In the present work a coupled system consisting by $ \psi $-Hilfer fractional order Langevin equations supplemented with nonlocal integral boundary conditions is studied. Existence and uniqueness results are obtained by using standard fixed point theorems. The obtained results are well illustrated by numerical examples.
Citation: Weerawat Sudsutad, Sotiris K. Ntouyas, Chatthai Thaiprayoon. Nonlocal coupled system for $ \psi $-Hilfer fractional order Langevin equations[J]. AIMS Mathematics, 2021, 6(9): 9731-9756. doi: 10.3934/math.2021566
In the present work a coupled system consisting by $ \psi $-Hilfer fractional order Langevin equations supplemented with nonlocal integral boundary conditions is studied. Existence and uniqueness results are obtained by using standard fixed point theorems. The obtained results are well illustrated by numerical examples.
| [1] | K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer, New York, 2010. |
| [2] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of the Fractional Differential Equations, North-Holland Mathematics Studies, 204, 2006. |
| [3] | V. Lakshmikantham, S. Leela, J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009. |
| [4] | K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, NewYork, 1993. |
| [5] | I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. |
| [6] | S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, 1993. |
| [7] | Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. |
| [8] | B. Ahmad, A. Alsaedi, S.K. Ntouyas, J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer, Cham, Switzerland, 2017. |
| [9] | R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. |
| [10] | R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, J. Chem. Phys., 284 (2002), 399–408. |
| [11] | R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouvill fractional derivatives, Frac. Calc. Appl. Anal., 12 (2009), 299–318. |
| [12] |
R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simulat., 44 (2017), 460–481. doi: 10.1016/j.cnsns.2016.09.006
|
| [13] |
J. Vanterler da C. Sousa, E. Capelas de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72–91. doi: 10.1016/j.cnsns.2018.01.005
|
| [14] |
J. Vanterler da C. Sousa, E. Capelas de Oliveira, Leibniz type rule: $\psi$-Hilfer fractional operator, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 305–311. doi: 10.1016/j.cnsns.2019.05.003
|
| [15] |
J. Vanterler da C. Sousa, E. Capelas de Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., 81 (2018), 50–56. doi: 10.1016/j.aml.2018.01.016
|
| [16] |
K. M. Furati, N. D. Kassim, N. E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616–1626. doi: 10.1016/j.camwa.2012.01.009
|
| [17] | H. Gu, J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257 (2015), 344–354. |
| [18] | J. Wang, Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266 (2015), 850–859. |
| [19] | S. Asawasamrit, A. Kijjathanakorn, S. K. Ntouyas, J. Tariboon, Nonlocal boundary value problems for Hilfer fractional differential equations, Bull. Korean Math. Soc., 55 (2018), 1639–1657. |
| [20] |
A. Mali, K. Kucche, Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations, Math. Meth. Appl. Sci., 43 (2020), 8608–8631. doi: 10.1002/mma.6521
|
| [21] | S. K. Ntouyas, D. Vivek, Existence and uniqueness results for sequential $\psi$-Hilfer fractional differential equations with multi-point boundary conditions, Acta Mathematica Universitatis Comenianae, 90 (2021), 171–185. |
| [22] | W. T. Coffey, Yu. P. Kalmykov, J. T. Waldron, The Langevin Equation, second ed., World Scientific, Singapore, 2004. |
| [23] | A. Alsaedi, S. K. Ntouyas, B. Ahmad, Existence results for Langevin fractional differential inclusions involving two fractional orders with four-point multi-term fractional integral boundary conditions, Abstr. Appl. Anal., 2013 (2013), 1–17. |
| [24] |
J. Tariboon, S. K. Ntouyas, Nonlinear second-order impulsive $q$-difference Langevin equation with boundary conditions, Bound. Value Probl., 2014 (2014), 85. doi: 10.1186/1687-2770-2014-85
|
| [25] | J. Tariboon, S. K. Ntouyas, C. Thaiprayoon, Nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions, Adv. Math. Phys., 2014 (2014), 1–15. |
| [26] |
Ch. Nuchpong, S. K. Ntouyas, D. Vivek, J. Tariboon, Nonlocal boundary value problems for $\psi$-Hilfer fractional-order Langevin equations, Bound. Value Probl., 2021 (2021), 1–12. doi: 10.1186/s13661-020-01478-2
|
| [27] |
C. Thaiprayoon, W. Sudsutad, S. K. Ntouyas, Mixed nonlocal boundary value problem for implicit fractional integro-differential equations via $\psi$-Hilfer fractional derivative, Adv. Differ. Equ., 2021 (2021), 1–24. doi: 10.1186/s13662-020-03162-2
|
| [28] | K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. |
| [29] | A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2005. |
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Weerawat Sudsutad, Sotiris K. Ntouyas, Chatthai Thaiprayoon. Nonlocal coupled system for $ \psi $-Hilfer fractional order Langevin equations[J]. AIMS Mathematics, 2021, 6(9): 9731-9756. doi: 10.3934/math.2021566
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