EXPLICIT SOLUTIONS OF THE CONFLUENT HYPERGEOMETRIC EQUATIN BY MEANS OF THE DIFFERINTEGRAL THEOREMS

Year 2016, Volume: 18 Issue: 54, 0 - 2, 01.09.2016

Ökkeş Öztürk

Abstract

In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator. Differintegral theory is used to solve some classes of differential equations and fractional differential equations. One of these equations is the confluent hypergeometric equation. In this paper, we intend to solve this equation by means of the differintegral theorems

Keywords

Fractional calculus, Differintegral, Confluent hypergeometric equation, Differintegral theorems, Generalized Leibniz rule

DİFERİNTEGRAL TEOREMLERİ YARDIMIYLA KONFLUENT HİPERGEOMETRİK DENKLEMİNİN AÇIK ÇÖZÜMLERİ

Abstract

Uygulamalı matematiğin bir alanı olan kesirli hesapta diferintegral, türev/integral operatörünün bir birleşimidir. Diferansiyel denklemlerin ve kesirli diferansiyel denklemlerin bazı sınıflarını çözmek için diferintegral teorisi kullanılmaktadır. Bu denklemlerden birisi konfluent hipergeometrik denklemidir. Bu makalede, diferintegral teoremleri yardımıyla bu denklemi çözmeyi hedefleriz

Keywords

Kesirli hesap, Diferintegral, Konfluent hipergeometrik denklemi, Diferintegral teoremleri, Genelleştirilmiş Leibniz kuralı

References

• Podlubny I. 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, USA: Academic Press, 1999.
• Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations, USA: John Wiley & Sons, 1993.
• Oldham KB, Spanier J. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, USA: Academic Press, 1974.
• Yilmazer R, Ozturk O. Explicit Solutions of Singular Differential Equation by means of Fractional Calculus Operators, Abstract and Applied Analysis, vol. 2013, 2013, 6 pages. [5]
• Tu ST, Chyan DK, Srivastava HM. Some Families of Ordinary and Partial Fractional Differintegral Equations, Integral Transform. Spec. Funct. vol. 11, 2001, p.291-302. [6]
• Nishimoto K. Kummer’s Twenty-Four Functions and N-Fractional Calculus, NonlinearAnalysis, Theory, Methods & Applications, vol. 30, 1997, p.1271-1282. [7]
• Yilmazer R. N-Fractional Calculus Operator method to a Modified Hydrogen Atom Equation, Math. Commun., vol. 15, 2010, p.489-501. [8]
• Whittaker ET, Watson GN. 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 Edition), Cambridge: Cambridge University Press, 1927. [9]
• Tricomi FG. Funzioni Ipergeometriche Confluenti, Roma: Edizioni Cremonese, 1954. [11]
• Watson GN. A Treatise on the Theory of Bessel Functions (Second Edition), Cambridge: Cambridge University Press, 1944.
• Yilmazer R, Inc M, Tchie, F, Baleanu D. Particular Solutions of the Confluent Hypergeometric Differential Equation by using the Nabla Fractional Calculus Operator. Entropy, vol. 18 (2), (2016), 6 pages.
• Virchenko N. On the Generalized Confluent Hypergeometric Function and Its Application, Fractional Calculus and Applied Analysis, vol. 9 (2), 2006, p.101-108.
• Srivastava HM, Saxena RK. Some Volterra-Type Fractional Integro-Differential Equations with a Multivariable Confluent Hypergeometric Function as Their Kernel. Journal of Integral Equations and Applications, vol. 17 (2), 2005, 199-217.
• Campos LMBC. On Some Solutions of the Extended Confluent Hypergeometric Differential Equation. Journal of Computational and Applied Mathematics vol. 137, 2001, p.177–200.
• Lin SD, Shyu JC, Nishimoto K, Srivastava HM. Explicit Solutions of Some General Families of Ordinary and Partial Differential Equations Associated with the Bessel Equation by means of Fractional Calculus, Journal of Fractional Calculus, vol. 25, 2004, p.33-45. [17]
• Bayın S. Mathematical Methods in Science and Engineering, USA: John Wiley & Sons, 2006.
• Yilmazer R, Bas E. Explicit Solutions of Fractional Schrödinger Equation via Fractional Calculus Operator, Int. J. Open Problems Compt. Math., vol. 5 (2), 2012, p.132-141.