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## Fractional operators with boundary points dependent kernels and integration by parts

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

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 & Continuous Dynamical Systems - S, 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] T. Abdeljawad and F. Jarad, Variational principles in the frame of certain generalized fractional derivatives, submitted.Google Scholar [12] Y. Adjabi, F. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] A. Atangana and D. Baleanu, New fractional derivative with non-local and non-singular kernel, Thermal Sci., 20 (2016), 757-763. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [21] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 73-85. Google Scholar [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. Google Scholar [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. Google Scholar [24] R. Hilfer, Applications Of Fractional Calculus In Physics, Word Scientific, Singapore, 2000. doi: 10.1142/9789812817747. Google Scholar [25] F. Jarad, D. 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. Google Scholar [26] F. Jarad, D. 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. Google Scholar [27] F. Jarad, T. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] F. Jarad, T. 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. Google Scholar [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. Google Scholar [33] U. N. Katugampola, A new approach to generalized fractional derivatives, Bul. Math. Anal.Appl., 6 (2014), 1-15. Google Scholar [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. Google Scholar [35] A. A. Kilbas, Hadamard type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204. Google Scholar [36] K. Ervin Lenzi, A. A. Tateishi and H. Ribeiro, The Role of Fractional Time-Derivative Operators on Anomalous Diffusion, Frontiers in Physics., 2017.Google Scholar [37] J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 87-92. Google Scholar [38] R. L. Magin, Fractional Calculus In Bioengineering, Begell House Publishers, 2006.Google Scholar [39] I. Podlubny, Fractional Differential Equations, Academic Press: San Diego CA, 1999. Google Scholar [40] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals And Derivatives: Theory And Applications, Gordon and Breach, Yverdon, 1993. Google Scholar [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. Google Scholar

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##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] T. Abdeljawad and F. Jarad, Variational principles in the frame of certain generalized fractional derivatives, submitted.Google Scholar [12] Y. Adjabi, F. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] A. Atangana and D. Baleanu, New fractional derivative with non-local and non-singular kernel, Thermal Sci., 20 (2016), 757-763. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [21] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 73-85. Google Scholar [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. Google Scholar [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. Google Scholar [24] R. Hilfer, Applications Of Fractional Calculus In Physics, Word Scientific, Singapore, 2000. doi: 10.1142/9789812817747. Google Scholar [25] F. Jarad, D. 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. Google Scholar [26] F. Jarad, D. 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. Google Scholar [27] F. Jarad, T. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] F. Jarad, T. 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. Google Scholar [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. Google Scholar [33] U. N. Katugampola, A new approach to generalized fractional derivatives, Bul. Math. Anal.Appl., 6 (2014), 1-15. Google Scholar [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. Google Scholar [35] A. A. Kilbas, Hadamard type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204. Google Scholar [36] K. Ervin Lenzi, A. A. Tateishi and H. Ribeiro, The Role of Fractional Time-Derivative Operators on Anomalous Diffusion, Frontiers in Physics., 2017.Google Scholar [37] J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 87-92. Google Scholar [38] R. L. Magin, Fractional Calculus In Bioengineering, Begell House Publishers, 2006.Google Scholar [39] I. Podlubny, Fractional Differential Equations, Academic Press: San Diego CA, 1999. Google Scholar [40] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals And Derivatives: Theory And Applications, Gordon and Breach, Yverdon, 1993. Google Scholar [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. Google Scholar
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