# Solving a nonhomogeneous linear system of interval differential equations

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## Abstract

In most application problems, the exact values of the input parameters are unknown, but the intervals in which these values lie can be determined. In such problems, the dynamics of the system are described by an interval-valued differential equation. In this study, we present a new approach to nonhomogeneous systems of interval differential equations. We consider linear differential equations with real coefficients, but with interval initial values and forcing terms that are sets of real functions. For each forcing term, we assume these real functions to be linearly distributed between two given real functions. We seek solutions not as a vector of interval-valued functions, as usual, but as a set of real vector functions. We develop a method to find the solution and establish an existence and uniqueness theorem. We explain our approach and solution method through an illustrative example. Further, we demonstrate the advantages of the proposed approach over the differential inclusion approach and the generalized differentiability approach.

## Keywords

Interval differential equation Linear system of differential equations Set of functions## Notes

### Acknowledgements

We are grateful to the editor and the anonymous reviewers for their comments and suggestions, which helped to improve the quality of this paper.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

## References

- Amrahov ŞE, Khastan A, Gasilov N, Fatullayev AG (2016) Relationship between Bede-Gal differentiable set-valued functions and their associated support functions. Fuzzy Sets Syst 295:57–71MathSciNetCrossRefMATHGoogle Scholar
- Aubin J-P, Frankowska H (1990) Set-valued analysis. Birkhäuser, BostonMATHGoogle Scholar
- Azzam-Laouir D, Boukrouk W (2015) Second-order set-valued differential equations with boundary conditions. J Fixed Point Theory Appl 17(1):99–121MathSciNetCrossRefMATHGoogle Scholar
- Banks HT, Jacobs MQ (1970) A differential calculus for multifunctions. J Math Anal Appl 29:246–272MathSciNetCrossRefMATHGoogle Scholar
- Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151:581–599MathSciNetCrossRefMATHGoogle Scholar
- Bede B, Stefanini L (2009) Numerical solution of interval differential equations with generalized Hukuhara differentiability. In: Proceedings of the joint 2009 International fuzzy systems association world congress and 2009 European society of fuzzy logic and technology conference , pp 730–735Google Scholar
- Blackwell B, Beck JV (2010) A technique for uncertainty analysis for inverse heat conduction problems. Int J Heat Mass Transf 53(4):753–759CrossRefMATHGoogle Scholar
- Blagodatskikh VI, Filippov AF (1986) Differential inclusions and optimal control. Proc Steklov Inst Math 169:199–259MATHGoogle Scholar
- Boese FG (1994) Stability in a special class of retarded difference-differential equations with interval-valued parameters. J Math Anal Appl 181(1):227–247MathSciNetCrossRefMATHGoogle Scholar
- Bridgland TF (1970) Trajectory integrals of set-valued functions. Pac J Math 33(1):43–68MathSciNetCrossRefMATHGoogle Scholar
- Chalco-Cano Y, Román-Flores H, Jiménez-Gamero MD (2011) Generalized derivative and \(\pi \)-derivative for set-valued functions. Fuzzy Sets Syst 181:2177–2188MathSciNetMATHGoogle Scholar
- Chalco-Cano Y, Rufián-Lizana A, Román-Flores H, Jiménez-Gamero MD (2013) Calculus for interval-valued functions using generalized Hukuhara derivative and applications. Fuzzy Sets Syst 219:49–67MathSciNetCrossRefMATHGoogle Scholar
- Dabbous T (2012) Identification for systems governed by nonlinear interval differential equations. J Ind Manag Optim 8(3):765–780MathSciNetCrossRefMATHGoogle Scholar
- Galanis GN, Bhaskar TG, Lakshmikantham V et al (2005) Set valued functions in Frechet spaces: continuity Hukuhara differentiability and applications to set differential equations. Nonlinear Anal Theory Methods Appl 61(4):559–575MathSciNetCrossRefMATHGoogle Scholar
- Gasilov N, Amrahov ŞE, Fatullayev AG (2011) A geometric approach to solve fuzzy linear systems of differential equations. Appl Math Inf Sci 5(3):484–499MathSciNetMATHGoogle Scholar
- Gasilov N, Amrahov ŞE, Fatullayev AG (2014a) Solution of linear differential equations with fuzzy boundary values. Fuzzy Sets Syst 257:169–183MathSciNetCrossRefMATHGoogle Scholar
- Gasilov NA, Fatullayev AG, Amrahov ŞE, Khastan A (2014b) A new approach to fuzzy initial value problem. Soft Comput 18:217–225CrossRefMATHGoogle Scholar
- Gasilov NA, Amrahov ŞE, Fatullayev AG, Hashimoglu IF (2015) Solution method for a boundary value problem with fuzzy forcing function. Inf Sci 317:349–368MathSciNetCrossRefMATHGoogle Scholar
- Hoa NV (2015) The initial value problem for interval-valued second-order differential equations under generalized H-differentiability. Inf Sci 311:119–148MathSciNetCrossRefGoogle Scholar
- Hoa NV, Phu ND, Tung TT et al (2014) Interval-valued functional integro-differential equations. Adv Differ Equ, Article number: 177Google Scholar
- Hukuhara M (1967) Intégration des applications mesurables dont la valeur est un compact convexe. Funkcial Ekvac 10:205–223MathSciNetMATHGoogle Scholar
- Hüllermeier E (1997) An approach to modeling and simulation of uncertain dynamical systems. Int J Uncertain Fuzziness Knowl Based Syst 5:117–137CrossRefMATHGoogle Scholar
- Komleva TA, Plotnikov AV, Skripnik NV (2008) Differential equations with set-valued solutions. Ukr Math J 60(10):1540–1556CrossRefMATHGoogle Scholar
- Lakshmikantham V, Sun Y (1991) Applications of interval-analysis to minimal and maximal solutions of differential equations. Appl Math Comput 41(1):77–87MathSciNetMATHGoogle Scholar
- Lakshmikantham V, Leela S, Vatsala AS (2003) Interconnection between set and fuzzy differential equations. Nonlinear Anal Theory Methods Appl 54(2):351–360MathSciNetCrossRefMATHGoogle Scholar
- Lakshmikantham V, Bhaskar TG, Devi JV (2006) Theory of set differential equations in metric spaces. Cambridge Scientific Publ, CambridgeMATHGoogle Scholar
- Lupulescu V (2013) Hukuhara differentiability of interval-valued functions and interval differential equations on time scales. Inf Sci 248:50–67MathSciNetCrossRefMATHGoogle Scholar
- Malinowski MT (2012) Interval Cauchy problem with a second type Hukuhara derivative. Inf Sci 213:94–105MathSciNetCrossRefMATHGoogle Scholar
- Malinowski MT (2012) Second type Hukuhara differentiable solutions to the delay set-valued differential equations. Appl Math Comput 218(18):9427–9437MathSciNetMATHGoogle Scholar
- Malinowski MT (2015) Fuzzy and set-valued stochastic differential equations with solutions of decreasing fuzziness. Adv Intell Syst Comput 315:105–112MATHGoogle Scholar
- Markov S (1979) Calculus for interval functions of a real variable. Computing 22:325–337MathSciNetCrossRefMATHGoogle Scholar
- Moore RE (1966) Interval analysis. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
- Pan L-X (2015) The numerical solution for the interval-valued differential equations. J Comput Anal Appl 19(4):632–641MathSciNetMATHGoogle Scholar
- Phu ND, An TV, Hoa NV et al (2014) Interval-valued functional differential equations under dissipative conditions. Adv Differ Equ, Article number: 198Google Scholar
- Plotnikov AV (2000) Differentiation of multivalued mappings. T-derivative. Ukr Math J 52(8):1282–1291MathSciNetCrossRefMATHGoogle Scholar
- Plotnikov AV, Skripnik NV (2014) Conditions for the existence of local solutions of set-valued differential equations with generalized derivative. Ukr Math J 65(10):1498–1513MathSciNetCrossRefMATHGoogle Scholar
- Plotnikova NV (2005) Systems of linear differential equations with \(\pi \)-derivative and linear differential inclusions. Sbornik Math 196(11):1677–1691MathSciNetCrossRefMATHGoogle Scholar
- Quang LT, Hoa NV, Phu ND et al (2016) Existence of extremal solutions for interval-valued functional integro-differential equations. J Intell Fuzzy Syst 30(6):3495–3512CrossRefMATHGoogle Scholar
- Sivasundaram S, Sun Y (1992) Application of interval analysis to impulsive differential equations. Appl Math Comput 47(2–3):201–210MathSciNetMATHGoogle Scholar
- Skripnik N (2012) Interval-valued differential equations with generalized derivative. Appl Math 2(4):116–120CrossRefGoogle Scholar
- Stefanini L (2010) A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst 161(11):1564–1584MathSciNetCrossRefMATHGoogle Scholar
- Stefanini L, Bede B (2009) Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Anal Theory Methods Appl 71(3–4):1311–1328MathSciNetCrossRefMATHGoogle Scholar
- Tao J, Zhang Z (2016) Properties of interval-valued function space under the gH-difference and their application to semi-linear interval differential equations. Adv Differ Equ 45:1–28MathSciNetGoogle Scholar
- Wang C, Qiu Z, He Y (2015) Fuzzy interval perturbation method for uncertain heat conduction problem with interval and fuzzy parameters. Int J Numer Methods Eng 104(5):330–346MathSciNetCrossRefMATHGoogle Scholar