Fractional Versions of the Fundamental Theorem of Calculus

ABSTRACT

The concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. Some properties involving the particular Riemann-Liouville integral are mentioned. By means of this concept we present the fractional derivatives, specifically, the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz versions are discussed. The so-called fundamental theorem of fractional calculus is presented and discussed in all these different versions.

The concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. Some properties involving the particular Riemann-Liouville integral are mentioned. By means of this concept we present the fractional derivatives, specifically, the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz versions are discussed. The so-called fundamental theorem of fractional calculus is presented and discussed in all these different versions.

Cite this paper

E. Grigoletto and E. Oliveira, "Fractional Versions of the Fundamental Theorem of Calculus,"*Applied Mathematics*, Vol. 4 No. 7, 2013, pp. 23-33. doi: 10.4236/am.2013.47A006.

E. Grigoletto and E. Oliveira, "Fractional Versions of the Fundamental Theorem of Calculus,"

References

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[2] K. B. Oldham and J. Spanier, “The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order,” Academic Press, New York, 1974.

[3] R. L. Bagley and P. J. Torvik, “A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,” Journal of Rheology, Vol. 27, No. 3, 1983, pp. 201-210. doi:10.1122/1.549724

[4] J. A. Tenreiro Machado, V. Kiryakova and F. Mainardi, “A Poster about the Recent History of Fractional Calculus,” Fractional Calculus & Applied Analysis, Vol. 13, No. 3, 2010, pp. 329-334.

[5] J. A. Tenreiro Machado, V. Kiryakova and F. Mainardi, “A Poster about the Old History of Fractional Calculus,” Fractional Calculus & Applied Analysis, Vol. 13, No. 4, 2010, pp. 447-454.

[6] J. A. Tenreiro Machado, V. Kiryakova and F. Mainardi, “Recent History of Fractional Calculus,” Communications in Nonlinear Science and Numerical Simulation, Vol. 16, No. 3, 2011, pp. 1140-1153. doi:10.1016/j.cnsns.2010.05.027

[7] K. Diethelm, “The Analysis of Fractional Differential Equations,” Springer Verlag, Berlin, Heidelberg, 2010. doi:10.1007/978-3-642-14574-2

[8] R. Hilfer, “Applications of Fractional Calculus in Physics,” World Scientific, Singapore City, 2000.

[9] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and Applications of Fractional Differential Equations,” Elsevier, Amsterdam, 2006.

[10] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, 1999.

[11] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional ntegrals and Derivatives: Theory and Applications,” Gordon and Breach Science Publishers, Amsterdam, 1993.

[12] V. E. Tarasov, “Fractional Dynamics, Applications of Fractional Calculus to Dynamics of Particles, Fields and Media,” Springer, Heidelberg, 2010.

[13] V. E. Tarasov, “Fractional Vector Calculus and Fractional Maxwell’s Equations,” Annals of Physics, Vol. 323, No. 11, 2008, pp. 2756-2778. doi:10.1016/j.aop.2008.04.005

[14] H. Vic Dannon, “The Fundamental Theorem of the Fractional Calculus and the Meaning of Fractional Derivatives,” Gauge Institute Journal, Vol. 5, No. 1, 2009, pp. 1-26.

[15] N. Heymans and I. Podlubny, “Physical Interpretation of Initial Conditions for Fractional Differential Equations with Riemann-Liouville Fractional Derivative,” Rheologica Acta, Vol. 45, No. 5, 2006, pp. 765-772. doi:10.1007/s00397-005-0043-5

[16] I. Podlubny, “Geometric and Physical Interpretation of Fractional Integral and Fractional Differentiation,” Journal of Fractional Calculus & Applied Analysis, Vol. 5, No. 4, 2002, pp. 367-386.

[17] A. Cabada and G. Wang, “Positive Solutions of Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions,” Journal of Mathematical Analysis and Applications, Vol. 389, No. 1, 2012, pp. 403-411. doi:10.1016/j.jmaa.2011.11.065

[18] R. Figueiredo Camargo, “Fractional Calculus and Applications (in Portuguese)” Doctoral Thesis, UNICAMP, Campinas, 2009.

[19] J. A. Tenreiro Machado, “Discrete-Time Fractional-Order Controllers,” Journal of Fractional Calculus & Applied Analysis, Vol. 4, No. 1, 2001, pp. 47-66.

[20] R. Figueiredo Camargo, E. Capelas de Oliveira and J. Vaz Jr., “On the Generalized Mittag-Leffler Function and Its Application in a Fractional Telegraph Equation,” Mathematical Physsics, Analysis & Geometry, Vol. 15, No. 1, 2012, pp. 1-16. doi:10.1007/s11040-011-9100-8

[21] F. Silva Costa and E. Capelas de Oliveira, “Fractional Wave-Diffusion Equation with Periodic Conditions,” Journal of Mathematical Physics, Vol. 53, 2012, Article ID: 123520. doi:10.1063/1.4769270

[22] H. Jafari and S. Momani, “Solving Fractional Diffusion and wave Equations by Modified Homotopy Perturbation Method,” Physics Letters A, Vol. 370, No. 5-6, 2007, pp. 388-396. doi:10.1016/j.physleta.2007.05.118

[23] E. Contharteze Grigoletto, “Fractional Differential Equations and the Mittag-Leffler Functions (in Portuguese),” Ph.D. Thesis, UNICAMP, Campinas, to Appear.

[1] [1] K. S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations,” John Wiley & Sons, Inc., New York, 1993.

[2] K. B. Oldham and J. Spanier, “The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order,” Academic Press, New York, 1974.

[3] R. L. Bagley and P. J. Torvik, “A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,” Journal of Rheology, Vol. 27, No. 3, 1983, pp. 201-210. doi:10.1122/1.549724

[4] J. A. Tenreiro Machado, V. Kiryakova and F. Mainardi, “A Poster about the Recent History of Fractional Calculus,” Fractional Calculus & Applied Analysis, Vol. 13, No. 3, 2010, pp. 329-334.

[5] J. A. Tenreiro Machado, V. Kiryakova and F. Mainardi, “A Poster about the Old History of Fractional Calculus,” Fractional Calculus & Applied Analysis, Vol. 13, No. 4, 2010, pp. 447-454.

[6] J. A. Tenreiro Machado, V. Kiryakova and F. Mainardi, “Recent History of Fractional Calculus,” Communications in Nonlinear Science and Numerical Simulation, Vol. 16, No. 3, 2011, pp. 1140-1153. doi:10.1016/j.cnsns.2010.05.027

[7] K. Diethelm, “The Analysis of Fractional Differential Equations,” Springer Verlag, Berlin, Heidelberg, 2010. doi:10.1007/978-3-642-14574-2

[8] R. Hilfer, “Applications of Fractional Calculus in Physics,” World Scientific, Singapore City, 2000.

[9] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and Applications of Fractional Differential Equations,” Elsevier, Amsterdam, 2006.

[10] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, 1999.

[11] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional ntegrals and Derivatives: Theory and Applications,” Gordon and Breach Science Publishers, Amsterdam, 1993.

[12] V. E. Tarasov, “Fractional Dynamics, Applications of Fractional Calculus to Dynamics of Particles, Fields and Media,” Springer, Heidelberg, 2010.

[13] V. E. Tarasov, “Fractional Vector Calculus and Fractional Maxwell’s Equations,” Annals of Physics, Vol. 323, No. 11, 2008, pp. 2756-2778. doi:10.1016/j.aop.2008.04.005

[14] H. Vic Dannon, “The Fundamental Theorem of the Fractional Calculus and the Meaning of Fractional Derivatives,” Gauge Institute Journal, Vol. 5, No. 1, 2009, pp. 1-26.

[15] N. Heymans and I. Podlubny, “Physical Interpretation of Initial Conditions for Fractional Differential Equations with Riemann-Liouville Fractional Derivative,” Rheologica Acta, Vol. 45, No. 5, 2006, pp. 765-772. doi:10.1007/s00397-005-0043-5

[16] I. Podlubny, “Geometric and Physical Interpretation of Fractional Integral and Fractional Differentiation,” Journal of Fractional Calculus & Applied Analysis, Vol. 5, No. 4, 2002, pp. 367-386.

[17] A. Cabada and G. Wang, “Positive Solutions of Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions,” Journal of Mathematical Analysis and Applications, Vol. 389, No. 1, 2012, pp. 403-411. doi:10.1016/j.jmaa.2011.11.065

[18] R. Figueiredo Camargo, “Fractional Calculus and Applications (in Portuguese)” Doctoral Thesis, UNICAMP, Campinas, 2009.

[19] J. A. Tenreiro Machado, “Discrete-Time Fractional-Order Controllers,” Journal of Fractional Calculus & Applied Analysis, Vol. 4, No. 1, 2001, pp. 47-66.

[20] R. Figueiredo Camargo, E. Capelas de Oliveira and J. Vaz Jr., “On the Generalized Mittag-Leffler Function and Its Application in a Fractional Telegraph Equation,” Mathematical Physsics, Analysis & Geometry, Vol. 15, No. 1, 2012, pp. 1-16. doi:10.1007/s11040-011-9100-8

[21] F. Silva Costa and E. Capelas de Oliveira, “Fractional Wave-Diffusion Equation with Periodic Conditions,” Journal of Mathematical Physics, Vol. 53, 2012, Article ID: 123520. doi:10.1063/1.4769270

[22] H. Jafari and S. Momani, “Solving Fractional Diffusion and wave Equations by Modified Homotopy Perturbation Method,” Physics Letters A, Vol. 370, No. 5-6, 2007, pp. 388-396. doi:10.1016/j.physleta.2007.05.118

[23] E. Contharteze Grigoletto, “Fractional Differential Equations and the Mittag-Leffler Functions (in Portuguese),” Ph.D. Thesis, UNICAMP, Campinas, to Appear.