
-
Previous Article
Collective behaviors of a Winfree ensemble on an infinite cylinder
- DCDS-B Home
- This Issue
-
Next Article
Properties of basins of attraction for planar discrete cooperative maps
Solving a system of linear differential equations with interval coefficients
Department of Computer Engineering, Baskent University, 06790, Turkey |
In this study, we consider a system of homogeneous linear differential equations, the coefficients and initial values of which are constant intervals. We apply the approach that treats an interval problem as a set of real (classical) problems. In previous studies, a system of linear differential equations with real coefficients, but with interval forcing terms and interval initial values was investigated. It was shown that the value of the solution at each time instant forms a convex polygon in the coordinate plane. The motivating question of the present study is to investigate whether the same statement remains true, when the coefficients are intervals. Numerical experiments show that the answer is negative. Namely, at a fixed time, the region formed by the solution's value is not necessarily a polygon. Moreover, this region can be non-convex.
The solution, defined in this study, is compared with the Hukuhara- differentiable solution, and its advantages are exhibited. First, under the proposed concept, the solution always exists and is unique. Second, this solution concept does not require a set-valued, or interval-valued derivative. Third, the concept is successful because it seeks a solution from a wider class of set-valued functions.
References:
[1] |
Ş. E. Amrahov, A. Khastan, N. Gasilov and A. G. Fatullayev,
Relationship between Bede-Gal differentiable set-valued functions and their associated support functions, Fuzzy Sets and Systems, 295 (2016), 57-71.
doi: 10.1016/j.fss.2015.12.002. |
[2] |
J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990. |
[3] |
Y. Chalco-Cano, A. Rufián-Lizana, H. Román-Flores and M. D. Jiménez-Gamero,
Calculus for interval-valued functions using generalized Hukuhara derivative and applications, Fuzzy Sets and Systems, 219 (2013), 49-67.
doi: 10.1016/j.fss.2012.12.004. |
[4] |
T. F. Filippova, Differential equations for ellipsoidal estimates of reachable sets for a class of control systems with nonlinearity and uncertainty, IFAC PapersOnLine, 51 (2018), 770-775, http:dx.doi.org/10.1016/j.ifacol.2018.11.452. Google Scholar |
[5] |
N. A. Gasilov and Ş. E. Amrahov,
On differential equations with interval coefficients, Mathematical Methods in the Applied Sciences, 43 (2020), 1825-1837.
doi: 10.1002/mma.6006. |
[6] |
N. A. Gasilov and Ş. E. Amrahov,
Solving a nonhomogeneous linear system of interval differential equations, Soft Computing, 22 (2018), 3817-3828.
doi: 10.1007/s00500-017-2818-x. |
[7] |
N. A. Gasilov and M. Kaya, A method for the numerical solution of a boundary value problem for a linear differential equation with interval parameters, International Journal of Computational Methods, 16 (2019), 1850115, 17 pp.
doi: 10.1142/S0219876218501153. |
[8] |
E. Hüllermeier,
An approach to modeling and simulation of uncertain dynamical systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5 (1997), 117-137.
doi: 10.1142/S0218488597000117. |
[9] |
R. B. Kearfott and V. Kreinovich,
Applications of interval computations: An introduction, Applications of Interval Computations, Appl. Optim., Kluwer Acad. Publ., Dordrecht, 3 (1996), 1-22.
doi: 10.1007/978-1-4613-3440-8_1. |
[10] |
V. Lakshmikantham, T. G. Bhaskar and J. V. Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge, 2006. |
[11] |
M. T. Malinowski,
Interval differential equations with a second type Hukuhara derivative, Applied Mathematics Letters, 24 (2011), 2118-2123.
doi: 10.1016/j.aml.2011.06.011. |
[12] |
R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009.
doi: 10.1137/1.9780898717716. |
[13] |
A. V. Plotnikov and T. A. Komleva,
On some properties of bundles of trajectories of a controlled bilinear inclusion, Ukrainian Mathematical Journal, 56 (2004), 586-600.
doi: 10.1007/s11253-005-0114-x. |
[14] |
A. V. Plotnikov and N. V. Skripnik,
Conditions for the existence of local solutions of set-valued differential equations with generalized derivative, Ukrainian Mathematical Journal, 65 (2014), 1498-1513.
doi: 10.1007/s11253-014-0875-1. |
[15] |
L. Stefanini and B. Bede,
Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal., 71 (2009), 1311-1328.
doi: 10.1016/j.na.2008.12.005. |
show all references
References:
[1] |
Ş. E. Amrahov, A. Khastan, N. Gasilov and A. G. Fatullayev,
Relationship between Bede-Gal differentiable set-valued functions and their associated support functions, Fuzzy Sets and Systems, 295 (2016), 57-71.
doi: 10.1016/j.fss.2015.12.002. |
[2] |
J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990. |
[3] |
Y. Chalco-Cano, A. Rufián-Lizana, H. Román-Flores and M. D. Jiménez-Gamero,
Calculus for interval-valued functions using generalized Hukuhara derivative and applications, Fuzzy Sets and Systems, 219 (2013), 49-67.
doi: 10.1016/j.fss.2012.12.004. |
[4] |
T. F. Filippova, Differential equations for ellipsoidal estimates of reachable sets for a class of control systems with nonlinearity and uncertainty, IFAC PapersOnLine, 51 (2018), 770-775, http:dx.doi.org/10.1016/j.ifacol.2018.11.452. Google Scholar |
[5] |
N. A. Gasilov and Ş. E. Amrahov,
On differential equations with interval coefficients, Mathematical Methods in the Applied Sciences, 43 (2020), 1825-1837.
doi: 10.1002/mma.6006. |
[6] |
N. A. Gasilov and Ş. E. Amrahov,
Solving a nonhomogeneous linear system of interval differential equations, Soft Computing, 22 (2018), 3817-3828.
doi: 10.1007/s00500-017-2818-x. |
[7] |
N. A. Gasilov and M. Kaya, A method for the numerical solution of a boundary value problem for a linear differential equation with interval parameters, International Journal of Computational Methods, 16 (2019), 1850115, 17 pp.
doi: 10.1142/S0219876218501153. |
[8] |
E. Hüllermeier,
An approach to modeling and simulation of uncertain dynamical systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5 (1997), 117-137.
doi: 10.1142/S0218488597000117. |
[9] |
R. B. Kearfott and V. Kreinovich,
Applications of interval computations: An introduction, Applications of Interval Computations, Appl. Optim., Kluwer Acad. Publ., Dordrecht, 3 (1996), 1-22.
doi: 10.1007/978-1-4613-3440-8_1. |
[10] |
V. Lakshmikantham, T. G. Bhaskar and J. V. Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge, 2006. |
[11] |
M. T. Malinowski,
Interval differential equations with a second type Hukuhara derivative, Applied Mathematics Letters, 24 (2011), 2118-2123.
doi: 10.1016/j.aml.2011.06.011. |
[12] |
R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009.
doi: 10.1137/1.9780898717716. |
[13] |
A. V. Plotnikov and T. A. Komleva,
On some properties of bundles of trajectories of a controlled bilinear inclusion, Ukrainian Mathematical Journal, 56 (2004), 586-600.
doi: 10.1007/s11253-005-0114-x. |
[14] |
A. V. Plotnikov and N. V. Skripnik,
Conditions for the existence of local solutions of set-valued differential equations with generalized derivative, Ukrainian Mathematical Journal, 65 (2014), 1498-1513.
doi: 10.1007/s11253-014-0875-1. |
[15] |
L. Stefanini and B. Bede,
Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal., 71 (2009), 1311-1328.
doi: 10.1016/j.na.2008.12.005. |



[1] |
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 |
[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] |
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 |
[4] |
Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021008 |
[5] |
Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 |
[6] |
Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 |
[7] |
Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181 |
[8] |
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 |
[9] |
Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 |
[10] |
Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 |
[11] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[12] |
Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200 |
[13] |
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 |
[14] |
Christophe Zhang. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021006 |
[15] |
Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068 |
[16] |
Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213 |
[17] |
Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995 |
[18] |
Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109 |
[19] |
Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25 |
[20] |
Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]