and integral. Math. Probl. Eng. 2014, 6 pages.
204. Zhang Y. Solving initial-boundary value problems for local fractional differential equation by local fractional Fourier series method. Abstr. Appl. Anal. 2014, 5 pages.
205. Chen W., Zhang X.-D., Korovsak D. Investigation on fractional and fractal derivative relaxation-oscillation models. Int. J. Nonlinear Sci. Numer. Simul. 2010, 11(1):3-10.
206. Yang A.-M., Zhang Y.-Z., Long Y. The Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar. Therm. Sci. 2013, 17(3):707-713.
207. Samko S.G., Kilbas A.A., Marichev O.I. Fractional Integrals and Derivatives: Theory and Applications 1993, Gordon and Breach, Yverdon.
208. Ortigueira M.D. Fractional central differences and derivatives. J. Vib. Control. 2008, 14(9-10):1255-1266.
209. Ortigueira M.D., Trujillo J.J. Generalized Grünwald-Letnikov fractional derivative and its Laplace and Fourier transforms. J. Comput. Nonlinear Dyn. 2011, 6(3).
210. Machado J.A.T. Fractional derivatives: probability interpretation and frequency response of rational approximations. Commun. Nonlinear Sci. Numer. Simul. 2009, 14(9):3492-3497.
211. Machado J.A.T. Fractional coins and fractional derivatives. Abstr. Appl. Anal. 2013, 5 pages.
212. Ortigueira M.D., Coito F. From differences to derivatives. Fract. Calc. Appl. Anal. 2004, 7(4):459.
213. Atangana A., Secer A. A note on fractional order derivatives and table of fractional derivatives of some special functions. Abstr. Appl. Anal. 2013, 8 pages.
214. Jumarie G. On the representation of fractional Brownian motion as an integral with respect to dt?. Appl. Math. Lett. 2005, 18(7):739-748.
215. Jumarie G. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 2006, 51(9):1367-1376.
216. Jumarie G. Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions. Appl. Math. Model. 2008, 32(5):836-859.
217. Jumarie G. Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for nondifferentiable functions. Appl. Math. Lett. 2009, 22(3):378-385.
218. Li C.-P., Zhao Z.-G. Introduction to fractional integrability and differentiability. Eur. Phys. J. 2011, 193(1):5-26.
219. Khalil R., Al-Horani M., Yousef A., Sababheh M. A new definition of fractional derivative. J. Comput. Appl. Math. 2014, 264:65-70.
220. Abdeljawad T. On conformable fractional calculus. J. Comput. Appl. Math. 2015, 279:57-66.
221. Sabzikar F., Meerschaert M.M., Chen J. Tempered fractional calculus. J. Comput. Phys. 2014, http://dx.doi.org/10.1016/j.jcp.2014.04.024.
222. Katugampola U.N. New approach to a generalized fractional integral. Appl. Math. Comput. 2011, 218(3):860-865.
223. Caputo M., Fabrizio M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1(2):73-85.
224. Losada J., Nieto J.J. Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1(2):87-92.
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235. Monje C.A., Chen Y., Vinagre B.M., Xue D., Feliu-Batlle V. Fractional-Order Systems and Controls: Fundamentals and Applications 2010, Springer Science and Business Media, New York.
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290. He J.-H. A new fractal derivation. Therm. Sci. 2011, 15(Suppl. 1):145-147.
291. He J.-H., Elagan S.K., Li Z.-B. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 2012, 376(4):257-259.
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302. Yang X.-J. Advanced Local Fractional Calculus and Its Applications 2012, World Science, New York.
303. Kolwankar K.M., Gangal A.D. Fractional differentiability of nowhere differentiable functions and dimensions. Chaos 1996, 6(4):505-513.
304. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations North-Holland Mathematical Studies 2006, 204. Elsevier (North-Holland) Science Publishers, Amsterdam, London, New York.
305. Sabatier J., Agrawal O.P., Machado J.A.T. Advances in Fractional Calculus 2007, Springer, Berlin, Heidelberg, New York.
306. Carpinteri A., Mainardi F. Fractals and Fractional Calculus in Continuum Mechanics 1997, Springer, New York.
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308. Kolwankar K.M., Gangal A.D. Hölder exponents of irregular signals and local fractional derivatives. Pramana 1997, 48(1):49-68.
309. Kolwankar K.M., Gangal A.D. Local fractional Fokker-Planck equation. Phys. Rev. Lett. 1998, 80(2):214.
310. Carpinteri A., Cornetti P. A fractional calculus approach to the description of stress and strain localization in fractal media. Chaos Soliton. Fract. 2002, 13(1):85-94.
311. Carpinteri A., Chiaia B., Cornetti P. Static-kinematic duality and the principle of virtual work in the mechanics of fractal media. Comput. Methods Appl. Mech. Eng. 2001, 191(1):3-19.
312. Carpinteri A., Chiaia B., Cornetti P. The elastic problem for fractal media: basic theory and finite element formulation. Comput. Struct. 2004, 82(6):499-508.
313. Carpinteri A., Cornetti P., Kolwankar K.M. Calculation of the tensile and flexural strength of disordered materials using fractional calculus. Chaos Soliton. Fract. 2004, 21(3):623-632.
314. Chen Y., Yan Y., Zhang K. On the local fractional derivative. J. Math. Anal. Appl. 2010, 362(1):17-33.
315. Adda F.B., Cresson J. About non-differentiable functions. J. Math. Anal. Appl. 2001, 263(2):721-737.
316. Babakhani A., Daftardar-Gejji V. On calculus of local fractional derivatives. J. Math. Anal. Appl. 2002, 270(1):66-79.
317. Yang X.-J. Local fractional integral transforms. Prog. Nonlinear Sci. 2011, 4(1):1-225.
318. Chen W. Time-space fabric underlying anomalous diffusion. Chaos Soliton. Fract. 2006, 28(4):923-929.
319. Chen W., Sun H., Zhang X., Korovsak D. Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl. 2010, 59(5):1754-1758.
320. He J.-H. A new fractal derivation. Therm. Sci. 2011, 15(Suppl. 1):145-147.
321. He J.-H., Elagan S.K., Li Z.-B. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 2012, 376(4):257-259.
322. Yang X.-J. Local Fractional Functional Analysis and Its Applications 2011, Asian Academic Publisher Limited, Hong Kong.
323. Yang X.-J., Srivastava H.M., He J.-H., Baleanu D. Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives. Phys. Lett. A 2013, 377(28):1696-1700.
324. Yang X.-J., Baleanu D., Srivastava H.M. Local fractional similarity solution for the diffusion equation defined on Cantor sets. Appl. Math. Lett. 2015, 47:54-60.
325. Oliveira E.C.D., Machado J.A.T. A review of definitions for fractional derivatives and integral. Math. Probl. Eng. 2014, 6 pages. Article ID 238459.
326. Zhang Y. Solving initial-boundary value problems for local fractional differential equation by local fractional Fourier series method. Abstr. Appl. Anal. 2014, 5 pages. Article ID 912464.
327. Chen W., Zhang X.-D., Korovsak D. Investigation on fractional and fractal derivative relaxation-oscillation models. Int. J. Nonlinear Sci. Numer. Simul. 2010, 11(1):3-10.
328. Yang A.-M., Zhang Y.-Z., Long Y. The Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar. Therm. Sci. 2013, 17(3):707-713.
329. Samko S.G., Kilbas A.A., Marichev O.I. Fractional Integrals and Derivatives: Theory and Applications 1993, Gordon and Breach, Yverdon.
330. Ortigueira M.D. Fractional central differences and derivatives. J. Vib. Control. 2008, 14(9-10):1255-1266.
331. Ortigueira M.D., Trujillo J.J. Generalized Grünwald-Letnikov fractional derivative and its Laplace and Fourier transforms. J. Comput. Nonlinear Dyn. 2011, 6(3). Article ID 034501.
332. Machado J.A.T. Fractional derivatives: probability interpretation and frequency response of rational approximations. Commun. Nonlinear Sci. Numer. Simul. 2009, 14(9):3492-3497.
333. Machado J.A.T. Fractional coins and fractional derivatives. Abstr. Appl. Anal. 2013, 5 pages. Article ID 205097.
334. Ortigueira M.D., Coito F. From differences to derivatives. Fract. Calc. Appl. Anal. 2004, 7(4):459.
335. Atangana A., Secer A. A note on fractional order derivatives and table of fractional derivatives of some special functions. Abstr. Appl. Anal. 2013, 8 pages. Article ID 279681.
336. Jumarie G. On the representation of fractional Brownian motion as an integral with respect to dt?. Appl. Math. Lett. 2005, 18(7):739-748.
337. Jumarie G. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 2006, 51(9):1367-1376.
338. Jumarie G. Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions. Appl. Math. Model. 2008, 32(5):836-859.
339. Jumarie G. Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for nondifferentiable functions. Appl. Math. Lett. 2009, 22(3):378-385.
340. Li C.-P., Zhao Z.-G. Introduction to fractional integrability and differentiability. Eur. Phys. J. 2011, 193(1):5-26.
341. Khalil R., Al-Horani M., Yousef A., Sababheh M. A new definition of fractional derivative. J. Comput. Appl. Math. 2014, 264:65-70.
342. Abdeljawad T. On conformable fractional calculus. J. Comput. Appl. Math. 2015, 279:57-66.
343. Sabzikar F., Meerschaert M.M., Chen J. Tempered fractional calculus. J. Comput. Phys. 2014, URL. http://dx.doi.org/10.1016/j.jcp.2014.04.024.
344. Katugampola U.N. New approach to a generalized fractional integral. Appl. Math. Comput. 2011, 218(3):860-865.
345. Caputo M., Fabrizio M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1(2):73-85.
346. Losada J., Nieto J.J. Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1(2):87-92.
347. Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations to Methods of Their Solution and Some of Their Applications 1998, vol. 198. Academic Press, New York.
348. Machado J.A.T., Kiryakova V., Mainardi F. A poster about the recent history of fractional calculus. Fract. Calc. Appl. Anal. 2010, 13(3):329-334.
349. Machado J.A.T., Kiryakova V., Mainardi F. Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(3):1140-1153.
350. Magin R.L. Fractional Calculus in Bioengineering 2006, 149. Begell House Publishers, Redding.
351. Kiryakova V.S. Generalized Fractional Calculus and Applications 1993, 301. Longman Scientific and Technical, Harlow (Essex).
352. Debnath L. A brief historical introduction to fractional calculus. Int. J. Math. Educ. Sci. Technol. 2004, 35(4):487-501.
353. Tarasov V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media 2011, Springer Science and Business Media, New York.
354. Ortigueira M.D. Fractional Calculus for Scientists and Engineers 2011, 84. Springer Science and Business Media, New York.
355. Meerschaert M.M., Sikorskii A. Stochastic Models for Fractional Calculus 2011, 43. Walter de Gruyter, New York.
356. Machado J.A.T., Galhano A.M., Trujillo J.J. On development of fractional calculus during the last fifty years. Scientometrics 2014, 98(1):577-582.
357. Monje C.A., Chen Y., Vinagre B.M., Xue D., Feliu-Batlle V. Fractional-Order Systems and Controls: Fundamentals and Applications 2010, Springer Science and Business Media, New York.
358. Baleanu D., Diethelm K., Scalas E., Trujillo J.J. Models and Numerical Methods 2012, 3:10-16. World Scientific, Singapore.
359. Caponetto R. Fractional Order Systems: Modeling and Control Applications 2010, 72. World Scientific, Singapore.
360. Anastassiou G.A. Fractional Differentiation Inequalities 2009, Springer, New York.
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