# American Institute of Mathematical Sciences

May  2021, 26(5): 2739-2747. doi: 10.3934/dcdsb.2020203

## Solving a system of linear differential equations with interval coefficients

 Department of Computer Engineering, Baskent University, 06790, Turkey

* Corresponding author: Nizami A. Gasilov, gasilov@baskent.edu.tr

Received  February 2020 Revised  April 2020 Published  May 2021 Early access  June 2020

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.

Citation: Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. 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. [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.

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##### 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. [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.
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)
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
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 $\overline{y}(t)$
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