N-Fractional Calculus Operator Nη Method Applied to a Gegenbauer Differential Equation

Yıl 2012, Cilt: 9 Sayı: 1, - , 01.02.2012

Reşat Yılmazer , Ökkeş Öztürk

Öz

By means of fractional calculus techniques, we find explicit solutions of the
Gegenbauer equation. We use the N-fractional calculus operator Nη method to derive the
solutions of these equations.

Anahtar Kelimeler

Fractional calculus, Gegenbauer equation, ordinary differential equation, generalized Leibniz rule

Kaynakça

• [1] K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons 1993.
• [2] K. Oldham and J. Spanier, The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order, Mathematics in Science and Engineering V, Academic Press 1974.
• [3] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Methods of Their Solution and Some of Their Applications, Mathematics in Science and Enginering 198, Academic Press, New York, London, Tokyo and Toronto 1999.
• [4] B. Ross (Editor), Fractional Calculus and Its Applications: Proceedings of the International Conference Held at the University of New Haven, June 1974, Lecture Notes in Mathematics 457, Springer, New York 1975.
• [5] S. G. Samko, A. A. Kilbas and O. I. Marichev, 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) Gordon and Breach Science Publishers 1993.
• [6] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific Publishing, New York 2012.
• [7] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York 2006.
• [8] S.-T. Tu, D.-K. Chyan and H. M. Srivastava, Some families of ordinary and partial fractional differintegral equations, Integral Transforms and Special Functions 11 (2001), 291–302.
• [9] K. Nishimoto, Kummer’s twenty-four functions and N-fractional calculus, Nonlinear Analysis, Theory, Methods & Applications 30 (1997), 1271–1282.
• [10] R. Yilmazer, N-fractional calculus operator Nµ method to a modified hydrogen atom equation, Mathematical Communications 15 (2010), 489–501.
• [11] D. Bˇaleanu, O. G. Mustafa and R. P. Agarwal, On L p -solutions for a class of sequential fractional differential equations, Applied Mathematics and Computation 218 (2011), 2074– 2081.
• [12] D. Baleanu and S. I. Vacaru, Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics, Central European Journal of Physics 9 (2011), 1267–1279.
• [13] D. Bˇaleanu, O. G. Mustafa and D. O’Regan, A Nagumo-like uniqueness theorem for fractional differential equations, Journal of Physics A: Mathematical and Theoretical 44 (2011), 392003.
• [14] D. Bˇaleanu, O. G. Mustafa and R. P. Agarwal, On the solution set for a class of sequential fractional differential equations, Journal of Physics A: Mathematical and Theoretical 43 (2010), 385209.
• [15] 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, Cambridge University Press, Cambridge 1927.
• [16] M. Hukuhara, Ordinary Differential Equations (in Japanese), Iwanami-Shoten, Tokyo 1941.
• [17] F. G. Tricomi, Funzioni Ipergeometriche Confluenti, Edizioni Cremonese, Roma 1954.
• [18] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge 1944.
• [19] S. Bayın, Mathematical Methods in Science and Engineering, John Wiley and Sons 2006.
• [20] E. S. Panakhov and R. Yilmazer, A Hochstadt-Lieberman theorem for hydrogen atom equation, Applied and Computational Mathematics 11 (2012), 74–80.
• [21] K. Nishimoto, An Essence of Nishimoto’s Fractional Calculus (Calculus of the 21st Century): Integrations and Differentiations of Arbitrary Order, Descartes Press, Koriyama 1991.
• [22] K. Nishimoto, Fractional Calculus I, II, III, IV and V, Descartes Press, Koriyama 1984, 1987, 1989, 1991 and 1996.