March  2020, 13(3): 695-708. doi: 10.3934/dcdss.2020038

Variational principles in the frame of certain generalized fractional derivatives

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

3. 

Department of Medical Research, China Medical University, Taichung 40402, Taiwan

4. 

Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

* Corresponding author

Received  August 2018 Revised  September 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. Variational principles in the frame of certain generalized fractional derivatives. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 695-708. doi: 10.3934/dcdss.2020038
References:
[1]

T. Abdeljawad and D. Baleanu, Fractional differences and integration by parts, J. Comput. Anal. Appl., 3 (2011), 574-582. 

[2]

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

[3]

T. Abdeljawad and D. Baleanu, On fractiona l derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9.

[4]

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.

[5]

O. P. Agrawal, Generalized Euler-Lagrange equations and transversality conditions for fvps in terms of caputo derivative, J. Vib. Control, 13 (2007), 1217-1237.  doi: 10.1177/1077546307077472.

[6]

O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), 368-379.  doi: 10.1016/S0022-247X(02)00180-4.

[7]

R. Almeida, Variational problems involving a Caputo-type fractional derivative, J. Optim. Theory Appl., 174 (2017), 276-294.  doi: 10.1007/s10957-016-0883-4.

[8]

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

[9]

D. Baleanu and J. J. Trujillo, On exact solutions of a class of fractional Euler-Lagrange equations, Nonlin. Dyn., 52 (2008), 331-335.  doi: 10.1007/s11071-007-9281-7.

[10]

D. Baleanu, T. Abdeljawad and F. Jarad, Fractional variational principles with delay, J. Phys. A: Math. and Theor., 41 (2008), 315403, 8 pp. doi: 10.1088/1751-8113/41/31/315403.

[11]

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

[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.

[13]

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

[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.

[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.

[16]

F. JaradD. Baleanu and T. Abdeljawad, Fractional variational principles with delay within Caputo derivatives, Rep. Math. Phys., 65 (2010), 17-28.  doi: 10.1016/S0034-4877(10)00010-8.

[17]

F. JaradD. Baleanu and T. Abdeljawad, Fractional variational optimal control problems with delayed arguments, Nonlinear Dyn., 62 (2010), 609-614.  doi: 10.1007/s11071-010-9748-9.

[18]

F. JaradT. Abdeljawad and D. Baleanu, Higher order fractional variational optimal control problems with delayed arguments, Appl. Math. Comput., 218 (2012), 9234-9240.  doi: 10.1016/j.amc.2012.02.080.

[19]

F. Jarad, T. Abdeljawad and D. Baleanu, On Riesz-Caputo formulation for sequential fractional variational principles, Abstract and Applied Analysis, 2012 (2012), Article ID 890396, 15 pages. doi: 10.1155/2012/890396.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

M. J. Lazo and D. F. M. Torres, Variational calculus with confromable fractional derivatives, IEEE/CAA J. Autom. Sinica, 4 (2017), 340-352.  doi: 10.1109/JAS.2016.7510160.

[25]

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

[26]

J. A. T. MachadoV. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140-1153.  doi: 10.1016/j.cnsns.2010.05.027.

[27]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006.

[28]

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

[29]

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

[30]

B. Van Brunt, The Calculus of Variations, Springer, New York, 2004. doi: 10.1007/b97436.

show all references

References:
[1]

T. Abdeljawad and D. Baleanu, Fractional differences and integration by parts, J. Comput. Anal. Appl., 3 (2011), 574-582. 

[2]

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

[3]

T. Abdeljawad and D. Baleanu, On fractiona l derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9.

[4]

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.

[5]

O. P. Agrawal, Generalized Euler-Lagrange equations and transversality conditions for fvps in terms of caputo derivative, J. Vib. Control, 13 (2007), 1217-1237.  doi: 10.1177/1077546307077472.

[6]

O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), 368-379.  doi: 10.1016/S0022-247X(02)00180-4.

[7]

R. Almeida, Variational problems involving a Caputo-type fractional derivative, J. Optim. Theory Appl., 174 (2017), 276-294.  doi: 10.1007/s10957-016-0883-4.

[8]

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

[9]

D. Baleanu and J. J. Trujillo, On exact solutions of a class of fractional Euler-Lagrange equations, Nonlin. Dyn., 52 (2008), 331-335.  doi: 10.1007/s11071-007-9281-7.

[10]

D. Baleanu, T. Abdeljawad and F. Jarad, Fractional variational principles with delay, J. Phys. A: Math. and Theor., 41 (2008), 315403, 8 pp. doi: 10.1088/1751-8113/41/31/315403.

[11]

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

[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.

[13]

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

[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.

[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.

[16]

F. JaradD. Baleanu and T. Abdeljawad, Fractional variational principles with delay within Caputo derivatives, Rep. Math. Phys., 65 (2010), 17-28.  doi: 10.1016/S0034-4877(10)00010-8.

[17]

F. JaradD. Baleanu and T. Abdeljawad, Fractional variational optimal control problems with delayed arguments, Nonlinear Dyn., 62 (2010), 609-614.  doi: 10.1007/s11071-010-9748-9.

[18]

F. JaradT. Abdeljawad and D. Baleanu, Higher order fractional variational optimal control problems with delayed arguments, Appl. Math. Comput., 218 (2012), 9234-9240.  doi: 10.1016/j.amc.2012.02.080.

[19]

F. Jarad, T. Abdeljawad and D. Baleanu, On Riesz-Caputo formulation for sequential fractional variational principles, Abstract and Applied Analysis, 2012 (2012), Article ID 890396, 15 pages. doi: 10.1155/2012/890396.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

M. J. Lazo and D. F. M. Torres, Variational calculus with confromable fractional derivatives, IEEE/CAA J. Autom. Sinica, 4 (2017), 340-352.  doi: 10.1109/JAS.2016.7510160.

[25]

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

[26]

J. A. T. MachadoV. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140-1153.  doi: 10.1016/j.cnsns.2010.05.027.

[27]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006.

[28]

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

[29]

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

[30]

B. Van Brunt, The Calculus of Variations, Springer, New York, 2004. doi: 10.1007/b97436.

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