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
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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 295:57–71MathSciNetCrossRefzbMATHGoogle Scholar
Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151:581–599MathSciNetCrossRefzbMATHGoogle 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):753–759CrossRefzbMATHGoogle Scholar
Blagodatskikh VI, Filippov AF (1986) Differential inclusions and optimal control. Proc Steklov Inst Math 169:199–259zbMATHGoogle Scholar
Boese FG (1994) Stability in a special class of retarded difference-differential equations with interval-valued parameters. J Math Anal Appl 181(1):227–247MathSciNetCrossRefzbMATHGoogle Scholar
Chalco-Cano Y, Román-Flores H, Jiménez-Gamero MD (2011) Generalized derivative and \(\pi \)-derivative for set-valued functions. Fuzzy Sets Syst 181:2177–2188MathSciNetzbMATHGoogle 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 219:49–67MathSciNetCrossRefzbMATHGoogle 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):559–575MathSciNetCrossRefzbMATHGoogle 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):484–499MathSciNetzbMATHGoogle 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):1498–1513MathSciNetCrossRefzbMATHGoogle Scholar
Plotnikova NV (2005) Systems of linear differential equations with \(\pi \)-derivative and linear differential inclusions. Sbornik Math 196(11):1677–1691MathSciNetCrossRefzbMATHGoogle 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):3495–3512CrossRefzbMATHGoogle Scholar
Sivasundaram S, Sun Y (1992) Application of interval analysis to impulsive differential equations. Appl Math Comput 47(2–3):201–210MathSciNetzbMATHGoogle Scholar
Skripnik N (2012) Interval-valued differential equations with generalized derivative. Appl Math 2(4):116–120CrossRefGoogle 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):1311–1328MathSciNetCrossRefzbMATHGoogle 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):330–346MathSciNetCrossRefzbMATHGoogle Scholar