THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER

Year 2019, Volume: 2 Issue: 2, 154 - 165, 29.07.2019

Elsayed Elsayed D. Vivek K. Kanagarajan

https://doi.org/10.33773/jum.577349

Cited By: 3

Abstract

In this paper, we consider boundary value problems for the following nonlinear implicit differential equations with complex order

 

D +x(t) = f t,x(t),D +x(t) , ? = m+i?, t ? J := [0,T],  ax(0)+bx(T) = c,

where D + is the Caputo fractional derivative of order ? ? C. Let ? ? R , 0 < ? < 1, m ? (0,1], and f : J ×R ? R is given continuous function. Here a,b,c are real constants with a+b  = 0. We derive the existence and stability of solution for a class of boundary value problem(BVP) for nonlinear fractional implicit differential equations(FIDEs) with complex order. The results are based upon the Banach contraction principle and Schaefer’s fixed point theorem.

Keywords

Implicit differential equation, Boundary value problem, Stirling asymptotic formula

References

• [1] C. Alsina, R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl., 2, (1998), 373-380.[2] R. Andriambololona, R. Tokiniaina, H. Rakotoson, Definitions of complex order integrals and complex order derivatives using operator approach, International Journal of Latest Research in Science and Technology, 1(4), (2012), 317-323.[3] S. Andras, J. J. Kolumban, On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Analysis, 82,(2013),1-11.[4] A.Arara, M.Benchohra, N.Hamidi, J.J.Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Analysis , 72 (2), (2010), 580-586.[5] Bertram Ross, Francis H. Northover, A use for a derivative of complex order in the fractional calculus, 9(4),(1977), 400-406.[6] Z. Bai, H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications , 311 (2), (2005), 495-505.[7] Z. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Analysis, 72 (2), (2010), 916-924.[8] C.S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Applied Mathematics Letters , 23, (2010), 1050-1055.[9] Z. Bai, H. Lu, Positive solutions for a boundary value problem of nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 311,(2005), 495-505.[10] M. Benchohra, J. E. Lazreg, Existence and Uniqueness results for nonlinear implicit frac-tional differential equations with boundary conditions, Romanian Journal of Mathematicsand Computer Science, 4, (2014), 60-72.[11] M. Benchohra, S. Bouriah, Existence and stability results for nonlinear boundary valueproblem for implicit differential equations of fractional order, Morccan Journal of Pure andApplied Analysis, 1(1), (2015), 22-37.[12] M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equa-tions with fractional order, Survey in Mathematics and its Applications, (3), (2008), 1-12.[13] Carla M. A. Pinto, J. A. Tenreiromachado. Complex order van der Pol oscillator. Nonlin-ear Dynamics, Springer Verlag, 65 (3), 2010, pp.247-254.[14] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.[15] R.Hilfer, Application of fractional Calculus in Physics, World Scientific, Singapore, 1999.[16] DH. Hyers, G. Isac, TM. Rassias, Stability of functional equation in several variables,Vol. 34, Progress in nonlinea differential equations their applications, Boston (MA):Birkhauser; 1998.[17] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. , USA27, (1941), 222-224.[18] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, In-ternational Journal of mathematics,23,(2012),doi:10.1142/S0129167X12500565.[19] S. M. Jung, Hyers-Ulam stability of linear differential equations of first order, AppliedMathematics Letters, 17,(2004),1135-1140.[20] A. A. Kilbas, H. M. Srivasta, J. J. Trujillo, Theory and application of fractional differentialequations, Elsevier B. V, Netherlands,(2016).[21] E.R. Love, Fractional derivatives of imaginary order, Journal of the London MathematicalSociety, 2(2-3), 241-259, (1971).[22] P. Muniyappan, S. Rajan, Hyers-Ulam-Rassias stability of fractional differential equation,International Journal of pure and Applied Mathematics, 102, (2015),631-642.[23] R.L. Magin, Fractional Calculus in Bioengineering, Begell House, 2006.[24] K.S.Miller, B.Ross, AnIntroductiontotheFractionalCalculusandFractionalDifferentialEquations, Wiley, New York, 1993.[25] Moustafa El-Shahed, Positive solutions for boundary value problem of nonlinear frac-tional differential equation, Abstract and Applied Analysis , 2007 (2007) Article ID 10368, 8pages.[26] A. Neamaty, M. Yadollahzadeh, R. Darzi, On fractional differential equation with com-plex order, Progress in fractional differential equations and Apllications, 1(3), (2015),223-227.[27] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.[28] Rabha W. Ibrahim, Ulam stability of boundary value problem, Kragujevac Journal of Math-ematics, 37(2) ,(2013), 287-297.[29] I. A. Rus, Ualm stabilities of ordinary differential equations in a Banach space, CarpathianJournal Mathematics, 26, (2010), 103-107.[30] S. G. Samko, A.A. Kilbas O. I. Marichev, Fractional Integrals and Derivatives-Theory andApplications, Gordon and Breach Science Publishers, Amsterdam , 1993.[31] Teodor M. Atanackovi, Sanja Konjik, Stevan Pilipovic, Dusan Zorica, Complex orderfractional derivatives in viscoelasticity, Mech Time-Depend Mater, 1, (2016), 1-21.[32] S.M. Ulam, A Collection of Mathematical Problems, Interscience, New York, 1960.[33] D. Vivek, K. Kanagarajan, S. Harikrishnan, Dynamics and stability results for fractionalintegro-differential equations with complex order, Discontinuity, Nonlinearity and Com-plexity, (2007), (Accepted manuscript, id: DNC-D-2017-0007).[34] J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differen-tial equations with Caputo derivative,Electronic Journal of Qualitative Theory of DifferentialEquations,63,(2011),1-10.[35] J. Wang, Y. Zhou, New concepts and results in stability of fractional differential equa-tions,Communications on Nonlinear Science and Numerical Simulations,17,(2012),2530-2538.[36] S. Zhang, Existence of solution for a boundary value problem of fractional order, ActaMathematica Scientia. Series B. English Edition , 26 (2), (2006), 220-228.