-
Previous Article
On some damped 2 body problems
- EECT Home
- This Issue
-
Next Article
Improved boundary regularity for a Stokes-Lamé system
Results on controllability of non-densely characterized neutral fractional delay differential system
1. | Department of Mathematics, Sri Eshwar College of Engineering(Autonomous), Coimbatore - 641 202, Tamil Nadu, India |
2. | Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chepauk, Chennai - 600 005, Tamil Nadu, India |
3. | Department of Mathematics, GMR Institute of Technology, Rajam - 532 127, Andhra Pradesh, India |
4. | Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawasir 11991, Saudi Arabia |
5. | Post Graduate and Research Department of Mathematics, Kongunadu Arts and Science College(Autonomous), Coimbatore - 641 029, Tamil Nadu, India |
This work establishes the controllability of nondense fractional neutral delay differential equation under Hille-Yosida condition in Banach space. The outcomes are derived with the aid of fractional calculus theory, semigroup operator theory and Schauder fixed point theorem. Theoretical results are verified through illustration.
References:
[1] |
L. Byszewski,
Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494-505.
doi: 10.1016/0022-247X(91)90164-U. |
[2] |
L. Byszewski and H. Akca,
On a mild solution of a semilinear functional-differential evolution nonlocal problem, J. Appl. Math. Stoch. Anal., 10 (1997), 265-271.
doi: 10.1155/S1048953397000336. |
[3] |
Y. K. Chang,
Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos Solitons Fractals, 33 (2007), 1601-1609.
doi: 10.1016/j.chaos.2006.03.006. |
[4] |
Y. K. Chang, A. Anguraj and M. Mallika Arjunan,
Existence results for non-densely defined neutral impulsive differential inclusions with nonlocal conditions, J. Appl. Math. Comput., 28 (2008), 79-91.
doi: 10.1007/s12190-008-0078-8. |
[5] |
X. Fu and X. Liu,
Controllability of non-densely defined neutral functional differential systems in abstract space, Chinese Ann. Math. B, 28 (2007), 243-252.
doi: 10.1007/s11401-005-0028-9. |
[6] |
B. Ghanbari, S. Kumar and R. Kumar, A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos Solitons Fractals, 133 (2020), 109619.
doi: 10.1016/j.chaos.2020.109619. |
[7] |
W. Gao, P. Veeresha, D. G. Prakasha, H. M. Baskonus and G. Yel, New numerical results for the time-fractional phi-four equation using a novel analytical approach, Symmetry, 12 (2020), 478.
doi: 10.3390/sym12030478. |
[8] |
E. P. Gatsori,
Controllability results for nondensely defined evolution differential inclusions with nonlocal conditions, J. Math. Anal. Appl., 297 (2004), 194-211.
doi: 10.1016/j.jmaa.2004.04.055. |
[9] |
H. Gu, Y. Zhou, B. Ahmad and A. Alsaedi,
Integral solutions of fractional evolution equations with nondense domain, Electronic J. Differ. Eq., 2017 (2017), 1-15.
|
[10] |
F. Jarad, T. Abdeljawad and D. Baleanu,
Fractional variational optimal control problems with delayed arguments, Nonlinear Dyn., 62 (2010), 609-614.
doi: 10.1007/s11071-010-9748-9. |
[11] |
F. Jarad, T. Abdeljawad and D. Baleanu,
Fractional variational principles with delay within caputo derivatives, Rep. Math. Phys., 65 (2010), 17-28.
doi: 10.1016/S0034-4877(10)00010-8. |
[12] |
V. Kavitha and M. M. Arjunan,
Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces, Nonlinear Anal. Hybri., 4 (2010), 441-450.
doi: 10.1016/j.nahs.2009.11.002. |
[13] |
H. Kellerman and M. Hieber,
Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.
doi: 10.1016/0022-1236(89)90116-X. |
[14] |
A. Kumar and D. N. Pandey, Controllability results for nondensely defined impulsive fractional differential equations in abstract space, Differ. Equ. Dyn. Syst., (2019).
doi: 10.1007/s12591-019-00471-1. |
[15] |
S. Kumar, R. Kumar, J. Singh, K. S. Nisar and D. Kumar, An efficient numerical scheme for fractional model of HIV-1 infection of $CD4^+$ T-cells with the effect of antiviral drug therapy, Alex. Eng. J., (2020).
doi: 10.1016/j.aej.2019.12.046. |
[16] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, in North-Holland Mathematics Studies, 204, Elsevier Science, Amsterdam, 2006. |
[17] |
K. D. Kucche and S. T. Sutar,
On Existence and stability results for nonlinear fractional delay differential equations, Bulletin of Parana's Mathematical Society, 36 (2018), 55-75.
doi: 10.5269/bspm.v36i4.33603. |
[18] |
N. I. Mahmudov, R. Murugesu, C. Ravichandran and V. Vijayakumar,
Approximate controllability results for fractional semilinear integro-differential inclusions in Hilbert spaces, Results Math., 71 (2017), 45-61.
doi: 10.1007/s00025-016-0621-0. |
[19] |
T. A. Maraaba, F. Jarad and D. Baleanu,
On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Sci. China Ser. A, 51 (2008), 1775-1786.
doi: 10.1007/s11425-008-0068-1. |
[20] |
T. A. Maraaba, D. Baleanu and F. Jarad, Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys., 49 (2008), 083507.
doi: 10.1063/1.2970709. |
[21] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York, Springer-verlag, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[22] |
I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, an Diego, 1999.
![]() |
[23] |
C. Ravichandran and J. J. Trujillo,
Controllability of impulsive fractional functional integro-differential equations in Banach spaces, J. Funct. Spaces, 2013 (2013), 1-8.
doi: 10.1155/2013/812501. |
[24] |
C. Ravichandran, K. Jothimani, H. M. Baskonus and N. Valliammal, New results on nondensely characterized integrodifferential equations with fractional order, Eur. Phys. J. Plus, 133 (2018), 1-10. Google Scholar |
[25] |
C. Ravichandran, N. Valliammal and J. J. Nieto,
New results on exact controllability of a class of neutral integrodifferential systems with state dependent delay in Banach spaces, J. Franklin Inst., 356 (2019), 1535-1565.
doi: 10.1016/j.jfranklin.2018.12.001. |
[26] |
S. J. Sadati, D. Baleanu, A. Ranjbar, R. Ghaderi and T. Abdeljawad,
Mittag-Leffler stability theorem for fractional nonlinear systems with delay, Abstr. Appl. Anal., 2010 (2010), 1-7.
doi: 10.1155/2010/108651. |
[27] |
R. Sakthivel, R. Ganesh and S. M. Anthoni,
Approximate controllability of fractional nonlinear differential inclusions, Appl. Math. Comput., 225 (2013), 708-717.
doi: 10.1016/j.amc.2013.09.068. |
[28] |
J. V. D. C. Sousa, E. C. de Oliveira and K. D. Kucche,
On the fractional functional differential equation with abstract volterra operator, B. Braz. Math.l Soc., 50 (2019), 803-822.
doi: 10.1007/s00574-019-00139-y. |
[29] |
P. Veeresha, D. G. Prakasha, H. M. Baskonus and G. Yel,
An efficient analytical approach for fractional Lakshmanan-Porsezian-Daniel model, Math. Methods Appl. Sci., 43 (2020), 4136-4155.
doi: 10.1002/mma.6179. |
[30] |
V. Vijayakumar, C. Ravichandran, R. Murugesu and J. J. Trujillo,
Controllability results for a class of fractional semilinear integro-differential inclusions via resolvent operators, Appl. Math. Comput., 247 (2014), 152-161.
doi: 10.1016/j.amc.2014.08.080. |
[31] |
V. Vijayakumar,
Approximate controllability results for nondensely defined fractional neutral differential inclusions with Hille Yosida operators, Internat. J. Control, 92 (2019), 2210-2222.
doi: 10.1080/00207179.2018.1433331. |
[32] |
J. Wang and Y. Zhou,
Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal. Real World Appl., 12 (2011), 3642-3653.
doi: 10.1016/j.nonrwa.2011.06.021. |
[33] |
B. Yan,
Boundary value problems on the half-line with impulses and infinite delay, J. Math. Anal. Appl., 259 (2001), 94-114.
doi: 10.1006/jmaa.2000.7392. |
[34] |
K. Yosida, Functional Analysis, 6$^th$ edition, Springer, Berlin, 1980. |
[35] |
Y. Zhou and F. Jiao,
Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.
doi: 10.1016/j.camwa.2009.06.026. |
[36] |
Y. Zhou, V. Vijayakumar, C. Ravichandran and R. Murugesu,
Controllability results for fractional order neutral functional differential inclusions with infinite delay, Fixed Point Theory, 18 (2017), 773-798.
doi: 10.24193/fpt-ro.2017.2.62. |
[37] |
Y. Zhou, V. Vijayakumar and R. Murugesu,
Controllability for fractional evolution inclusions without compactness, Evol. Equ. Control The., 4 (2015), 507-524.
doi: 10.3934/eect.2015.4.507. |
[38] |
Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.
doi: 10.1142/9069. |
[39] |
Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier, New York, 2015. |
show all references
References:
[1] |
L. Byszewski,
Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494-505.
doi: 10.1016/0022-247X(91)90164-U. |
[2] |
L. Byszewski and H. Akca,
On a mild solution of a semilinear functional-differential evolution nonlocal problem, J. Appl. Math. Stoch. Anal., 10 (1997), 265-271.
doi: 10.1155/S1048953397000336. |
[3] |
Y. K. Chang,
Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos Solitons Fractals, 33 (2007), 1601-1609.
doi: 10.1016/j.chaos.2006.03.006. |
[4] |
Y. K. Chang, A. Anguraj and M. Mallika Arjunan,
Existence results for non-densely defined neutral impulsive differential inclusions with nonlocal conditions, J. Appl. Math. Comput., 28 (2008), 79-91.
doi: 10.1007/s12190-008-0078-8. |
[5] |
X. Fu and X. Liu,
Controllability of non-densely defined neutral functional differential systems in abstract space, Chinese Ann. Math. B, 28 (2007), 243-252.
doi: 10.1007/s11401-005-0028-9. |
[6] |
B. Ghanbari, S. Kumar and R. Kumar, A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos Solitons Fractals, 133 (2020), 109619.
doi: 10.1016/j.chaos.2020.109619. |
[7] |
W. Gao, P. Veeresha, D. G. Prakasha, H. M. Baskonus and G. Yel, New numerical results for the time-fractional phi-four equation using a novel analytical approach, Symmetry, 12 (2020), 478.
doi: 10.3390/sym12030478. |
[8] |
E. P. Gatsori,
Controllability results for nondensely defined evolution differential inclusions with nonlocal conditions, J. Math. Anal. Appl., 297 (2004), 194-211.
doi: 10.1016/j.jmaa.2004.04.055. |
[9] |
H. Gu, Y. Zhou, B. Ahmad and A. Alsaedi,
Integral solutions of fractional evolution equations with nondense domain, Electronic J. Differ. Eq., 2017 (2017), 1-15.
|
[10] |
F. Jarad, T. Abdeljawad and D. Baleanu,
Fractional variational optimal control problems with delayed arguments, Nonlinear Dyn., 62 (2010), 609-614.
doi: 10.1007/s11071-010-9748-9. |
[11] |
F. Jarad, T. Abdeljawad and D. Baleanu,
Fractional variational principles with delay within caputo derivatives, Rep. Math. Phys., 65 (2010), 17-28.
doi: 10.1016/S0034-4877(10)00010-8. |
[12] |
V. Kavitha and M. M. Arjunan,
Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces, Nonlinear Anal. Hybri., 4 (2010), 441-450.
doi: 10.1016/j.nahs.2009.11.002. |
[13] |
H. Kellerman and M. Hieber,
Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.
doi: 10.1016/0022-1236(89)90116-X. |
[14] |
A. Kumar and D. N. Pandey, Controllability results for nondensely defined impulsive fractional differential equations in abstract space, Differ. Equ. Dyn. Syst., (2019).
doi: 10.1007/s12591-019-00471-1. |
[15] |
S. Kumar, R. Kumar, J. Singh, K. S. Nisar and D. Kumar, An efficient numerical scheme for fractional model of HIV-1 infection of $CD4^+$ T-cells with the effect of antiviral drug therapy, Alex. Eng. J., (2020).
doi: 10.1016/j.aej.2019.12.046. |
[16] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, in North-Holland Mathematics Studies, 204, Elsevier Science, Amsterdam, 2006. |
[17] |
K. D. Kucche and S. T. Sutar,
On Existence and stability results for nonlinear fractional delay differential equations, Bulletin of Parana's Mathematical Society, 36 (2018), 55-75.
doi: 10.5269/bspm.v36i4.33603. |
[18] |
N. I. Mahmudov, R. Murugesu, C. Ravichandran and V. Vijayakumar,
Approximate controllability results for fractional semilinear integro-differential inclusions in Hilbert spaces, Results Math., 71 (2017), 45-61.
doi: 10.1007/s00025-016-0621-0. |
[19] |
T. A. Maraaba, F. Jarad and D. Baleanu,
On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Sci. China Ser. A, 51 (2008), 1775-1786.
doi: 10.1007/s11425-008-0068-1. |
[20] |
T. A. Maraaba, D. Baleanu and F. Jarad, Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys., 49 (2008), 083507.
doi: 10.1063/1.2970709. |
[21] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York, Springer-verlag, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[22] |
I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, an Diego, 1999.
![]() |
[23] |
C. Ravichandran and J. J. Trujillo,
Controllability of impulsive fractional functional integro-differential equations in Banach spaces, J. Funct. Spaces, 2013 (2013), 1-8.
doi: 10.1155/2013/812501. |
[24] |
C. Ravichandran, K. Jothimani, H. M. Baskonus and N. Valliammal, New results on nondensely characterized integrodifferential equations with fractional order, Eur. Phys. J. Plus, 133 (2018), 1-10. Google Scholar |
[25] |
C. Ravichandran, N. Valliammal and J. J. Nieto,
New results on exact controllability of a class of neutral integrodifferential systems with state dependent delay in Banach spaces, J. Franklin Inst., 356 (2019), 1535-1565.
doi: 10.1016/j.jfranklin.2018.12.001. |
[26] |
S. J. Sadati, D. Baleanu, A. Ranjbar, R. Ghaderi and T. Abdeljawad,
Mittag-Leffler stability theorem for fractional nonlinear systems with delay, Abstr. Appl. Anal., 2010 (2010), 1-7.
doi: 10.1155/2010/108651. |
[27] |
R. Sakthivel, R. Ganesh and S. M. Anthoni,
Approximate controllability of fractional nonlinear differential inclusions, Appl. Math. Comput., 225 (2013), 708-717.
doi: 10.1016/j.amc.2013.09.068. |
[28] |
J. V. D. C. Sousa, E. C. de Oliveira and K. D. Kucche,
On the fractional functional differential equation with abstract volterra operator, B. Braz. Math.l Soc., 50 (2019), 803-822.
doi: 10.1007/s00574-019-00139-y. |
[29] |
P. Veeresha, D. G. Prakasha, H. M. Baskonus and G. Yel,
An efficient analytical approach for fractional Lakshmanan-Porsezian-Daniel model, Math. Methods Appl. Sci., 43 (2020), 4136-4155.
doi: 10.1002/mma.6179. |
[30] |
V. Vijayakumar, C. Ravichandran, R. Murugesu and J. J. Trujillo,
Controllability results for a class of fractional semilinear integro-differential inclusions via resolvent operators, Appl. Math. Comput., 247 (2014), 152-161.
doi: 10.1016/j.amc.2014.08.080. |
[31] |
V. Vijayakumar,
Approximate controllability results for nondensely defined fractional neutral differential inclusions with Hille Yosida operators, Internat. J. Control, 92 (2019), 2210-2222.
doi: 10.1080/00207179.2018.1433331. |
[32] |
J. Wang and Y. Zhou,
Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal. Real World Appl., 12 (2011), 3642-3653.
doi: 10.1016/j.nonrwa.2011.06.021. |
[33] |
B. Yan,
Boundary value problems on the half-line with impulses and infinite delay, J. Math. Anal. Appl., 259 (2001), 94-114.
doi: 10.1006/jmaa.2000.7392. |
[34] |
K. Yosida, Functional Analysis, 6$^th$ edition, Springer, Berlin, 1980. |
[35] |
Y. Zhou and F. Jiao,
Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.
doi: 10.1016/j.camwa.2009.06.026. |
[36] |
Y. Zhou, V. Vijayakumar, C. Ravichandran and R. Murugesu,
Controllability results for fractional order neutral functional differential inclusions with infinite delay, Fixed Point Theory, 18 (2017), 773-798.
doi: 10.24193/fpt-ro.2017.2.62. |
[37] |
Y. Zhou, V. Vijayakumar and R. Murugesu,
Controllability for fractional evolution inclusions without compactness, Evol. Equ. Control The., 4 (2015), 507-524.
doi: 10.3934/eect.2015.4.507. |
[38] |
Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.
doi: 10.1142/9069. |
[39] |
Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier, New York, 2015. |
[1] |
Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021016 |
[2] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[3] |
Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021023 |
[4] |
Seddigheh Banihashemi, Hossein Jafaria, Afshin Babaei. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021025 |
[5] |
Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002 |
[6] |
Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021020 |
[7] |
Burcu Gürbüz. A computational approximation for the solution of retarded functional differential equations and their applications to science and engineering. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021069 |
[8] |
Mohamed Ouzahra. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021081 |
[9] |
K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 461-485. doi: 10.3934/naco.2020038 |
[10] |
Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021020 |
[11] |
V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066 |
[12] |
Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021058 |
[13] |
Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, , () : -. doi: 10.3934/era.2021024 |
[14] |
Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002 |
[15] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[16] |
Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021019 |
[17] |
Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021021 |
[18] |
Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami. On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021033 |
[19] |
Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201 |
[20] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
2019 Impact Factor: 0.953
Tools
Article outline
[Back to Top]