Advertisement

Mittag-Leffler Functions and Fractional Calculus

Keywords

Fractional Derivative Fractional Calculus Fractional Differential Equation Fractional Integral Fractional Diffusion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agarwal, R. P. (1953). A Propos d’une note de M. Pierre Humbert, C. R. Acad. Sci. Paris, 296, 2031–2032.Google Scholar
  2. Agarwal, R. P. (1963). Generalized Hypergeometric Series, Asia Publishing House, Bombay, London and New York.Google Scholar
  3. Caputo, M. (1969). Elasticitá e Dissipazione, Zanichelli, Bologna.Google Scholar
  4. Dzherbashyan, M.M. (1966). Integral Transforms and Representation of Functions in Complex Domain (in Russian), Nauka, Moscow.Google Scholar
  5. Erdélyi, A. (1950-51). On some functional transformations, Univ. Politec. Torino, Rend. Sem. Mat. 10, 217–234.Google Scholar
  6. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1953). Higher Transcendental Functions, Vol. 1, McGraw - Hill, New York, Toronto and London.Google Scholar
  7. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1954). Tables of Integral Transforms, Vol. 1, McGraw - Hill, New York, Toronto and London.Google Scholar
  8. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1954a). Tables of Integral Transforms, Vol. 2, McGraw - Hill, New York, Toronto and London.Google Scholar
  9. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1955). Higher Transcendental Functions, Vol. 3, McGraw - Hill, New York, Toronto and London.MATHGoogle Scholar
  10. Fox, C. (1963). Integral transforms based upon fractional integration, Proc. Cambridge Philos. Soc., 59, 63–71.CrossRefMathSciNetGoogle Scholar
  11. Haubold, H. J. and Mathai, A. M. (2000). The fractional kinetic equation and thermonuc1ear functions, Astrophysics and Space Science, 273, 53–63.MATHCrossRefADSGoogle Scholar
  12. Hilfer, R. (Ed.). (2000). Applications of Fractional Calculus in Physics, World Scientific, Singapore.MATHGoogle Scholar
  13. Kalla, S. L. and Saxena, R. K. (1969). Integral operators involving hypergeometric functions, Math. Zeitschr., 108, 231–234.MATHCrossRefMathSciNetGoogle Scholar
  14. Kilbas. A. A. and Saigo, M. (1998). Fractional calculus of the H-function, Fukuoka Univ. Science Reports, 28, 41–51.MATHMathSciNetGoogle Scholar
  15. Kilbas, A. A. and Saigo, M. (1996). On Mittag- Leffler type function, fractional calculus operators and solutions of integral equations, Integral Transforms and Special Functions, 4, 355–370.MATHCrossRefMathSciNetGoogle Scholar
  16. Kilbas, A. A, Saigo, M. and Saxena, R. K. (2002). Solution of Volterra integrodifferential equations with generalized Mittag-Leffler function in the kernels, J. Integral Equations and Applications, 14, 377–396.MATHCrossRefMathSciNetGoogle Scholar
  17. Kilbas, A. A., Saigo, M. and Saxena, R. K. (2004). Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms and Special Functions, 15, 31–49.MATHCrossRefMathSciNetGoogle Scholar
  18. Kober, H. (1940). On fractional integrals and derivatives, Quart. J. Math. Oxford, Ser. ll, 193–211.Google Scholar
  19. Love, E. R (1967). Some integral equations involving hypergeometric functions,Proc. Edin. Math. Soc., 15(2), 169–198.MATHCrossRefMathSciNetGoogle Scholar
  20. Mathai, A. M. and Saxena, R. K. (1973). Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Lecture Notes in Mathematics, 348, Springer- Verlag, Berlin, Heidelberg.MATHGoogle Scholar
  21. Mathai, A. M. and Saxena, R. K. (1978). The H-function with Applications in Statistics and Other Disciplines, John Wiley and Sons, New York - London - Sydney.MATHGoogle Scholar
  22. Miller, K. S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.MATHGoogle Scholar
  23. Mittag-Leffler, G. M. (1903). Sur la nouvelle fonction E α(x), C. R. Acad. Sci. Paris, (Ser. II) 137, 554–558.Google Scholar
  24. Oldham, K. B. and Spanier, J. (1974). The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York.MATHGoogle Scholar
  25. Podlubny, 1. (1999). Fractional Differential Equations, Academic Press, San Diego.MATHGoogle Scholar
  26. Podlubny, 1. (2002). Geometric and physical interpretations of fractional integration and fractional differentiation, Frac. Calc. Appl. Anal., 5(4), 367–386.MATHMathSciNetGoogle Scholar
  27. Prabhakar, T. R. (1971). A singular integral equation with a generalized Mittag- Leffler function in the kernel,Yokohama Math. J., 19, 7–15.MATHMathSciNetGoogle Scholar
  28. Ross, B. (1994). A formula for the fractional integration and differentiation of (a + b x)c, J. Fract. Calc., 5, 87–89.MATHADSMathSciNetGoogle Scholar
  29. Saigo, M. (1978). A remark on integral operators involving the Gauss hypergeometric function,Math. Reports of College of Gen. Edu., Kyushu University, 11, 135–143.MathSciNetGoogle Scholar
  30. Saigo, M. and Raina, R. K. (1988). Fractional calculus operators associated with a general class of polynomials, Fukuoka Univ. Science Reports, 18, 15–22.MATHADSMathSciNetGoogle Scholar
  31. Samko, S. G., Kilbas, A. A. and Marichev, 0. 1. (1993). Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Reading.Google Scholar
  32. Saxena, R. K. (1967). On fractional integration operators, Math. Zeitsch., 96, 288–291.MATHCrossRefGoogle Scholar
  33. Saxena, R. K. (2002). Certain properties of generalized Mittag-Leffler function, Proceedings of the Third Annual Conference of the Society for Special Functions and Their Applications, Varanasi, March 4-6, 75–81.Google Scholar
  34. Saxena, R. K. (2003). Alternative derivation of the solution of certain integro-differential equations of Volterra-type, Ganita Sandesh, 17(1), 51–56.MathSciNetMATHGoogle Scholar
  35. Saxena,R. K. (2004). On a unified fractional generalization of free electron laser equation, Vijnana Parishad Anusandhan Patrika, 47(l), 17–27.MathSciNetMATHGoogle Scholar
  36. Saxena, R. K. and Kumbhat, R. K. (1973). A generalization of Kober operators, Vijnana Parishad Anusandhan Patrika, 16, 31–36.MathSciNetGoogle Scholar
  37. Saxena, R. K. and Kumbhat, R. K. (1974). Integral operators involving H-function, Indian J. Pure appl. Math., 5, 1–6.MATHMathSciNetGoogle Scholar
  38. Saxena, R. K. and Kumbhat, R. K. (1975). Some properties of generalized Kober operators, Vijnana Parishad Anusandhan Patrika, 18, 139–150.MathSciNetMATHGoogle Scholar
  39. Saxena, R. K, Mathai, A. M and Haubold, H. J. (2002). On fractional kinetic equations, Astrophysics and Space Science, 282, 281–287.CrossRefADSGoogle Scholar
  40. Saxena, R. K, Mathai, A. M and Haubold, H. J. (2004). On generalized fractional kinetic equations, Physica A , 344, 657–664.CrossRefADSMathSciNetGoogle Scholar
  41. Saxena, R. K, Mathai, A. M. and Haubold, H. J. (2004). Unified fractional kinetic equations and a fractional diffusion equation, Astrophysics and Space Science, 290, 241–245.MATHCrossRefADSGoogle Scholar
  42. Saxena, R. K. and Nishimoto, K. (2002). On a fractional integral formula of Saigo operator, J. Fract. Calc., 22, 57–58.MATHMathSciNetGoogle Scholar
  43. Saxena, R. K. and Saigo, M. (2005). Certain properties of fractional calculus operators associated with generalized Mittag-Leffler function. Frac. Calc. Appl. Ana1., 8(2), 141–154.MATHMathSciNetGoogle Scholar
  44. Sneddon, I. N. (1975). The Use in Mathematical Physics of Erdélyi- Kober Operators and Some of Their Applications, Lecture Notes in Mathematics (Edited by B. Ross), 457, 37–79.CrossRefMathSciNetGoogle Scholar
  45. Srivastava, H. M. and Saxena, R. K. (2001). Operators of fractional integration and their applications, Appl. Math. Comput., 118, 1–52.MATHCrossRefMathSciNetGoogle Scholar
  46. Srivastava, H. M. and Karlsson, P. W. (1985). Multiple Gaussian Hypergeometric Series, Ellis Horwood, Chichester, U.K.MATHGoogle Scholar
  47. Stein, E. M. (1970). Singular Integrals and Differential Properties of Functions, Princeton University Press, New Jersey.Google Scholar
  48. Wiman, A. (1905). Uber den Fundamental satz in der Theorie de Funktionen E α(x). Acta Math., 29, 191–201.MATHCrossRefMathSciNetGoogle Scholar
  49. Weyl, H. (1917). Bemerkungen zum Begriff des Differentialquotienten gebrochener Ordnung, Vierteljahresschr. Naturforsch. Gen. Zurich, 62, 296–302.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Personalised recommendations