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

Applied Mathematics and Computation

Volumn 187, Issue 1 SPEC. ISS., 2007, Pages 280-289

  Lin, Shy Der ^a   Tsai, Yen Shung ^a   Wang, Pin Yu ^b  

^a CHUNG YUAN CHRISTIAN UNIVERSITY   (Taiwan)

^b NANYA INSTITUTE OF TECHNOLOGY   (Taiwan)

Author keywords

Analytic functions; Bessel's equation; Differintegral equations; Fractional calculus; Generalized Leibniz rule; Index law; Legendre equations; Legendre polynomials; Linearity property; Ordinary and partial differential equations; Power series solutions; Principal value

Indexed keywords

COMPUTATIONAL COMPLEXITY; INTEGRAL EQUATIONS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIALS;

BESSEL'S EQUATIONS; FRACTIONAL CALCULUS; GENERALIZED LEIBNIZ RULE; INDEX LAW; LEGENDRE EQUATIONS; LEGENDRE POLYNOMIALS; LINEARITY PROPERTY; POWER-SERIES SOLUTIONS; PRINCIPAL VALUE;

DIFFERENTIATION (CALCULUS);

EID: 34247515503     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2006.08.152     Document Type: Article

References (24)

1. Alawneh A., Odibat Z., and Nishimoto K. Negative power series solutions for a class of ordinary differential equations using N-fractional calculus. J. Fract. Calc. 21 (2002) 29-37
2. Carracedo C.M., and Alix M.S. The Theory of Fractional Powers of Operators. North-Holland Mathematical Studies vol. 187 (2001), Elsevier (North-Holland) Science Publishers, Amsterdam
3. Erdélyi A., Magnus W., Oberhettinger F., and Tricomi F.G. Higher Transcendental Functions vols. I and II (1953), McGraw-Hill Book Company, New York
4. In: Hilfer R. (Ed). Applications of Fractional Calculus in Physics (2000), World Scientific Publishing Company, Singapore
5. Ince E.L. Ordinary Differential Equations (1927), Longmans, Green and Company, London Reprinted by Dover Publications, New York, 1956
6. Kilbas A.A., Srivastava H.M., and Trujillo J.J. Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies vol. 204 (2006), Elsevier (North-Holland) Science Publishers, Amsterdam
7. Kiryakova V. Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics vol. 301 (1994), Longman Scientific and Technical, Harlow
8. Lin S.-D., Ling W.-C., Nishimoto K., and Srivastava H.M. A simple fractional-calculus approach to the solutions of the Bessel differential equation of general order and some of its applications. Comput. Math. Appl. 49 (2005) 1487-1498
9. Lin S.-D., and Nishimoto K. N-Method to a generalized associated Legendre equation. J. Fract. Calc. 14 (1998) 95-111
10. Lin S.-D., and Nishimoto K. New finding of particular solutions for a generalized associated Legendre equation. J. Fract. Calc. 18 (2000) 9-37
11. Lin S.-D., Tu S.-T., and Srivastava H.M. Explicit solutions of certain ordinary differential equations by means of fractional calculus. J. Fract. Calc. 20 (2001) 35-43
12. Miller K.S., and Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), A Wiley-Interscience Publication, John Wiley and Sons, New York
13. Nishimoto K. Fractional Calculus vols. I, II, III and IV (1984), Descartes Press, Koriyama 1987, 1989 and 1991
14. Nishimoto K. An Essence of Nishimoto's Fractional Calculus (Calculus of the 21st Century): Integrations and Differentiations of Arbitrary Order (1991), Descartes Press, Koriyama
15. Odibat Z., Al-Lamy R., and Nishimoto K. Negative power series solutions of a class of differential equations including the ones of Legendre and Chebyshev. J. Fract. Calc. 28 (2005) 53-63
16. Ozaktas H.M., Zalevsky Z., and Kutay M.A. The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley Series in Pure and Applied Optics (2000), John Wiley and Sons, New York
17. Podlubny I. 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 (1999), Academic Press, New York
18. Samko S.G., Kilbas A.A., and Marichev O.I. Fractional Integrals and Derivatives: Theory and Applications, Translated from the Russian: Integrals and Derivatives of Fractional Order and Some of Their Applications ("Nauka i Tekhnika", Minsk, 1987) (1993), Gordon and Breach Science Publishers, Reading
19. Srivastava H.M., and Buschman R.G. Theory and Applications of Convolution Integral Equations. Kluwer Series on Mathematics and Its Applications vol. 79 (1992), Kluwer Academic Publishers, Dordrecht
20. Srivastava H.M., Owa S., and Nishimoto K. Some fractional differintegral equations. J. Math. Anal. Appl. 106 (1985) 360-366
21. Tu S.-T., Chyan D.-K., and Srivastava H.M. Some families of ordinary and partial fractional differintegral equations. Integral Transf. Spec. Funct. 11 (2001) 291-302
22. Wang P.-Y., Lin S.-D., and Srivastava H.M. Remarks on a simple fractional-calculus approach to the solutions of the Bessel differential equation of general order and some of its applications. Comput. Math. Appl. 51 (2006) 105-114
23. Watson G.N. A Treatise on the Theory of Bessel Functions. second ed. (Reprinted) (1966), Cambridge University Press, Cambridge
24. Whittaker E.T., and Watson G.N. 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 ed. (Reprinted) (1973), Cambridge University Press, Cambridge