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Numerical solutions of fuzzy differential equations. (English) Zbl 0939.65086

The authors study the explicit Euler approximation of a Cauchy problem for fuzzy (ordinary) differential equations. By the embedding approach, the original problem in a Banach space with supremum norm is replaced by two parametric, first-order ordinary differential equations. These differential equations can be solved by classical numerical algorithms. The authors present a pointwise convergence theorem for the standard Euler method in the given Banach space under uniform boundedness of second derivatives, Lipschitz-continuous right hand sides and for equidistant stepsize selection. 5 examples (both linear and nonlinear) are exhibited and discussed in view of the implementation of the explicit Euler approximation. A standard, two-dimensional error estimation is given in terms of the global Lipschitz constant, the length of integration interval \(T-t_0\), and the equidistant stepsize \(h\).

MSC:

65J15 Numerical solutions to equations with nonlinear operators
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
65J10 Numerical solutions to equations with linear operators
34G10 Linear differential equations in abstract spaces
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