Solving a system of linear differential equations with interval coefficients

Nizami A. Gasilov

doi:10.3934/dcdsb.2020203

Article Contents

2021, Volume 26, Issue 5: 2739-2747. Doi: 10.3934/dcdsb.2020203

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Solving a system of linear differential equations with interval coefficients

Nizami A. Gasilov^,

Department of Computer Engineering, Baskent University, 06790, Turkey

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

Received: February 2020

Revised: April 2020

Early access: June 2020

Published: May 2021

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Abstract

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.

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Mathematics Subject Classification: Primary:93B03, 34A30;Secondary:65G40.

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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)

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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

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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) $

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