### AN ELEMENTARY AND INTRODUCTORY APPROACH TO FRACTIONAL CALCULUS AND ITS APPLICATIONS

#### Abstract

A b s t r a c t: The subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It does indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this paper* is to present a brief elementary and introductory approach to the theory of fractional calculus and its applications especially in developing solutions of certain interesting families of ordinary and partial fractional differintegral equations. Relevant connections of some of the results presented in this lecture with those obtained in many other earlier works on this subject will also be indicated.

#### Keywords

#### Full Text:

PDF#### References

C. M. Carracedo and M. S. Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematical Studies, Vol. 187, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2001.

M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractals, 14 (2002), 433–440.

M. M. El-Borai, Semigroups and some nonlinear fractional differential equations, Appl. Math. Comput. 149 (2004), 823–831.

M. M. El-Borai, The fundamental solutions for fractional evolution equations of parabolic type, J. Appl. Math. Stochast. Anal. 2004 (2004), 197–211.

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vols. I and II, McGraw-Hill Book Company, New York, Toronto and London, 1953.

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. II, McGraw-Hill Book Company, New York, Toronto and London, 1954.

M. Fukuhara, Ordinary Differential Equations, Vol. II, Iwanami Shoten, Tokyo, 1941 (Japanese).

L. Galué, N-fractional calculus operator ν method applied to some second order nonhomogeneous equations, J. Fract. Calc. 16 (1999), 85–97.

R. Gorenflo, F. Mainardi and H. M. Srivastava, Special functions in fractional relaxation- oscillation and fractional diffusion-wave phenomena, Proceedings of the Eighth International Colloquium on Differential Equations, (Plovdiv, Bulgaria; August 18–23, 1997) (D. Bainov, Editor), VSP Publishers, Utrecht and Tokyo, 1998, pp. 195–202.

R. Gorenflo and S. Vessela, Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics, Vol. 1461, Springer-Verlag, Berlin, Heidelberg, New York and London, 1991.

R. Hilfer (Editor), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000.

E. L. Ince, Ordinary Differential Equations, Longmans, Green and Company, London, 1927; Reprinted by Dover Publications, New York, 1956.

R. N. Kalia (Editor), Recent Advances in Fractional Calculus, Global Publishing Company, Sauk Rapids (Minnesota), 1993.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.

V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics, Vol. 301, Longman Scientific and Technical, Harlow (Essex), 1993.

S.-D. Lin, W.-C. Ling, K. Nishimoto and H. M. Srivastava, A simple fractionalcalculus approach to the solutions of the Bessel differential equation of general order and some of its applications, Comput. Math. Appl. 49 (2005), 1487–1498.

S.-D. Lin and K. Nishimoto, N-Method to a generalized associated Legendre equation, J. Fract. Calc. 14 (1998), 95–111.

S.-D. Lin and K. Nishimoto, New finding of particular solutions for a generalized associated Legendre equation, J. Fract. Calc. 18 (2000), 9–37.

S.-D. Lin, K. Nishimoto, T. Miyakoda and H. M. Srivastava, Some differintegral formulas for power, composite and rational functions, J. Fract. Calc. 32 (2000), 87–98.

S.-D. Lin, H. M. Srivastava, S.-T. Tu and P.-Y. Wang, Some families of linear ordinary and partial differential equations solvable by means of fractional calculus, Internat. J. Differential Equations Appl. 4 (2002), 405–421.

S.-D. Lin, Y.-S. Tsai and P.-Y. Wang, Explicit solutions of a certain class of associated Legendre equations by means of fractional calculus, Appl. Math. Comput. 187 (2007), 280–289.

S.-D. Lin, S.-T. Tu, I-C. Chen and H. M. Srivastava, Explicit solutions of a certain family of fractional differintegral equations, Hyperion Sci. J. Ser. A Math. Phys. Electric. Engrg. 2 (2001), 85–90.

S.-D. Lin, S.-T. Tu and H. M. Srivastava, Explicit solutions of certain ordinary differential equations by means of fractional calculus, J. Fract. Calc. 20 (2001), 35–43.

S.-D. Lin, S.-T. Tu and H. M. Srivastava, Certain classes of ordinary and partial differential equations solvable by means of fractional calculus, Appl. Math. Comput. 131 (2002), 223–233.

S.-D. Lin, S.-T. Tu and H. M. Srivastava, Explicit solutions of some classes of non- Fuchsian differential equations by means of fractional calculus, J. Fract. Calc. 21 (2002), 49–60.

A. C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions, Pitman Research Notes in Mathematics, Vol. 31, Pitman Publishing Limited, London, 1979.

A. C. McBride and G. F. Roach (Editors), Fractional Calculus (Glasgow, Scotland; August 5–11, 1984), Pitman Research Notes in Mathematics, Vol. 138, Pitman Publishing Limited, London, 1985.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley and Sons, New York, Chichester, Brisbane, Toronto and Singapore, 1993.

K. Nishimoto, Fractional Calculus, Vols. I, II, III, IV, and V, Descartes Press, Koriyama, 1984, 1987, 1989, 1991, and 1996.

K. Nishimoto (Editor), Fractional Calculus and Its Applications (Tokyo; May 29 – June 1, 1989), Nihon University (College of Engineering), Koriyama, 1990.

K. Nishimoto, An Essence of Nishimoto ‘s Fractional Calculus Calculus of the 21st Century): Integrations and Differentiations of Arbitrary Order, Descartes Press, Koriyama, 1991.

K. Nishimoto, Operator ν method to nonhomogeneous Gauss and Bessel equations, J. Fract. Calc. 9 (1996), 1–15.

K. Nishimoto, J. Aular de Duran and L. Galué, N-Fractional calculus operator Nν method to nonhomogeneous Fukuhara equations. I, J. Fract. Calc. 9 (1996), 23–31.

K. Nishimoto and S. Salinas de Romero, N-Fractional calculus operator Nν method to nonhomogeneous and homogeneous Whittaker equations. I, J. Fract. Calc. 9 (1996), 17–22.

K. Nishimoto, S. Salinas de Romero, J. Matera and A. I. Prieto, N-Method to the homogeneous Whittaker equations, J. Fract. Calc. 15 (1999), 13–23.

K. Nishimoto, S. Salinas de Romero, J. Matera and A. I. Prieto, N-Method to the homogeneous Whittaker equations (Revise and Supplement), J. Fract. Calc. 16 (1999), 123–128.

K. Nishimoto, H. M. Srivastava and S.-T. Tu, Application of fractional calculus in solving certain classes of Fuchsian differential equations, J. College Engrg. Nihon Univ. Ser. B 32 (1991), 119–126.

K. Nishimoto, H. M. Srivastava and S.-T. Tu, Solutions of some second-order linear differential equations by means of fractional calculus, J. College Engrg. Nihon Univ. Ser. B 33 (1992), 15–25.

K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York and London, 1974.

H. M. Ozaktas, Z. Zalevsky and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, Wiley Series in Pure and Applied Optics, John Wiley and Sons, New York, Chichester, Brisbane, Toronto and Singapore, 2000.

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, Vol. 198, Academic Press, New York, London, Sydney, Tokyo and Toronto, 1999.

A. I. Prieto, S. Salinas de Romero and H. M. Srivastava, Some fractional calculus results involving the generalized Lommel-Wright and related functions, Appl. Math. Lett., 20 (2007), 17–22.

B. Ross (Editor), Fractional Calculus and Its Applications (West Haven, Connecticut; June 15–16, 1974), Lecture Notes in Mathematics, Vol. 457, Springer-Verlag, Berlin, Heidelberg and New York, 1975.

B. Rubin, Fractional Integrals and Potentials, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific and Technical, Harlow (Essex), 1996.

S. Salinas de Romero and K. Nishimoto, N-Fractional calculus operator Nν method to nonhomogeneous and homogeneous Whittaker equations. II (Some illustrative examples), J. Fract. Calc., 12 (1997), 29–35.

S. Salinas de Romero and H. M. Srivastava, An application of the N-fractional calculus operator method to a modified Whittaker equation, Appl. Math. Comput., 115 (2000), 11–21.

S. G. Samko, A. A. Kilbas and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications (in Russian), „Nauka i Tekhnika“, Minsk, 1987.

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Reading, Tokyo, Paris, Berlin and Langhorne (Pennsylvania), 1993.

H. M. Srivastava and R. G. Buschman, Convolution Integral Equations with Special Function Kernels, Halsted Press, John Wiley and Sons, New York, 1977.

H. M. Srivastava and R. G. Buschman, Theory and Applications of Convolution Integral Equations, Kluwer Series on Mathematics and Its Applications, Vol. 79, Kluwer Academic Publishers, Dordrecht, Boston and London, 1992.

H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi and Madras, 1982.

H. M. Srivastava and B. R. K. Kashyap, Special Functions in Queuing Theory and Related Stochastic Processes, Academic Press, New York, London, and Toronto, 1982.

H. M. Srivastava, S.-D. Lin, Y.-T. Chao and P.-Y. Wang, Explicit solutions of a certain class differential equations by means of fractional calculus, Russian J. Math. Phys., 14 (2007), 357–365.

H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1984.

H. M. Srivastava and S. Owa (Editors), Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989.

H. M. Srivastava and S. Owa (Editors), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992.

H. M. Srivastava, S. Owa and K. Nishimoto, Some fractional differintegral equations, J. Math. Anal. Appl., 106 (1985), 360–366.

F. G. Tricomi, Funzioni Ipergeometriche Confluenti, Edizioni Cremonese, Rome, 1954.

S.-T. Tu, D.-K. Chyan and H. M. Srivastava, Some families of ordinary and partial fractional differintegral equations, Integral Transform. Spec. Funct., 11 (2001), 291–302.

S.-T. Tu, Y.-T. Huang, I-C. Chen and H. M. Srivastava, A certain family of fractional differintegral equations, Taiwanese J. Math., 4 (2000), 417–426.

S.-T. Tu, S.-D. Lin, Y.-T. Huang and H. M. Srivastava, Solutions of a certain class of fractional differintegral equations, Appl. Math. Lett., 14 (2) (2001), 223–229.

S.-T. Tu, S.-D. Lin and H. M. Srivastava, Solutions of a class of ordinary and partial differential equations via fractional calculus, J. Fract. Calc., 18 (2000), 103– 110.

P.-Y. Wang, S.-D. Lin and H. M. Srivastava, Explicit solutions of Jacobi and Gauss differential equations by means of operators of fractional calculus, Appl. Math. Comput., 199 (2008), 760–769.

P.-Y. Wang, S.-D. Lin and S.-T. Tu, A survey of fractional-calculus approaches the solutions of the Bessel differential equation of general order, Appl. Math. Comput., 187 (2007), 544–555.

G. N. Watson, A Treatise on the Theory of Bessel Functions, Second edition, Cambridge University Press, Cambridge, London and New York, 1944.

E. T. Whittaker and G. N. Watson, 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, Fourth edition, Cambridge University Press, Cambridge, London and New York, 1927.

DOI: http://dx.doi.org/10.20903/csnmbs.masa.2008.29.1-2.11

### Refbacks

- There are currently no refbacks.

This work is licensed under a Creative Commons Attribution 4.0 International License.

**Contact details**

Bul. Krste Misirkov br.2

1000 Skopje, Republic of Macedonia

**Tel.**++389 2 3235-400

**cell:**++389 71 385-106

**mail:**manu@manu.edu.mk

**About the journal**

CSNMBS is a part of the MASA

**Contribution**series. Published by the Section Natural, Mathematical and Biotechnical Sciences.

**About this site**

Maintained by the Researh center for Materials and Enviroment - MANU/MASA.

Site (including the theme) set, adapted by MASA - CSIT.