Figure 1.
The values of numerical solution of Example 1 at different time instants: $ t = 0 $ (upper left quarter), $ t = 0.2 $ (upper right quarter), $ t = 0.4 $ (lower left quarter), and $ t = 0.6 $ (lower right quarter)
-
Previous Article
Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency
- DCDS-B Home
- This Issue
-
Next Article
Traveling waves in quadratic autocatalytic systems with complexing agent
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.
Keywords:
Interval differential equation,
system of differential equations,
linear differential equation,
numerical solution.
Mathematics Subject Classification:
Primary:93B03, 34A30;Secondary:65G40.
Citation:
Nizami A. Gasilov.
Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020203
![]()
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. |
Figure 2.
Solutions of Example 2, obtained by two methods, at $ t = 0 $ (upper left quarter), $ t = 0.2 $ (upper right quarter), $ t = 0.4 $ (lower left quarter), and $ t = 0.6 $ (lower right quarter). The continuous lines represent the numerical solution, obtained by the proposed method, while the dashed lines represent the Hukuhara-differentiable solution
Figure 3.
The Hukuhara-differentiable solution $ X(t) = \left[ \underline{x}(t),\ \overline{x}(t)\right] $ and $ Y(t) = \left[ \underline{y}(t),\ \overline{y}(t)\right] $ for Example 2. At the left half, the lower and upper lines represent $ \underline{x}(t) $ and $ \overline{x}(t) $ , respectively. The lines at the right half represent $ \underline{y}(t) $ and $ \overline{y}(t) $
| [1] |
Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 |
| [2] |
Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 |
| [3] |
Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020317 |
| [4] |
Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326 |
| [5] |
Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020440 |
| [6] |
Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 |
| [7] |
Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050 |
| [8] |
Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020468 |
| [9] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
| [10] |
Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065 |
| [11] |
Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020047 |
| [12] |
Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321 |
| [13] |
John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044 |
| [14] |
Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042 |
| [15] |
Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 |
| [16] |
Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180 |
| [17] |
Liang Huang, Jiao Chen. The boundedness of multi-linear and multi-parameter pseudo-differential operators. Communications on Pure & Applied Analysis, 2021, 20 (2) : 801-815. doi: 10.3934/cpaa.2020291 |
| [18] |
Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 |
| [19] |
Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 |
| [20] |
Aisling McGlinchey, Oliver Mason. Observations on the bias of nonnegative mechanisms for differential privacy. Foundations of Data Science, 2020, 2 (4) : 429-442. doi: 10.3934/fods.2020020 |
2019 Impact Factor: 1.27
Tools
Article outline
Figures and Tables
[Back to Top]