Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2019, Cilt: 2 Sayı: 2, 154 - 165, 29.07.2019
https://doi.org/10.33773/jum.577349

Öz

Kaynakça

  • [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.

THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER

Yıl 2019, Cilt: 2 Sayı: 2, 154 - 165, 29.07.2019
https://doi.org/10.33773/jum.577349

Öz

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.

Kaynakça

  • [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.
Toplam 1 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Araştırma Makalesi
Yazarlar

Elsayed Elsayed 0000-0003-0894-8472

D. Vivek

K. Kanagarajan

Yayımlanma Tarihi 29 Temmuz 2019
Gönderilme Tarihi 13 Haziran 2019
Kabul Tarihi 3 Aralık 2019
Yayımlandığı Sayı Yıl 2019 Cilt: 2 Sayı: 2

Kaynak Göster

APA Elsayed, E., Vivek, D., & Kanagarajan, K. (2019). THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER. Journal of Universal Mathematics, 2(2), 154-165. https://doi.org/10.33773/jum.577349
AMA Elsayed E, Vivek D, Kanagarajan K. THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER. JUM. Temmuz 2019;2(2):154-165. doi:10.33773/jum.577349
Chicago Elsayed, Elsayed, D. Vivek, ve K. Kanagarajan. “THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER”. Journal of Universal Mathematics 2, sy. 2 (Temmuz 2019): 154-65. https://doi.org/10.33773/jum.577349.
EndNote Elsayed E, Vivek D, Kanagarajan K (01 Temmuz 2019) THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER. Journal of Universal Mathematics 2 2 154–165.
IEEE E. Elsayed, D. Vivek, ve K. Kanagarajan, “THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER”, JUM, c. 2, sy. 2, ss. 154–165, 2019, doi: 10.33773/jum.577349.
ISNAD Elsayed, Elsayed vd. “THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER”. Journal of Universal Mathematics 2/2 (Temmuz 2019), 154-165. https://doi.org/10.33773/jum.577349.
JAMA Elsayed E, Vivek D, Kanagarajan K. THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER. JUM. 2019;2:154–165.
MLA Elsayed, Elsayed vd. “THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER”. Journal of Universal Mathematics, c. 2, sy. 2, 2019, ss. 154-65, doi:10.33773/jum.577349.
Vancouver Elsayed E, Vivek D, Kanagarajan K. THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER. JUM. 2019;2(2):154-65.