Soft Computing

, Volume 22, Issue 12, pp 3817–3828 | Cite as

Solving a nonhomogeneous linear system of interval differential equations



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.


Interval differential equation Linear system of differential equations Set of functions 



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.


  1. 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
  2. Aubin J-P, Frankowska H (1990) Set-valued analysis. Birkhäuser, BostonMATHGoogle Scholar
  3. 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
  4. Banks HT, Jacobs MQ (1970) A differential calculus for multifunctions. J Math Anal Appl 29:246–272MathSciNetCrossRefMATHGoogle Scholar
  5. 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
  6. 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
  7. 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
  8. Blagodatskikh VI, Filippov AF (1986) Differential inclusions and optimal control. Proc Steklov Inst Math 169:199–259MATHGoogle Scholar
  9. 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
  10. Bridgland TF (1970) Trajectory integrals of set-valued functions. Pac J Math 33(1):43–68MathSciNetCrossRefMATHGoogle Scholar
  11. 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
  12. 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
  13. Dabbous T (2012) Identification for systems governed by nonlinear interval differential equations. J Ind Manag Optim 8(3):765–780MathSciNetCrossRefMATHGoogle Scholar
  14. 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
  15. 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
  16. Gasilov N, Amrahov ŞE, Fatullayev AG (2014a) Solution of linear differential equations with fuzzy boundary values. Fuzzy Sets Syst 257:169–183MathSciNetCrossRefMATHGoogle Scholar
  17. Gasilov NA, Fatullayev AG, Amrahov ŞE, Khastan A (2014b) A new approach to fuzzy initial value problem. Soft Comput 18:217–225CrossRefMATHGoogle Scholar
  18. 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
  19. Hoa NV (2015) The initial value problem for interval-valued second-order differential equations under generalized H-differentiability. Inf Sci 311:119–148MathSciNetCrossRefGoogle Scholar
  20. Hoa NV, Phu ND, Tung TT et al (2014) Interval-valued functional integro-differential equations. Adv Differ Equ, Article number: 177Google Scholar
  21. Hukuhara M (1967) Intégration des applications mesurables dont la valeur est un compact convexe. Funkcial Ekvac 10:205–223MathSciNetMATHGoogle Scholar
  22. Hüllermeier E (1997) An approach to modeling and simulation of uncertain dynamical systems. Int J Uncertain Fuzziness Knowl Based Syst 5:117–137CrossRefMATHGoogle Scholar
  23. Komleva TA, Plotnikov AV, Skripnik NV (2008) Differential equations with set-valued solutions. Ukr Math J 60(10):1540–1556CrossRefMATHGoogle Scholar
  24. 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
  25. Lakshmikantham V, Leela S, Vatsala AS (2003) Interconnection between set and fuzzy differential equations. Nonlinear Anal Theory Methods Appl 54(2):351–360MathSciNetCrossRefMATHGoogle Scholar
  26. Lakshmikantham V, Bhaskar TG, Devi JV (2006) Theory of set differential equations in metric spaces. Cambridge Scientific Publ, CambridgeMATHGoogle Scholar
  27. Lupulescu V (2013) Hukuhara differentiability of interval-valued functions and interval differential equations on time scales. Inf Sci 248:50–67MathSciNetCrossRefMATHGoogle Scholar
  28. Malinowski MT (2012) Interval Cauchy problem with a second type Hukuhara derivative. Inf Sci 213:94–105MathSciNetCrossRefMATHGoogle Scholar
  29. Malinowski MT (2012) Second type Hukuhara differentiable solutions to the delay set-valued differential equations. Appl Math Comput 218(18):9427–9437MathSciNetMATHGoogle Scholar
  30. Malinowski MT (2015) Fuzzy and set-valued stochastic differential equations with solutions of decreasing fuzziness. Adv Intell Syst Comput 315:105–112MATHGoogle Scholar
  31. Markov S (1979) Calculus for interval functions of a real variable. Computing 22:325–337MathSciNetCrossRefMATHGoogle Scholar
  32. Moore RE (1966) Interval analysis. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
  33. Pan L-X (2015) The numerical solution for the interval-valued differential equations. J Comput Anal Appl 19(4):632–641MathSciNetMATHGoogle Scholar
  34. Phu ND, An TV, Hoa NV et al (2014) Interval-valued functional differential equations under dissipative conditions. Adv Differ Equ, Article number: 198Google Scholar
  35. Plotnikov AV (2000) Differentiation of multivalued mappings. T-derivative. Ukr Math J 52(8):1282–1291MathSciNetCrossRefMATHGoogle Scholar
  36. 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
  37. Plotnikova NV (2005) Systems of linear differential equations with \(\pi \)-derivative and linear differential inclusions. Sbornik Math 196(11):1677–1691MathSciNetCrossRefMATHGoogle Scholar
  38. 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
  39. Sivasundaram S, Sun Y (1992) Application of interval analysis to impulsive differential equations. Appl Math Comput 47(2–3):201–210MathSciNetMATHGoogle Scholar
  40. Skripnik N (2012) Interval-valued differential equations with generalized derivative. Appl Math 2(4):116–120CrossRefGoogle Scholar
  41. Stefanini L (2010) A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst 161(11):1564–1584MathSciNetCrossRefMATHGoogle Scholar
  42. 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
  43. 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
  44. 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

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Baskent UniversityAnkaraTurkey
  2. 2.Computer Engineering DepartmentAnkara UniversityAnkaraTurkey

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