Iterative solution of quadratic fractional integral equation involving generalized Mittag Leffler function
Downloads
Abstract
The present paper verifies and validates the factual and estimated outcomes of a certain quadratic fractional
integral equation involving the generalized Mittag-Leffler function via algorithm that representatively embodies
successive estimations under fragile partial fractional Lipschitz and compactness type circumstances. The
paper also validate the existence and convergence of a nonlinear quadratic fractional integral equation with the
generlized Mittag Leffler function which is the generalization of Mittag-Leffler function, on a closed and bounded
interval of the real line with the help of some conditions.
Keywords:
Quadratic fractional integral equation, Fractional derivatives and Integrals.Mathematics Subject Classification:
Mathematics- Pages: 57-63
- Date Published: 01-01-2021
- Vol. 9 No. 01 (2021): Malaya Journal of Matematik (MJM)
A. Babakhani, V. Daftardar-Gejji, Existence of positive solutions of nonlinear fractional differential equations, $J$. Math. Anal. Appl. 278 (2003) 434-442.
M. Cichon, A.M.A. El-Sayed, H.A.H. Salem, Existence theorem for nonlinear functional integral equations of fractional orders, Comment. Math. 41 (2001) 59-67.
M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311 (2005) 112-119.
A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary order, Nonlinear Anal. 33 (1998) 181186.
A.A. Kilbas, J.J. Trujillo, Differential equations of fractional order: Methods, results, problems I, Appl. Anal. 78 (2001) 153-192.
K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
J. Banas, J.R. Rodriguez, K. Sadarangani, On a class of Urysohn-Stieltjes quadratic integral equations and their applications, J. Comput. Appl. Math. 113 (2000) 35-50.
S. Hu, M. Khavanin, W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Anal. 34 (1989) 261-266.
C.T. Kelly, Approximation of solutions of some quadratic integral equations in transport theory, J. Integral Equations 4 (1982) 221-237.
B. Cahlon, M. Eskin, Existence theorems for an integral equation of the Chandrasekhar H-equation with perturba(s)) dson, J. Math. Anal. Appl. 83 (1981) 159-171.
S. Chandrasekhar, Radiative Transfer, Oxford Univ. Press, London, 1950.
J.H. He, Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering'98, Dalian, China, 1998, pp. 288 291.
J.H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol. 15 (2) (1999) 86-90.
R.L. Bagley, P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheology 27 (3) (1983) 201-210.
R.L. Bagley, P.J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J. 23 (6) (1985) 918-925.
F. Mainardi, Fractional calculus: 'Some basic problems in continuum and statistical mechanics', in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Verlag, New York, 1997, pp. 291-348.
B. Mandelbrot, Some noises with $1 / mathrm{f}$ spectrum, a bridge between direct current and white noise, IEEE Trans. Inform. Theory 13 (2) (1967) 289-298.
Y.A. Rossikhin, M.V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev. 50 (1997) $15-67$
R.T. Baillie, Long memory processes and fractional integration in econometrics, J. Econometrics 73 (1996) 5-59.
R.L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng. 32 (1) (2004) 1-104.
R.L. Magin, Fractional calculus in bioengineering-part 2 , Crit. Rev. Biomed. Eng. 32 (2) (2004) 105-193.
R.L. Magin, Fractional calculus in bioengineering-part 3 , Crit. Rev. Biomed. Eng. 32 (3/4) (2004) 194-377.
R. Metzler, J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004) $161-208$.
T.S. Chow, Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Phys. Lett. A $342(2005) 148-155$
Agarwal R.P. A propos d'une note M. Pierre Humbert,C.R.Acad.Sci. Paris 236 (1953),pp.2031-2032.
Chouhan Amit and Satishsaraswat, Some Rmearks on Generalized Mittag-Leffler Function and Fractional operators ,IJMMAC Vol2, No.2, pp. 131-139.
GM.Mittag - Leffler, Sur la nouvelle function of $E_alpha(x)$ ,C.R.Acad. Sci. Paris 137 (1903), pp.554-558.
Humbert P. and Agarwal R.P., Sur la function de Mittag - Leffler et quelquesunes deses generalizations, Bull. Sci.Math. (2)77(1953), pp.180-186.
Shukla A.K. and Prajapati J.C., On a generalization of Mittag - Leffler function and its properties, J.Math.Anal.Appl.336(2007),pp.79-81 .
Salim T.O. and Faraj O., A generalization of MittagLeffler function and Integral operator associated with the Fractional calculus, Journal of Fractional Calculus and Applications,3(5) (2012),pp.1-13.
Wiman A., Uber de fundamental satz in der theorie der funktionen, Acta Math. 29(1905),pp.191-201.
B.C. Dhage, A nonlinear alternative in Banach algebras with applications to functional differential equations, Nonlinear Funct. Anal. & Appl. 8 (2004), 563-575.
B.C. Dhage, Fixed point theorems in ordered Banach algebras and applications, PanAmer. Math. J. 9(4) (1999), 93-102.
B.C. Dhage, Partially condensing mappings in ordered normed linear spaces and applications to functional integral equations, Tamkang J. Math. 45 (4) (2014), 397-426.
B.C. Dhage, Nonlinear $mathscr{D}$-set-contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations, Malaya J. Mat. $3(1)(2015), 62-85$.
T. R. Prabhakar, A singular integral equation with a generalized Mittal-Leffler function in the Kernel, Yokohama Math.J.,19(1971), 7-15.
Similar Articles
- P. Amsini, R. Uma Rani, Interval type 2 intuitionistic FCM cluster with spatial information algorithm applied for histopathology images , Malaya Journal of Matematik: Vol. 9 No. 01 (2021): Malaya Journal of Matematik (MJM)
You may also start an advanced similarity search for this article.
Metrics
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 MJM
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.