Yıl 2019,
Cilt: 1 Sayı: 1, 35 - 43, 18.01.2019
S. Harıkrıshnan
,
Elsayed Elsayed
,
K. Kanagarajan
Kaynakça
- [1] A. Ahmadkhanlu, M. Jahanshahi, On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, B. Iran. Math. Soc., 38, (2012), 241-252.[2] N. Benkhettou, A. Hammoudi, D. F. M. Torres, Existence and uniqueness of solution for a fractional Riemann-lioville initial value problem on time scales, Journal of King Saud University-Science, 28,(2016),87-92.[3] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh˜auser, Boston, 2003.[4] M. Bohner, A. Peterson, Dtnamica equations on times scale, Birkhauser Boston, Boston, MA.[5] R.P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35, (1999), 3-22.[6] K.M. Furati, M.D. Kassim, N.e-. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64, (2012), 1616-1626.[7] A. Granas, and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.[8] S. Harikrishnan, Kamal Shah, Dumitru Baleanu, K. Kanagarajan, Note on the solution of random differential equations via ψ-Hilfer fractional derivative, Advances in Difference Equations, 2018, 2018:224.[9] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 23,(2012).[10] V. Lupulescu, S.K. Ntouyas, Random fractional differential equations International electronic journal of pure and applied mathematics, 4(2), 2012, 119-136.[11] H. Gu, J.J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 15(2015), 344-354.[12] R.Hilfer, Application of fractional Calculus in Physics, World Scientific, Singapore, 1999.[13] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.[14] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Sci. Publishers, Yverdon, 1993.[15] T. T. Soong, Random Differential Equations in Science and Engineering. Academic Press, New York, 1973.[16] K.M. Kolwankar, A.D. Gangal, Fractional differentiability of nowhere differentiable functions and dimension, Chaos, 6 (1996) 505-513.[17] J. Vanterler da C. Sousa, E. Capelas de Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., In Press, Accepted Manuscript-2018.[18] J. Vanterler da C. Sousa, E. Capelas de Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., Accepted Manuscript-2018.[19] J. Vanterler da C. Sousa, D. Santos Oliveira, E. Capelas de Oliveira, On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation, Math. Meth. Appl. Sci., (2018) 113.[20] E. Capelas de Oliveira, J. Vanterler da C. Sousa, UlamHyersRassias stability for a class of fractional integro-differential equations, Results Math., (2018), 73:111.[21] J. Vanterler da C. Sousa, Kishor D. Kucche, E. Capelas de Oliveira, Stability of ψ-Hilfer impulsive fractional differential equations, Appl. Math. Lett., (2018).[22] D. Vivek, K. Kanagarajan, E.M. Elsayed, Some existence and stability results for Hilferfractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15, (2018), 1-15.[23] H. Vu, Random fractional functional differential equations, International Journal of Nonlinear Analysis and Applications, 7(2), (2016), 253-267.[24] J.Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron J. Qual. Theory Differ. Equ., 63, (2011), 1-10.
ON TIMES SCALE FRACTIONAL ORDER DIFFERENTIAL EQUATION INVOLVING RANDOM VARIABLE
Yıl 2019,
Cilt: 1 Sayı: 1, 35 - 43, 18.01.2019
S. Harıkrıshnan
,
Elsayed Elsayed
,
K. Kanagarajan
Öz
In this paper, nonlocal and boundary value problems(BVP) of fractional differential equations involving random walk on times scale is discussed. The sufficient conditions for existence and uniqueness of dynamical systems are obtained using standard fixed point methods. The stability of solutions is made sure by Ulam-Hyers stability method.
Kaynakça
- [1] A. Ahmadkhanlu, M. Jahanshahi, On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, B. Iran. Math. Soc., 38, (2012), 241-252.[2] N. Benkhettou, A. Hammoudi, D. F. M. Torres, Existence and uniqueness of solution for a fractional Riemann-lioville initial value problem on time scales, Journal of King Saud University-Science, 28,(2016),87-92.[3] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh˜auser, Boston, 2003.[4] M. Bohner, A. Peterson, Dtnamica equations on times scale, Birkhauser Boston, Boston, MA.[5] R.P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35, (1999), 3-22.[6] K.M. Furati, M.D. Kassim, N.e-. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64, (2012), 1616-1626.[7] A. Granas, and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.[8] S. Harikrishnan, Kamal Shah, Dumitru Baleanu, K. Kanagarajan, Note on the solution of random differential equations via ψ-Hilfer fractional derivative, Advances in Difference Equations, 2018, 2018:224.[9] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 23,(2012).[10] V. Lupulescu, S.K. Ntouyas, Random fractional differential equations International electronic journal of pure and applied mathematics, 4(2), 2012, 119-136.[11] H. Gu, J.J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 15(2015), 344-354.[12] R.Hilfer, Application of fractional Calculus in Physics, World Scientific, Singapore, 1999.[13] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.[14] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Sci. Publishers, Yverdon, 1993.[15] T. T. Soong, Random Differential Equations in Science and Engineering. Academic Press, New York, 1973.[16] K.M. Kolwankar, A.D. Gangal, Fractional differentiability of nowhere differentiable functions and dimension, Chaos, 6 (1996) 505-513.[17] J. Vanterler da C. Sousa, E. Capelas de Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., In Press, Accepted Manuscript-2018.[18] J. Vanterler da C. Sousa, E. Capelas de Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., Accepted Manuscript-2018.[19] J. Vanterler da C. Sousa, D. Santos Oliveira, E. Capelas de Oliveira, On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation, Math. Meth. Appl. Sci., (2018) 113.[20] E. Capelas de Oliveira, J. Vanterler da C. Sousa, UlamHyersRassias stability for a class of fractional integro-differential equations, Results Math., (2018), 73:111.[21] J. Vanterler da C. Sousa, Kishor D. Kucche, E. Capelas de Oliveira, Stability of ψ-Hilfer impulsive fractional differential equations, Appl. Math. Lett., (2018).[22] D. Vivek, K. Kanagarajan, E.M. Elsayed, Some existence and stability results for Hilferfractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15, (2018), 1-15.[23] H. Vu, Random fractional functional differential equations, International Journal of Nonlinear Analysis and Applications, 7(2), (2016), 253-267.[24] J.Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron J. Qual. Theory Differ. Equ., 63, (2011), 1-10.