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

* 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 & Continuous Dynamical Systems - S, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. 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]

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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

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

[22]

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

[23]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

show all references

References:
[1]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. 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]

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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

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

[22]

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

[23]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

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