• Previous Article
    Variational principles in the frame of certain generalized fractional derivatives
  • DCDS-S Home
  • This Issue
  • Next Article
    Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative
doi: 10.3934/dcdss.2020039

Generalized fractional derivatives and Laplace transform

1. 

Department of Mathematics, Çankaya University 06790, Ankara, Turkey

2. 

Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia

* Corresponding author

Received  August 2018 Revised  October 2018 Published  March 2019

Fund Project: The second author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17

In this article, we study generalized fractional derivatives that contain kernels depending on a function on the space of absolute continuous functions. We generalize the Laplace transform in order to be applicable for the generalized fractional integrals and derivatives and apply this transform to solve some ordinary differential equations in the frame of the fractional derivatives under discussion.

Citation: Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020039
References:
[1]

T. Abdeljawad, D. Baleanu and F. Jarad, Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys., 49 (2008), 083507, 11pp. doi: 10.1063/1.2970709. Google Scholar

[2]

T. AbdeljawadF. Jarad and D. Baleanu, On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Sci. China Ser. A: Math, 51 (2008), 1775-1786. doi: 10.1007/s11425-008-0068-1. Google Scholar

[3]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017, 1098-1107. doi: 10.22436/jnsa.010.03.20. Google Scholar

[4]

T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Difference Equ., 2017 (2017), Paper No. 78, 9 pp. doi: 10.1186/s13662-017-1126-1. Google Scholar

[5]

Y. AdjabiF. JaradD. Baleanu and T. Abdeljawad, On Cauchy problems with Caputo-Hadamard fractional derivatives, J. Comput. Anal. Appl., 21 (2016), 661-681. Google Scholar

[6]

Y. AdjabiF. Jarad and T. Abdeljawad, On generalized fractional operators and a Gronwall type inequality with applications, Filomat, 31 (2017), 5457-5473. doi: 10.2298/FIL1717457A. Google Scholar

[7]

R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonl. Sci. Numer. Simult., 44 (2017), 460-481. doi: 10.1016/j.cnsns.2016.09.006. Google Scholar

[8]

A. Atangana and D. Baleanu, New fractional derivative with non-local and non-singular kernel, Thermal Sci., 20 (2016), 757-763. Google Scholar

[9]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85. Google Scholar

[10]

V. Daftardar-Gejji and H. Jaffari, Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives, J. Math. Anal. Appl., 328 (2007), 1026-1033. doi: 10.1016/j.jmaa.2006.06.007. Google Scholar

[11]

D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204 (1996), 609-625. doi: 10.1006/jmaa.1996.0456. Google Scholar

[12]

Y. Y. Gambo, F. Jarad, T. Abdeljawad and D. Baleanu, On Caputo modification of the Hadamard fractional derivative, Adv. Difference Equ., 2014 (2014), 12pp. doi: 10.1186/1687-1847-2014-10. Google Scholar

[13]

R. Hilfer, Applications of Fractional Calculus in Physics, Word Scientific, Singapore, 2000. doi: 10.1142/9789812817747. Google Scholar

[14]

F. Jarad, T. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivative, Adv. Difference Equ., 2012 (2012), 8pp. doi: 10.1186/1687-1847-2012-142. Google Scholar

[15]

F. JaradT. Abdeljawad and D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10 (2017), 2607-2619. doi: 10.22436/jnsa.010.05.27. Google Scholar

[16]

F. Jarad, E. Uğurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Adv. Difference Equ., 2017 (2017), Paper No. 247, 16 pp. doi: 10.1186/s13662-017-1306-z. Google Scholar

[17]

U. N. Katugampola, New approach to generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865. doi: 10.1016/j.amc.2011.03.062. Google Scholar

[18]

U. N. Katugampola, A new approach to generalized fractional derivatives, Bul. Math. Anal.Appl., 6 (2014), 1-15. Google Scholar

[19]

A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Application of Fractional Fifferential Equations, North Holland Mathematics Studies, 204, Amsterdam, 2006. Google Scholar

[20]

A. A. Kilbas, Hadamard type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204. Google Scholar

[21]

C. F. Lorenzoand and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam., 29 (2002), 57-98. doi: 10.1023/A:1016586905654. Google Scholar

[22]

R. L. Magin, Fractional calculus in Bioengineering, House Publishers, Redding, 2006.Google Scholar

[23]

D. S. Oliveira and E. Capelas de Oliveira, On a Caputo-type fractional derivatives, Available from: http://www.ime.unicamp.br/sites/default/files/pesquisa/relatorios/rp-2017-13.pdf. doi: 10.1515/apam-2017-0068. Google Scholar

[24]

I. Podlubny, Fractional Differential Equations, Academic Press: San Diego CA, 1999. Google Scholar

[25]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993. Google Scholar

[26]

J. Vanterler daC. Sousa and E. Capelas de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonl. Sci. Numer. Simult., 60 (2018), 72-91. doi: 10.1016/j.cnsns.2018.01.005. Google Scholar

[27]

J. Vanterler da C. Sousa and E. Capelas de Oliveira, A Gronwall inequality and the Cauchy-tupe problem by means of $\psi$-Hilfer operator, preprint, arXiv: 1709.03634.Google Scholar

[28]

J. Vanterler da C. Sousa and E. Capelas de Oliveira, A new fractional derivative of variable order with non-singular order and fractional differential equations, preprint, arXiv: 1712.06506.Google Scholar

[29]

X. J. YangH. M. Srivastava and J. A. T. Machado, A new fractional derivatives without singular kernel: Application to the modelling of the steady heat flow, Therm. Sci., 20 (2016), 753-756. Google Scholar

show all references

References:
[1]

T. Abdeljawad, D. Baleanu and F. Jarad, Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys., 49 (2008), 083507, 11pp. doi: 10.1063/1.2970709. Google Scholar

[2]

T. AbdeljawadF. Jarad and D. Baleanu, On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Sci. China Ser. A: Math, 51 (2008), 1775-1786. doi: 10.1007/s11425-008-0068-1. Google Scholar

[3]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017, 1098-1107. doi: 10.22436/jnsa.010.03.20. Google Scholar

[4]

T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Difference Equ., 2017 (2017), Paper No. 78, 9 pp. doi: 10.1186/s13662-017-1126-1. Google Scholar

[5]

Y. AdjabiF. JaradD. Baleanu and T. Abdeljawad, On Cauchy problems with Caputo-Hadamard fractional derivatives, J. Comput. Anal. Appl., 21 (2016), 661-681. Google Scholar

[6]

Y. AdjabiF. Jarad and T. Abdeljawad, On generalized fractional operators and a Gronwall type inequality with applications, Filomat, 31 (2017), 5457-5473. doi: 10.2298/FIL1717457A. Google Scholar

[7]

R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonl. Sci. Numer. Simult., 44 (2017), 460-481. doi: 10.1016/j.cnsns.2016.09.006. Google Scholar

[8]

A. Atangana and D. Baleanu, New fractional derivative with non-local and non-singular kernel, Thermal Sci., 20 (2016), 757-763. Google Scholar

[9]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85. Google Scholar

[10]

V. Daftardar-Gejji and H. Jaffari, Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives, J. Math. Anal. Appl., 328 (2007), 1026-1033. doi: 10.1016/j.jmaa.2006.06.007. Google Scholar

[11]

D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204 (1996), 609-625. doi: 10.1006/jmaa.1996.0456. Google Scholar

[12]

Y. Y. Gambo, F. Jarad, T. Abdeljawad and D. Baleanu, On Caputo modification of the Hadamard fractional derivative, Adv. Difference Equ., 2014 (2014), 12pp. doi: 10.1186/1687-1847-2014-10. Google Scholar

[13]

R. Hilfer, Applications of Fractional Calculus in Physics, Word Scientific, Singapore, 2000. doi: 10.1142/9789812817747. Google Scholar

[14]

F. Jarad, T. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivative, Adv. Difference Equ., 2012 (2012), 8pp. doi: 10.1186/1687-1847-2012-142. Google Scholar

[15]

F. JaradT. Abdeljawad and D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10 (2017), 2607-2619. doi: 10.22436/jnsa.010.05.27. Google Scholar

[16]

F. Jarad, E. Uğurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Adv. Difference Equ., 2017 (2017), Paper No. 247, 16 pp. doi: 10.1186/s13662-017-1306-z. Google Scholar

[17]

U. N. Katugampola, New approach to generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865. doi: 10.1016/j.amc.2011.03.062. Google Scholar

[18]

U. N. Katugampola, A new approach to generalized fractional derivatives, Bul. Math. Anal.Appl., 6 (2014), 1-15. Google Scholar

[19]

A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Application of Fractional Fifferential Equations, North Holland Mathematics Studies, 204, Amsterdam, 2006. Google Scholar

[20]

A. A. Kilbas, Hadamard type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204. Google Scholar

[21]

C. F. Lorenzoand and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam., 29 (2002), 57-98. doi: 10.1023/A:1016586905654. Google Scholar

[22]

R. L. Magin, Fractional calculus in Bioengineering, House Publishers, Redding, 2006.Google Scholar

[23]

D. S. Oliveira and E. Capelas de Oliveira, On a Caputo-type fractional derivatives, Available from: http://www.ime.unicamp.br/sites/default/files/pesquisa/relatorios/rp-2017-13.pdf. doi: 10.1515/apam-2017-0068. Google Scholar

[24]

I. Podlubny, Fractional Differential Equations, Academic Press: San Diego CA, 1999. Google Scholar

[25]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993. Google Scholar

[26]

J. Vanterler daC. Sousa and E. Capelas de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonl. Sci. Numer. Simult., 60 (2018), 72-91. doi: 10.1016/j.cnsns.2018.01.005. Google Scholar

[27]

J. Vanterler da C. Sousa and E. Capelas de Oliveira, A Gronwall inequality and the Cauchy-tupe problem by means of $\psi$-Hilfer operator, preprint, arXiv: 1709.03634.Google Scholar

[28]

J. Vanterler da C. Sousa and E. Capelas de Oliveira, A new fractional derivative of variable order with non-singular order and fractional differential equations, preprint, arXiv: 1712.06506.Google Scholar

[29]

X. J. YangH. M. Srivastava and J. A. T. Machado, A new fractional derivatives without singular kernel: Application to the modelling of the steady heat flow, Therm. Sci., 20 (2016), 753-756. Google Scholar

[1]

Fahd Jarad, Thabet Abdeljawad. Variational principles in the frame of certain generalized fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 695-708. doi: 10.3934/dcdss.2020038

[2]

Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 723-739. doi: 10.3934/dcdss.2020040

[3]

Tatiana Odzijewicz. Generalized fractional isoperimetric problem of several variables. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2617-2629. doi: 10.3934/dcdsb.2014.19.2617

[4]

Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Zakia Hammouch, Dumitru Baleanu. A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 975-993. doi: 10.3934/dcdss.2020057

[5]

Mostafa El Haffari, Ahmed Roubi. Prox-dual regularization algorithm for generalized fractional programs. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1991-2013. doi: 10.3934/jimo.2017028

[6]

Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial & Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723

[7]

Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361

[8]

Mehar Chand, Jyotindra C. Prajapati, Ebenezer Bonyah, Jatinder Kumar Bansal. Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 539-560. doi: 10.3934/dcdss.2020030

[9]

Amir Khan, Asaf Khan, Tahir Khan, Gul Zaman. Extension of triple Laplace transform for solving fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 755-768. doi: 10.3934/dcdss.2020042

[10]

Ruiyang Cai, Fudong Ge, Yangquan Chen, Chunhai Kou. Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019033

[11]

Pierre Aime Feulefack, Jean Daniel Djida, Atangana Abdon. A new model of groundwater flow within an unconfined aquifer: Application of Caputo-Fabrizio fractional derivative. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3227-3247. doi: 10.3934/dcdsb.2018317

[12]

Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023

[13]

Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062

[14]

Hongjie Dong, Dong Li. On a generalized maximum principle for a transport-diffusion model with $\log$-modulated fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3437-3454. doi: 10.3934/dcds.2014.34.3437

[15]

Ram U. Verma. General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 333-339. doi: 10.3934/naco.2011.1.333

[16]

Anurag Jayswal, Ashish Kumar Prasad, Izhar Ahmad. On minimax fractional programming problems involving generalized $(H_p,r)$-invex functions. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1001-1018. doi: 10.3934/jimo.2014.10.1001

[17]

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93

[18]

Hayat Zouiten, Ali Boutoulout, Delfim F. M. Torres. Regional enlarged observability of Caputo fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1017-1029. doi: 10.3934/dcdss.2020060

[19]

G. M. Bahaa. Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 485-501. doi: 10.3934/dcdss.2020027

[20]

Miaohua Jiang. Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 967-983. doi: 10.3934/dcds.2015.35.967

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (125)
  • HTML views (589)
  • Cited by (0)

Other articles
by authors

[Back to Top]