Explicit solutions of a certain class of associated Legendre equations by means of fractional calculus

In recent years, many authors have demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. The main object of the present paper is to show how several recent contributions on this subject, involving a certain class of associated Legendre equations, can be obtained (in a unified manner) by suitably applying some general theorems on particular solutions of a certain family of linear ordinary fractional differintegral equations.

[1]  Edmund Taylor Whittaker,et al.  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 , 1920, Nature.

[2]  W. Ames.  Mathematics in Science and Engineering , 1999 .

[3]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[4]  H. Srivastava,et al.  Some families of ordinary and partial fractional differintegral equations , 2001 .

[5]  H. Srivastava,et al.  A simple fractional-calculus approach to the solutions of the Bessel differential equation of general order and some of its applications , 2005 .

[6]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[7]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[8]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

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

[10]  Hari M. Srivastava,et al.  Remarks on a simple fractional-calculus approach to the solutions of the bessel differential equation of general order and some of its applications , 2006, Computers and Mathematics with Applications.

[11]  C. Carracedo,et al.  The theory of fractional powers of operators , 2001 .

[12]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[13]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[14]  P. Hobbs.  Wiley Series in Pure and Applied Optics , 2008 .

[15]  Z. Zalevsky,et al.  The Fractional Fourier Transform: with Applications in Optics and Signal Processing , 2001 .

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

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

[18]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[19]  H. Srivastava,et al.  Some fractional differintegral equations , 1985 .

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