-
Previous Article
Electromagnetic waves described by a fractional derivative of variable and constant order with non singular kernel
- DCDS-S Home
- This Issue
-
Next Article
Abundant novel solutions of the conformable Lakshmanan-Porsezian-Daniel model
Lyapunov type inequality in the frame of generalized Caputo derivatives
1. | Department of Mathematics, Çankaya University 06790, Ankara, Turkey |
2. | Department of Mathematics, Faculty of Sciences, University of M'hamed Bougara, Boumerdes, 35000, Algeria |
3. | Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia |
4. | Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan |
5. | Department of Applied mathematics, Palestine Technical University-Kadoorie, Palestine |
6. | College of Engineering, Al Ain University of Science and Technology, Al Ain, UAE, College of Science, Tafila Technical University, Tafila, Jordan |
In this paper, we establish the Lyapunov-type inequality for boundary value problems involving generalized Caputo fractional derivatives that unite the Caputo and Caputo-Hadamrad fractional derivatives. An application about the zeros of generalized types of Mittag-Leffler functions is given.
References:
[1] |
T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), Paper No. 130, 11 pp.
doi: 10.1186/s13660-017-1400-5. |
[2] |
T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Difference Equ., 2017 (2017), Paper No. 313, 11 pp.
doi: 10.1186/s13662-017-1285-0. |
[3] |
T. Abdeljawad, Q. M. Al-Mdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dyn. Nat. Soc., 2017 (2017), Art. ID 4149320, 8 pp.
doi: 10.1155/2017/4149320. |
[4] |
T. Abdeljawad, J. Alzabut and F. Jarad, A generalized Lyapunov-type inequality in the frame of conformable derivatives, Adv. Difference Equ., 2017 (2017), Paper No. 321, 10 pp.
doi: 10.1186/s13662-017-1383-z. |
[5] |
T. Abdeljawad, B. Benli and D. Baleanu, A generalized $q$-Mittag-Leffler function by $q$-Captuo fractional linear equations, Abstr. Appl. Anal., 2012 (2012), Article ID 546062, 11 pp.
doi: 10.1155/2012/546062. |
[6] |
T. Abdeljawad, F. Jarad, S. F. Mallak and J. Alzabut, Lyapunov type inequalities via fractional proportional derivatives and application on the free zero disc of Kilbas-Saigo generalized Mittag-Leffler functions, Eur. Phys. J. Plus, 134 (2019), 247.
doi: 10.1140/epjp/i2019-12772-1. |
[7] |
T. Abdeljawad and F. Madjidi,
A Lyaponuv inequality for fractional difference operators with discrete Mittag-Leffler kernel of order $2\leq \alpha < 5/2$, Eur. Phys. J. Spec. Top., 226 (2017), 3355-3368.
|
[8] |
A. Atangana and D. Baleanu,
New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Sci., 20 (2016), 763-769.
|
[9] |
A. Atangana and J. F. Gómez-Aguilar,
Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos Solitons Fractals, 102 (2017), 285-294.
doi: 10.1016/j.chaos.2017.03.022. |
[10] |
D. Çakmak,
Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput., 216 (2010), 368-373.
doi: 10.1016/j.amc.2010.01.010. |
[11] |
M. Caputo and M. Fabrizio,
A new definition of fractional derivative without singular kerne, Prog. Frac. Diff. Appl., 1 (2015), 73-85.
|
[12] |
S. Clark and D. Hinton,
A Liapunov inequality for linear Hamiltonian systems, Math. Inequal. Appl., 1 (1998), 201-209.
doi: 10.7153/mia-01-18. |
[13] |
B. Cuahutenango-Barro, M. A. Taneco-Hernández and J. F. Gómez-Aguilar,
On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel, Chaos Solitons Fractals, 115 (2018), 283-299.
doi: 10.1016/j.chaos.2018.09.002. |
[14] |
K. Diethelm, The Analysis of Fractional Differential Equations, , Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-14574-2. |
[15] |
R. A. C. Ferreira,
A Lyapunov-type inequality for a fractional initial value problem, Fract. Calc. Appl. Anal., 16 (2013), 978-984.
doi: 10.2478/s13540-013-0060-5. |
[16] |
R. A. C. Ferreira,
Lyapunov-type inequalities for some sequential fractional boundary value problems, Adv. Dyn. Syst. Appl., 11 (2016), 33-43.
|
[17] |
R. A. C. Ferreira,
On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function, J. Math. Anal. Appl., 412 (2014), 1058-1063.
doi: 10.1016/j.jmaa.2013.11.025. |
[18] |
F. Jarad, T. 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. |
[19] |
F. Jarad, T. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivative, Adv. Difference Equ., 2012, (2012), 142, 8 pp.
doi: 10.1186/1687-1847-2012-142. |
[20] |
F. Jarad, T. Abdeljawad and Z. Hammouch,
On a class of ordinary differential equations in the frame of Atagana-Baleanu fractional derivative, Chaos Solitons Fractals, 117 (2018), 16-20.
doi: 10.1016/j.chaos.2018.10.006. |
[21] |
M. Jleli and B. Samet, Lyapunov-type inequalities for fractional boundary value problems equation with fractional initial conditions, Electron. J. Differential Equations, 2015 (2015), 11 pp. |
[22] |
J. F. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, Eur. Phys. J. Plus, 132 (2017), 13.
doi: 10.1140/epjp/i2017-11293-3. |
[23] |
J. F. Gómez-Aguilar, A. Atangana and V. F. Morales-Delgado, Electrical circuits RC, LC and RL described by Atangana-Baleanu fractional derivatives, Int. J. Circ. theor. Appl., 45 (2017), 1514–1533.
doi: 10.1002/cta.2348. |
[24] |
J. F. Gómez-Aguilar, H. Yépez-Martínez, R. F. Escobar-Jiménez, C. M. Astorga-Zaragoza and J. Reyes-Reyes,
Analytical and numerical solutions of electrical circuits described by fractional derivatives, Appl. Math. Model., 40 (2016), 9079-9094.
doi: 10.1016/j.apm.2016.05.041. |
[25] |
R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, Springer Heidelberg New York Dordrecht London, 2014.
doi: 10.1007/978-3-662-43930-2. |
[26] |
U. N. Katugampola,
New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.
doi: 10.1016/j.amc.2011.03.062. |
[27] |
U. N. Katugampola,
A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.
|
[28] |
A. A. Kilbas,
Hadamard type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204.
|
[29] |
A. A. Kilbas and M. Sa${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over i} }}$go,
Fractional integrals and derivatives of Mittag-Leffler type function (Russian), Dokl. Akad. Nauk Belarusi, 39 (1995), 22-26.
|
[30] |
A. A. Kilbas and M. Saigo,
On solutions of integral equations of Abel-Volterra type, Differential Integral Equations, 8 (1995), 993-1011.
|
[31] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, , Elsevier, Amsterdam, 2006. |
[32] |
A. M. Liapunov, Problème général de la stabilitie du mouvement, Ann. of Math. Stud., 17, Princeton Univ. Press, Princeton, N. J., 1949. |
[33] |
Q. Ma, C. Ma and J. Wang,
A Lyapunov-type inequality for a fractional differential equation with Hadamard derivative, J. Math. Inequal., 11 (2017), 135-141.
doi: 10.7153/jmi-11-13. |
[34] |
G. M. Mittag-Leffler,
Sur la nouvelle fonction $E_{\alpha }\left(z\right) $, C. R. Acad. Sci. Paris, 137 (1903), 554-558.
|
[35] |
N. Parhi and S. Panigrahi,
A Lyapunov-type integral inequality for higher order differential equations, Math. Slovaca, 52 (2002), 31-46.
|
[36] |
J. P. Pinasco, Lyapunov-Type Inequalities, Springer Briefs in Mathematics, Springer, New York, 2013.
doi: 10.1007/978-1-4614-8523-0. |
[37] |
I. Podlubny, Fractional Differential Equations,, Academic Press, an Diego, California, 1999.
![]() ![]() |
[38] |
T. R. Prabhakar,
A singular integral equation with a generalised Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.
|
[39] |
J. Rongand and C. Bai, Lyapunov-type inequality for afractional differential equation with fractional boundary conditions, Adv. Difference Equ., 2015, (2015), 82, 10 pp.
doi: 10.1186/s13662-015-0430-x. |
[40] |
X. Yang,
On Lyapunov-type inequality for certain higher-order differential equations, Appl. Math. Comput., 134 (2003), 307-317.
doi: 10.1016/S0096-3003(01)00285-5. |
[41] |
X. Yang and K. Lo,
Lyapunov-type inequality for a class of even-order differential equations, Appl. Math. Comput., 215 (2010), 3884-3890.
doi: 10.1016/j.amc.2009.11.032. |
[42] |
H. Ye, J. Gao and Y. Ding,
A generalized Lyapunov inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.
|
[43] |
H. Yépez-Martínez and J. F. Gómez-Aguilar,
A new modified definition of Caputo-Fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method (MHAM), J. Comput. Appl. Math., 346 (2019), 247-260.
doi: 10.1016/j.cam.2018.07.023. |
[44] |
H. Yépez-Martínez, J. F. Gómez-Aguilar, I. O. Sosa, J. M. Reyes and J. Torres-Jiménez,
The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Rev. Mexicana Fís., 62 (2016), 310-316.
|
show all references
References:
[1] |
T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), Paper No. 130, 11 pp.
doi: 10.1186/s13660-017-1400-5. |
[2] |
T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Difference Equ., 2017 (2017), Paper No. 313, 11 pp.
doi: 10.1186/s13662-017-1285-0. |
[3] |
T. Abdeljawad, Q. M. Al-Mdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dyn. Nat. Soc., 2017 (2017), Art. ID 4149320, 8 pp.
doi: 10.1155/2017/4149320. |
[4] |
T. Abdeljawad, J. Alzabut and F. Jarad, A generalized Lyapunov-type inequality in the frame of conformable derivatives, Adv. Difference Equ., 2017 (2017), Paper No. 321, 10 pp.
doi: 10.1186/s13662-017-1383-z. |
[5] |
T. Abdeljawad, B. Benli and D. Baleanu, A generalized $q$-Mittag-Leffler function by $q$-Captuo fractional linear equations, Abstr. Appl. Anal., 2012 (2012), Article ID 546062, 11 pp.
doi: 10.1155/2012/546062. |
[6] |
T. Abdeljawad, F. Jarad, S. F. Mallak and J. Alzabut, Lyapunov type inequalities via fractional proportional derivatives and application on the free zero disc of Kilbas-Saigo generalized Mittag-Leffler functions, Eur. Phys. J. Plus, 134 (2019), 247.
doi: 10.1140/epjp/i2019-12772-1. |
[7] |
T. Abdeljawad and F. Madjidi,
A Lyaponuv inequality for fractional difference operators with discrete Mittag-Leffler kernel of order $2\leq \alpha < 5/2$, Eur. Phys. J. Spec. Top., 226 (2017), 3355-3368.
|
[8] |
A. Atangana and D. Baleanu,
New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Sci., 20 (2016), 763-769.
|
[9] |
A. Atangana and J. F. Gómez-Aguilar,
Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos Solitons Fractals, 102 (2017), 285-294.
doi: 10.1016/j.chaos.2017.03.022. |
[10] |
D. Çakmak,
Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput., 216 (2010), 368-373.
doi: 10.1016/j.amc.2010.01.010. |
[11] |
M. Caputo and M. Fabrizio,
A new definition of fractional derivative without singular kerne, Prog. Frac. Diff. Appl., 1 (2015), 73-85.
|
[12] |
S. Clark and D. Hinton,
A Liapunov inequality for linear Hamiltonian systems, Math. Inequal. Appl., 1 (1998), 201-209.
doi: 10.7153/mia-01-18. |
[13] |
B. Cuahutenango-Barro, M. A. Taneco-Hernández and J. F. Gómez-Aguilar,
On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel, Chaos Solitons Fractals, 115 (2018), 283-299.
doi: 10.1016/j.chaos.2018.09.002. |
[14] |
K. Diethelm, The Analysis of Fractional Differential Equations, , Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-14574-2. |
[15] |
R. A. C. Ferreira,
A Lyapunov-type inequality for a fractional initial value problem, Fract. Calc. Appl. Anal., 16 (2013), 978-984.
doi: 10.2478/s13540-013-0060-5. |
[16] |
R. A. C. Ferreira,
Lyapunov-type inequalities for some sequential fractional boundary value problems, Adv. Dyn. Syst. Appl., 11 (2016), 33-43.
|
[17] |
R. A. C. Ferreira,
On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function, J. Math. Anal. Appl., 412 (2014), 1058-1063.
doi: 10.1016/j.jmaa.2013.11.025. |
[18] |
F. Jarad, T. 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. |
[19] |
F. Jarad, T. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivative, Adv. Difference Equ., 2012, (2012), 142, 8 pp.
doi: 10.1186/1687-1847-2012-142. |
[20] |
F. Jarad, T. Abdeljawad and Z. Hammouch,
On a class of ordinary differential equations in the frame of Atagana-Baleanu fractional derivative, Chaos Solitons Fractals, 117 (2018), 16-20.
doi: 10.1016/j.chaos.2018.10.006. |
[21] |
M. Jleli and B. Samet, Lyapunov-type inequalities for fractional boundary value problems equation with fractional initial conditions, Electron. J. Differential Equations, 2015 (2015), 11 pp. |
[22] |
J. F. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, Eur. Phys. J. Plus, 132 (2017), 13.
doi: 10.1140/epjp/i2017-11293-3. |
[23] |
J. F. Gómez-Aguilar, A. Atangana and V. F. Morales-Delgado, Electrical circuits RC, LC and RL described by Atangana-Baleanu fractional derivatives, Int. J. Circ. theor. Appl., 45 (2017), 1514–1533.
doi: 10.1002/cta.2348. |
[24] |
J. F. Gómez-Aguilar, H. Yépez-Martínez, R. F. Escobar-Jiménez, C. M. Astorga-Zaragoza and J. Reyes-Reyes,
Analytical and numerical solutions of electrical circuits described by fractional derivatives, Appl. Math. Model., 40 (2016), 9079-9094.
doi: 10.1016/j.apm.2016.05.041. |
[25] |
R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, Springer Heidelberg New York Dordrecht London, 2014.
doi: 10.1007/978-3-662-43930-2. |
[26] |
U. N. Katugampola,
New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.
doi: 10.1016/j.amc.2011.03.062. |
[27] |
U. N. Katugampola,
A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.
|
[28] |
A. A. Kilbas,
Hadamard type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204.
|
[29] |
A. A. Kilbas and M. Sa${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over i} }}$go,
Fractional integrals and derivatives of Mittag-Leffler type function (Russian), Dokl. Akad. Nauk Belarusi, 39 (1995), 22-26.
|
[30] |
A. A. Kilbas and M. Saigo,
On solutions of integral equations of Abel-Volterra type, Differential Integral Equations, 8 (1995), 993-1011.
|
[31] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, , Elsevier, Amsterdam, 2006. |
[32] |
A. M. Liapunov, Problème général de la stabilitie du mouvement, Ann. of Math. Stud., 17, Princeton Univ. Press, Princeton, N. J., 1949. |
[33] |
Q. Ma, C. Ma and J. Wang,
A Lyapunov-type inequality for a fractional differential equation with Hadamard derivative, J. Math. Inequal., 11 (2017), 135-141.
doi: 10.7153/jmi-11-13. |
[34] |
G. M. Mittag-Leffler,
Sur la nouvelle fonction $E_{\alpha }\left(z\right) $, C. R. Acad. Sci. Paris, 137 (1903), 554-558.
|
[35] |
N. Parhi and S. Panigrahi,
A Lyapunov-type integral inequality for higher order differential equations, Math. Slovaca, 52 (2002), 31-46.
|
[36] |
J. P. Pinasco, Lyapunov-Type Inequalities, Springer Briefs in Mathematics, Springer, New York, 2013.
doi: 10.1007/978-1-4614-8523-0. |
[37] |
I. Podlubny, Fractional Differential Equations,, Academic Press, an Diego, California, 1999.
![]() ![]() |
[38] |
T. R. Prabhakar,
A singular integral equation with a generalised Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.
|
[39] |
J. Rongand and C. Bai, Lyapunov-type inequality for afractional differential equation with fractional boundary conditions, Adv. Difference Equ., 2015, (2015), 82, 10 pp.
doi: 10.1186/s13662-015-0430-x. |
[40] |
X. Yang,
On Lyapunov-type inequality for certain higher-order differential equations, Appl. Math. Comput., 134 (2003), 307-317.
doi: 10.1016/S0096-3003(01)00285-5. |
[41] |
X. Yang and K. Lo,
Lyapunov-type inequality for a class of even-order differential equations, Appl. Math. Comput., 215 (2010), 3884-3890.
doi: 10.1016/j.amc.2009.11.032. |
[42] |
H. Ye, J. Gao and Y. Ding,
A generalized Lyapunov inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.
|
[43] |
H. Yépez-Martínez and J. F. Gómez-Aguilar,
A new modified definition of Caputo-Fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method (MHAM), J. Comput. Appl. Math., 346 (2019), 247-260.
doi: 10.1016/j.cam.2018.07.023. |
[44] |
H. Yépez-Martínez, J. F. Gómez-Aguilar, I. O. Sosa, J. M. Reyes and J. Torres-Jiménez,
The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Rev. Mexicana Fís., 62 (2016), 310-316.
|
[1] |
Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039 |
[2] |
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 |
[3] |
Miloud Moussai. Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives. Numerical Algebra, Control and Optimization, 2021 doi: 10.3934/naco.2021021 |
[4] |
Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3803-3819. doi: 10.3934/dcdss.2021019 |
[5] |
Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 867-880. doi: 10.3934/dcdss.2020050 |
[6] |
John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283 |
[7] |
Maria Fărcăşeanu, Mihai Mihăilescu, Denisa Stancu-Dumitru. Perturbed fractional eigenvalue problems. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6243-6255. doi: 10.3934/dcds.2017270 |
[8] |
Shakir Sh. Yusubov, Elimhan N. Mahmudov. Optimality conditions of singular controls for systems with Caputo fractional derivatives. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021182 |
[9] |
Mehmet Yavuz, Necati Özdemir. Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 995-1006. doi: 10.3934/dcdss.2020058 |
[10] |
Jean Daniel Djida, Juan J. Nieto, Iván Area. Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 609-627. doi: 10.3934/dcdss.2020033 |
[11] |
Raziye Mert, Thabet Abdeljawad, Allan Peterson. A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2417-2434. doi: 10.3934/dcdss.2020171 |
[12] |
Behzad Ghanbari, Devendra Kumar, Jagdev Singh. An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3577-3587. doi: 10.3934/dcdss.2020428 |
[13] |
Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial and Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723 |
[14] |
James Scott, Tadele Mengesha. A fractional Korn-type inequality. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3315-3343. doi: 10.3934/dcds.2019137 |
[15] |
Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations and Control Theory, 2022, 11 (1) : 239-258. doi: 10.3934/eect.2021001 |
[16] |
Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3703-3718. doi: 10.3934/dcdss.2021020 |
[17] |
Antonio Coronel-Escamilla, José Francisco Gómez-Aguilar. A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 561-574. doi: 10.3934/dcdss.2020031 |
[18] |
Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure and Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657 |
[19] |
Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2475-2487. doi: 10.3934/dcdss.2020139 |
[20] |
Ricardo Almeida, M. Luísa Morgado. Optimality conditions involving the Mittag–Leffler tempered fractional derivative. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 519-534. doi: 10.3934/dcdss.2021149 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]