Fractional Relaxation Equations and a Cauchy Formula for Repeated Integration of the Resolvent

Yıl 2018, Cilt: 2 Sayı: 1, 11 - 32, 25.03.2018

Leigh C. Becker İoannis K. Purnaras

https://doi.org/10.31197/atnaa.379282

Öz

Cauchy’s formula for repeated integration is shown to be valid for the function
R(t) = 􀀀(q)tq􀀀1Eq;q(􀀀􀀀(q)tq)
where and q are given positive constants with q 2 (0; 1), 􀀀 is the Gamma function, and Eq;q is a Mittag-
Leffler function. The function R is important in the study of Volterra integral equations because it is the
unique continuous solution of the so-called resolvent equation
R(t) = tq􀀀1 􀀀
Z t
0
(t 􀀀 s)q􀀀1R(s) ds
on the interval (0;1). This solution, commonly called the resolvent, is used to derive a formula for the
unique continuous solution of the Riemann-Liouville fractional relaxation equation
Dqx(t) = 􀀀ax(t) + g(t) (a > 0)
on the interval [0;1) when g is a given polynomial. This formula is used to solve a generalization of the
equation of motion of a falling body. The last example shows that the solution of a fractional relaxation
equation may be quite elementary despite the complexity of the resolvent.

Anahtar Kelimeler

Cauchy’s formula for repeated integration, fractional differential equations, Mittag-Leffler functions, relaxation equations, resolvents, Riemann-Liouville operators, Volterra integral equations

Kaynakça

• M. Abramowitz and I. A. Stegun, (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 2nd printing, National Bureau of Standards, Applied Mathematical Series 55, 1964.
• T. M. Apostol, Mathematical Analysis, second ed., Addison-Wesley, Reading MA, 1974.
• L. C. Becker, Properties of the resolvent of a linear Abel integral equation: implications for a complementary fractional equation, Electron. J. Qual. Theory Differ. Equ., No. 64 (2016), 1–38.
• L. C. Becker, T. A. Burton, and I. K. Purnaras, Complementary equations: A fractional differential equation and a Volterra integral equation, Electron. J. Qual. Theory Differ. Equ., No. 12 (2015), 1–24.
• L. C. Becker, Resolvents for weakly singular kernels and fractional differential equations, Nonlinear Anal.: TMA 75 (2012), 4839–4861.
• K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Heidelberg, 2010.
• A. Friedman, On integral equations of Volterra type, J. d’Analyse Math., Vol. XI (1963), 381–413.
• W. Fulks, Advanced Calculus, 2nd ed., John Wiley & Sons, New York, 1969.
• I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Seventh Edition, Elsevier Academic Press, 2007.
• A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies 204, Elsevier, 2006.
• F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons and Fractals, Vol. 7, No. 9 (1996), 1461–1477.
• F. Mainardi and R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math., 118 (2000), 283–299.
• J.B. Marion, Classical Dynamics of Particles and Systems, Academic Press, New York, 1965.
• Miller, R. K., Nonlinear Volterra Integral Equations, Benjamin, Menlo Park, CA, 1971.
• I.P. Natanson, Theory of Functions of a Real Variable, Vol. II, Frederick Ungar Publishing Co., New York, 1961.
• K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover, Mineola, NY, 2006.
• I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Vol. 198, Academic Press, San Diego, 1999.
• J. J. Rosales, M. Guía, F. Gómez, F. Aguilar, and J. Martinez, Two dimensional fractional projectile motion in a resisting medium, Cent. Eur. J. Phys., 12(7) (2014), 517–520.