Operators of fractional integration and their applications

The main purpose of this paper is to present a systematic (and historical) account of the investigations carried out by various authors in the field of fractional calculus and its applications. Several interesting results, relevant to the present investigation, are also considered.

[1]  C. Fox.  An application of fractional integration to chain transform theory. , 1958 .

[3]  H. Srivastava,et al.  Some triple series and triple integral equations , 1990 .

[4]  西本 勝之.  Certain Classes of Infinite Series Summable by Means of Fractional Calculus , 1989 .

[5]  西本 勝之.  An Application of Riemann-Liouville Operator in the Unification of Certain Functional Relations , 1991 .

[6]  I. Podlubny.  Fractional differential equations , 1998 .

[7]  H. Srivastava.  Fractional derivatives of the H-function of several variables*1 , 1985 .

[8]  L. Campos.  On an approach to complex analysis via differintegrations , 1993 .

[9]  E. R. Love,et al.  Fractional Derivatives of Imaginary Order , 1971 .

[10]  P. G. Rooney.  On the Ranges of Certain Fractional Integrals , 1972, Canadian Journal of Mathematics.

[11]  K. Gupta,et al.  On composition of some general fractional integral operators , 1994 .

[12]  Hari M. Srivastava,et al.  Current topics in analytic function theory , 1992 .

[13]  H. Srivastava,et al.  Evaluation of a certain class of Eulerian integrals , 1993 .

[14]  E. Love.  Some Integral Equations Involving Hypergeometric Functions , 1967, Proceedings of the Edinburgh Mathematical Society.

[15]  A. Mcbride,et al.  Fractional calculus and integral transforms of generalized functions , 1979 .

[16]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[17]  Hari M. Srivastava,et al.  A certain family of infinite series associated with Digamma functions , 1991 .

[18]  R. K. Saxena.  On fractional integration operators , 1967 .

[19]  An inversion formula for the Varma transform , 1966 .

[20]  Hari M. Srivastava,et al.  Convolution integral equations, with special function kernels , 1977 .

[21]  Bertram Ross.  Serendipity in Mathematics or How one is Led to Discover that , 1983 .

[22]  M. Riesz.  L'intégrale de Riemann-Liouville et le problème de Cauchy , 1949 .

[23]  Fractional Derivatives and Expansion Formulas Involving H-Functions of One and More Variables , 1997 .

[24]  Applications of Laplace transforms and their inverses , 1972 .

[25]  Hari M. Srivastava,et al.  The exact solution of certain differential equations of fractional order by using operational calculus , 1995 .

[26]  Hari M. Srivastava,et al.  A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients , 1992 .

[27]  Hari M. Srivastava,et al.  A class of distortion theorems involving certain operators of fractional calculus , 1988 .

[28]  Hari M. Srivastava,et al.  The integration of certain products of the multivariableH-function with a general class of polynomials , 1983 .

[29]  A. M. Mathai,et al.  Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences , 1973 .

[30]  K. Gupta,et al.  An integral involving generalized function of two variables , 1972 .

[31]  H. Srivastava,et al.  A Theorem Relating a Certain Generalized Weyl Fractional Integral with the Laplace Transform and a Class of Whittaker Transforms , 1990 .

[32]  S. Kalla.  Operators of fractional integration , 1980 .

[33]  Fractional Calculus and Its Applications Involving Certain Classes of Functional Relations , 1993 .

[34]  Hari M. Srivastava.  A Simple Algorithm for the Evaluation of a Class of Generalized Hypergeometric Series , 1992 .

[35]  C. Fox.  A FORMAL SOLUTION OF CERTAIN DUAL INTEGRAL EQUATIONS , 1965 .

[36]  T. P. Higgins.  A child's garden of special functions , 1975 .

[37]  An inversion formula for a kernel involving a Mellin-Barnes type integral , 1966 .

[38]  B. Ross,et al.  A BRIEF HISTORY AND EXPOSITION OF THE FUNDAMENTAL THEORY OF FRACTIONAL CALCULUS , 1975 .

[39]  A linear fractional calculus operator and its applications to certain subclasses of analytic functions , 1989 .

[40]  H. M. Srivastava,et al.  Fractional calculus and the sum of powers of natural numbers , 1991 .

[41]  H. Srivastava,et al.  Fractional integration of the H-function of several variables , 1995 .

[42]  Hari M. Srivastava,et al.  Some integrals involving a general class of polynomials and the multivariable H-function , 1986 .

[43]  Hari M. Srivastava,et al.  Theory and Applications of Convolution Integral Equations , 1992 .

[44]  H. T. Davis.  The Application of Fractional Operators to Functional Equations , 1927 .

[45]  V. Kiryakova.  Generalized Fractional Calculus and Applications , 1993 .

[46]  Hari M. Srivastava,et al.  Fractional Calculus and the Sums of Certain Families of Infinite Series , 1995 .

[47]  E. Love.  Two more hypergeometric integral equations , 1967, Mathematical Proceedings of the Cambridge Philosophical Society.

[48]  Megumi Saigo,et al.  A remark on integral operators involving the Gauss hypergeometric functions , 1978 .

[49]  Hari M. Srivastava,et al.  Fractional calculus operators and their applications involving power functions and summation of series , 1997 .

[50]  Fractional integral operators involving a general class of polynomials , 1990 .

[51]  A pair of unsymmetrical Fourier kernels , 1965 .

[52]  Arak M. Mathai,et al.  The H-function with applications in statistics and other disciplines , 1978 .

[53]  B. Ross,et al.  Fractional Calculus and Its Applications , 1975 .

[54]  A. W. Kemp,et al.  A treatise on generating functions , 1984 .

[55]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[56]  Hari M. Srivastava,et al.  Multiplication of fractional calculus operators and boundary value problems involving the Euler-Darboux equation , 1987 .

[57]  On the formal solution of dual integral equations , 1967 .

[58]  A. Erdélyi,et al.  Fractional Integration and Dual Integral Equations , 1962, Canadian Journal of Mathematics.

[59]  Hari M. Srivastava,et al.  A certain subclass of analytic functions associated with operators of fractional calculus , 1996 .

[60]  M. Saigo,et al.  Application of Generalized Fractional Calculus Operators in the Solution of Certain Dual Integral Equations , 1993 .

[61]  Hari M. Srivastava,et al.  Some applications of fractional calculus operators to certain subclasses of prestarlike functions with negative coefficients , 1995 .

[62]  C. Fox.  Integral transforms based upon fractional integration , 1963 .

[63]  M. Saigo,et al.  Fractional Calculus Operators Involving the Gauss Function on Spaces Fp,μand F'p,μ , 1990 .

[64]  Henry E. Fettis,et al.  Erratum: Tables of integral transforms. Vol. I, II (McGraw-Hill, New York, 1954) by A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi , 1973 .

[65]  Nguyen Thanh Hai,et al.  The Double Mellin-Barnes Type Integrals and Their Applications to Convolution Theory , 1991 .

[66]  E. Love.  Two index laws for fractional integrals and derivatives , 1972, Journal of the Australian Mathematical Society.

[67]  Shyam L. Kalla,et al.  Integral operators involving hypergeometric functions , 1969 .

[68]  M. Saigo,et al.  Starlike and convex functions involving a certain fractional integral operator , 1989 .

[69]  Hari M. Srivastava,et al.  The convolution method for the development of new leibniz rules involving fractional derivatives and of their integral analogues , 1993 .

[70]  I. Podlubny.  Riesz potential and Riemann-Liouville fractional integrals and derivatives of Jacobi polynomials , 1997 .

[71]  Hari M. Srivastava,et al.  A certain class of infinite sums associated with Digamma functions , 1999, Appl. Math. Comput..

[72]  Megumi Saigo,et al.  CERTAIN PROPERTIES OF OPERATORS OF FRACTIONAL INTEGRATION ASSOCIATED WITH MELLIN AND LAPLACE TRANSFORMATIONS , 1992 .

[73]  Megumi Saigo,et al.  On mittag-leffler type function, fractional calculas operators and solutions of integral equations , 1996 .

[74]  A simplified overview of certain relations among infinite series that arose in the context of fractional calculus , 1991 .

[75]  Thomas J. Osler,et al.  Fractional Derivatives and Special Functions , 1976 .

[76]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[77]  F. Mainardi.  The fundamental solutions for the fractional diffusion-wave equation , 1996 .

[78]  Thomas J. Osler,et al.  Fundamental properties of fractional derivatives via pochhammer integrals , 1975 .

[79]  C. Fox.  The $G$ and $H$ functions as symmetrical Fourier kernels , 1961 .

[80]  H. Srivastava.  A multilinear generating function for the Konhauser sets of biorthogonal polynomials suggested by the Laguerre polynomials , 1985 .

[81]  H. Srivastava,et al.  A subclass of analytic functions defined by using certain operators of fractional calculus , 1995 .

[82]  R. K. Raina,et al.  Some subclasses of analytic functions associated with fractional calculus operators , 1999 .

[83]  W. Lamb.  Fractional powers of operators defined on a Fréchet space , 1984, Proceedings of the Edinburgh Mathematical Society.

[84]  Hari M. Srivastava,et al.  Univalent and Starlike Generalized Hypergeometric Functions , 1987, Canadian Journal of Mathematics.

[85]  S. Lacroix.  Traité du calcul différentiel et du calcul intégral , 1800 .

[86]  R. Bagley,et al.  On the Fractional Calculus Model of Viscoelastic Behavior , 1986 .

[87]  Fractional integration and certain dual integral equations , 1967 .

[88]  B. Braaksma,et al.  Asymptotic expansions and analytic continuations for a class of Barnes-integrals , 1964 .

[89]  Hari M. Srivastava,et al.  The H-functions of one and two variables, with applications , 1982 .

[90]  P. W. Karlsson,et al.  Multiple Gaussian hypergeometric series , 1985 .

[91]  Thomas J. Osler,et al.  Leibniz Rule for Fractional Derivatives Generalized and an Application to Infinite Series , 1970 .

[92]  C. Fox.  An inversion formula for the kernel Kv(x) , 1965, Mathematical Proceedings of the Cambridge Philosophical Society.

[93]  西本 勝之.  Nishimoto's fractional calculus : calculus in the 21st century : integrations and differentiations of arbitrary order , 1991 .

[94]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .