March  2020, 13(3): 351-375. doi: 10.3934/dcdss.2020020

Fractional operators with boundary points dependent kernels and integration by parts

1. 

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

2. 

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

3. 

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

Received  April 2018 Revised  May 2018 Published  March 2019

Recently, U. N. Katugampola presented some generalized fractional integrals and derivatives by iterating a $ t^{\rho-1}- $weighted integral, $ \rho>0 $. The case $ \rho = 1 $ produces Riemann and Caputo fractional derivatives and the limiting case $ \rho\rightarrow 0^+ $ results in Hadamard type fractional operators. In this article, we discuss the differences between a new class of nonlocal generalized fractional derivatives generated by iterating left and right type conformable integrals weighted by $ (t-a)^{\rho-1} $ and $ (b-t)^{\rho-1} $ and the ones introduced by Katugampola. In fact, we will present very different integration by parts formulas by presenting new mixed left and right generalized fractional operators with boundary points dependent kernels. The properties of this new class of mixed fractional operators are analyzed in newly defined function spaces as well.

Citation: Thabet Abdeljawad. Fractional operators with boundary points dependent kernels and integration by parts. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 351-375. doi: 10.3934/dcdss.2020020
References:
[1]

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.

[2]

T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Journal of Reports in Mathematical Physics, 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9.

[3]

T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, Journal of Inequalities and Applications, 2017 (2017), Paper No. 130, 11 pp. doi: 10.1186/s13660-017-1400-5.

[4]

T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Advances in Difference Equations, 2017 (2017), Paper No. 313, 11 pp. doi: 10.1186/s13662-017-1285-0.

[5]

T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, Journal of Computational and Applied Mathematics, 339 (2018), 218-230.  doi: 10.1016/j.cam.2017.10.021.

[6]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Advances in Difference Equations, 2016 (2016), Paper No. 232, 18 pp. doi: 10.1186/s13662-016-0949-5.

[7]

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

[8]

T. Abdeljawad and D. Baleanu, Monotonicity results for a nabla fractional difference operator with discrete Mittag-Leffler kernels, Chaos, Solitons and Fractals, 102 (2017), 106-110.  doi: 10.1016/j.chaos.2017.04.006.

[9]

T. Abdeljawad, Q. M. Al-Mdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dynamics in Nature and Society, 2017 (2017), Article ID 4149320, 8 pages. doi: 10.1155/2017/4149320.

[10]

T. Abdeljawad, On conformable fractional calculus, Journal of Comput. and Appl. Math., 279 (2015), 57-66.  doi: 10.1016/j.cam.2014.10.016.

[11]

T. Abdeljawad and F. Jarad, Variational principles in the frame of certain generalized fractional derivatives, submitted.

[12]

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

[13]

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.

[14]

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

[15]

A. Atangana and D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J. Eng. Mech., (2016), D4016005.  doi: 10.1061/(ASCE)EM.1943-7889.0001091.

[16]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.

[17]

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

[18]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.

[19]

A.Atangana and J. F. Gomez Aguila, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 41 (2018), 315403, 8 pp.

[20]

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

[21]

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

[22]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11. 

[23]

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.

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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.

[29]

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.

[30]

F. Jarad, E. Uǧurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Advances in Difference Equations, 2017 (2017), Paper No. 247, 16 pp. doi: 10.1186/s13662-017-1306-z.

[31]

F. JaradT. Abdeljawad and D. Baleanu, On the generalized fractional derivatives and their Caputo modification, Journal of Nonlinear Sciences and Applications, 10 (2017), 2607-2619.  doi: 10.22436/jnsa.010.05.27.

[32]

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

[33]

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

[34]

A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory And Application Of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

[35]

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

[36]

K. Ervin Lenzi, A. A. Tateishi and H. Ribeiro, The Role of Fractional Time-Derivative Operators on Anomalous Diffusion, Frontiers in Physics., 2017.

[37]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 87-92. 

[38]

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

[39]

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

[40]

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

[41]

J. Tariboon, S. K. Ntouyas and P. Agarwal, New concepts of fractional quabtum calculus and applications to impulsive fractional q-difference equations, Adv. Differ. Eqns., 2015 (2015), 19pp. doi: 10.1186/s13662-014-0348-8.

show all references

References:
[1]

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.

[2]

T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Journal of Reports in Mathematical Physics, 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9.

[3]

T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, Journal of Inequalities and Applications, 2017 (2017), Paper No. 130, 11 pp. doi: 10.1186/s13660-017-1400-5.

[4]

T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Advances in Difference Equations, 2017 (2017), Paper No. 313, 11 pp. doi: 10.1186/s13662-017-1285-0.

[5]

T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, Journal of Computational and Applied Mathematics, 339 (2018), 218-230.  doi: 10.1016/j.cam.2017.10.021.

[6]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Advances in Difference Equations, 2016 (2016), Paper No. 232, 18 pp. doi: 10.1186/s13662-016-0949-5.

[7]

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

[8]

T. Abdeljawad and D. Baleanu, Monotonicity results for a nabla fractional difference operator with discrete Mittag-Leffler kernels, Chaos, Solitons and Fractals, 102 (2017), 106-110.  doi: 10.1016/j.chaos.2017.04.006.

[9]

T. Abdeljawad, Q. M. Al-Mdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dynamics in Nature and Society, 2017 (2017), Article ID 4149320, 8 pages. doi: 10.1155/2017/4149320.

[10]

T. Abdeljawad, On conformable fractional calculus, Journal of Comput. and Appl. Math., 279 (2015), 57-66.  doi: 10.1016/j.cam.2014.10.016.

[11]

T. Abdeljawad and F. Jarad, Variational principles in the frame of certain generalized fractional derivatives, submitted.

[12]

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

[13]

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.

[14]

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

[15]

A. Atangana and D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J. Eng. Mech., (2016), D4016005.  doi: 10.1061/(ASCE)EM.1943-7889.0001091.

[16]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.

[17]

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

[18]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.

[19]

A.Atangana and J. F. Gomez Aguila, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 41 (2018), 315403, 8 pp.

[20]

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

[21]

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

[22]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11. 

[23]

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.

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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.

[29]

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.

[30]

F. Jarad, E. Uǧurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Advances in Difference Equations, 2017 (2017), Paper No. 247, 16 pp. doi: 10.1186/s13662-017-1306-z.

[31]

F. JaradT. Abdeljawad and D. Baleanu, On the generalized fractional derivatives and their Caputo modification, Journal of Nonlinear Sciences and Applications, 10 (2017), 2607-2619.  doi: 10.22436/jnsa.010.05.27.

[32]

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

[33]

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

[34]

A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory And Application Of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

[35]

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

[36]

K. Ervin Lenzi, A. A. Tateishi and H. Ribeiro, The Role of Fractional Time-Derivative Operators on Anomalous Diffusion, Frontiers in Physics., 2017.

[37]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 87-92. 

[38]

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

[39]

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

[40]

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

[41]

J. Tariboon, S. K. Ntouyas and P. Agarwal, New concepts of fractional quabtum calculus and applications to impulsive fractional q-difference equations, Adv. Differ. Eqns., 2015 (2015), 19pp. doi: 10.1186/s13662-014-0348-8.

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