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Improved predictor-corrector method for solving fuzzy initial value problems. (English) Zbl 1165.65042

This paper is concerned with the numerical solution of initial value problems for first order fuzzy differential equations. First of all, after introducing some notations and definitions of fuzzy number, fuzzy set-valued mapping and (Hukuhara- and Seikkala-) differentiability, the Authors define the spline interpolation of a set of data ${\left({t}_{i},{u}_{i}\right)}_{i=1}^{n}$ where ${t}_{i}$ are real distinct numbers and ${u}_{i}$ fuzzy numbers.

Now, taking into account the integral equality $y\left(t\right)=y\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}{y}^{\text{'}}\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds$ that holds for fuzzy set valued differentiable functions $y\left(t\right)$ in both of the above senses, an explicit three step method based on interpolation of ${y}^{\text{'}}$-values is derived and also similarly an implicit two step method. Further, a predictor corrector three step method based in the above methods is proposed. Standard stability and convergence proofs are provided. The paper ends with the numerical results of some elementary linear problems.

##### MSC:
 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L05 Initial value problems for ODE (numerical methods) 34A30 Linear ODE and systems, general 26E50 Fuzzy real analysis 65L20 Stability and convergence of numerical methods for ODE