全部 标题 作者
关键词 摘要

OALib Journal期刊
ISSN: 2333-9721
费用：99美元
投递稿件

查看量下载量

Fractional Calculus, Fractional Differential Equations and Applications

DOI: 10.4236/oalib.1106244, PP. 1-9

Subject Areas: Mathematical Analysis

Keywords: Fractional Derivative, Fractional Differentiation, Factorial for the Integer Numbers, Riemann-Liouville Fractional Derivative

Full-Text   Cite this paper   Add to My Lib

Abstract

In this paper, we describe two approaches to the definition of fractional derivatives. We investigate the accuracy of the analysis method for solving the fractional order problem. We also give some improvements for the proof of the existence and uniqueness of the solution in fractional differential equations. Treatment of a fractional derivative operator has been made associated with the extended Appell hypergeometric functions of two or three variables and Lauricella hypergeometric function of three variables.

 Cite this paper

Mu’lla, M. A. M. (2020). Fractional Calculus, Fractional Differential Equations and Applications. Open Access Library Journal, 7, e6244. doi: http://dx.doi.org/10.4236/oalib.1106244.

References

[1]  Caponetto, R. (2010) Fractional Order Systems (Modelling and Control Applications). World Scientific, Singapore. https://doi.org/10.1142/7709
[2]  Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon.
[3]  Olver, F.W.J., Lozier, D.W., Boisvert, R.F. and Clark, C.W. (2010) NIST Handbook of Mathematical Functions [with 1 CD-ROM (Windows, Macintosh and UNIX)]. Cambridge University Press, Cambridge.
[4]  Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1981) Higher Transcendental Functions. Vol. 1, McGraw-Hill Book Corp., New York.
[5]  Diethelm, K. (2010) The Analysis of Fractional Differential Equations, an Application Oriented, Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics, Springer, Berlin. https://doi.org/10.1007/978-3-642-14574-2_8
[6]  Hilfer, R. (2003) Applications of Fractional Calculus in Physics. World Scientific, Singapore.
[7]  Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. North-Holland, Amsterdam.
[8]  Agarwal, P., Nieto, J.J. and Luo, M.J. (2017) Extended Riemann-Liouville Type Fractional Derivative Operator with Applications. Applied Mathematics & Information Sciences, 15, 12-29. https://doi.org/10.1515/math-2017-0137
[9]  Rainville, E.D. (1971) Special Functions. Macmillan Co., New York.
[10]  Chatterjee, A. (2005) Statistical Origins of Fractional Derivatives in Viscoelasticity. Journal of Sound and Vibration, 284, 1239-1245. https://doi.org/10.1016/j.jsv.2004.09.019
[11]  Diethelm, K. and Ford, N.J. (2004) Multi-Order Fractional Differential Equations and Their Numerical Solution. Applied Mathematics and Computation, 154, 621-640. https://doi.org/10.1016/S0096-3003(03)00739-2
[12]  Kochubei, A.N. (2009) Fractional Differential Equations: α-Entire Solutions, Regular and Irregular Singularities. Fractional Calculus and Applied Analysis, 12, 135-158.
[13]  Lubich, C. (1986) Discretized Fractional Calculus. SIAM Journal on Mathematical Analysis, 17, 704-719. https://doi.org/10.1137/0517050
[14]  Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.
[15]  Diethelm, K. (2007) Smoothness Properties of Solutions of Caputo-Type Fractional Differential Equations. Fractional Calculus and Applied Analysis, 10, 151-160.
[16]  Herrmann, R. (2011) Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore. https://doi.org/10.1142/8072
[17]  Mainardi, F. (2010) Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific, Singapore. https://doi.org/10.1142/p614
[18]  Diethelm, K. (2008) Multi-Term Fractional Differential Equations, Multi-Order Fractional Differential Systems and Their Numerical Solution. Journal Européen des Systèmes Automatisés, 42, 665-676. https://doi.org/10.3166/jesa.42.665-676
[19]  Miller, K.S. and Ross, B. (1993) An Introduction to Fractional Calculus and Fractional Differential Equations. Wiley, New York.
[20]  Baleanu, D., Guvenc, Z.B. and Machado, J.A.T. (2010) New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Berlin. https://doi.org/10.1007/978-90-481-3293-5
[21]  Luchko, Y. and Gorenflo, R. (1999) An Operational Method for Solving Fractional Differential Equations with the Caputo Derivatives. Acta Mathematica Vietnamica, 24, 207-233.
[22]  Baleanu, D., Agarwal, P., Parmar, R.K., Alqurashi, M.M. and Salahshour, S. (2015) Extended Riemann-Liouville Fractional Derivative Operator and Its Applications. Journal of Nonlinear Sciences and Applications, 10, 2914-2924. https://doi.org/10.22436/jnsa.010.06.06
[23]  ？zarslan, M.A. and ？zergin, E. (2010) Some Generating Relations for Extended Hypergeometric Functions via Generalized Fractional Derivative Operator. Mathematical and Computer Modelling, 52, 1825-1833. https://doi.org/10.1016/j.mcm.2010.07.011
[24]  Magin, R.L. (2011) Fractional Calculus Models of Complex Dynamics in Biological Tissues. Computers & Mathematics with Applications, 59, 1586-1593. https://doi.org/10.1016/j.camwa.2009.08.039

Full-Text


comments powered by Disqus

Contact Us

service@oalib.com

QQ:3279437679

WhatsApp +8615387084133

WeChat 1538708413




©2008-2020 OALib, Inc | Public user content licensed CC-BY 4.0 unless otherwise specified. OALib ISSN Online: 2333-9721 OALib ISSN Print: 2333-9705