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On the Numerical Solution of Ordinary, Interval and Fuzzy Differential Equations by Use of F-Transform

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Department of Economics and Management, University of Pisa, Via C. Ridolfi, 10, 56124 Pisa (PI), Italy
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Department of Finance, Faculty of Economics, VŠB—Technical University of Ostrava, Sokolská tr. 33, 70121 Ostrava, Czech Republic
3
DESP, Department of Economics, Society, Politics, University of Urbino Carlo Bo, Via A. Saffi 42, 61029 Urbino, Italy
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Author to whom correspondence should be addressed.
Axioms 2020, 9(1), 15; https://doi.org/10.3390/axioms9010015
Received: 24 November 2019 / Revised: 27 January 2020 / Accepted: 28 January 2020 / Published: 5 February 2020
(This article belongs to the Special Issue Fuzzy Transforms and Their Applications)
An interesting property of the inverse F-transform f ^ of a continuous function f on a given interval [ a , b ] says that the integrals of f ^ and f on [ a , b ] coincide. Furthermore, the same property can be established for the restrictions of the functions to all subintervals [ a , p k ] of the fuzzy partition of [ a , b ] used to define the F-transform. Based on this fact, we propose a new method for the numerical solution of ordinary differential equations (initial-value ordinary differential equation (ODE)) obtained by approximating the derivative x · ( t ) via F-transform, then computing (an approximation of) the solution x ( t ) by exact integration. For an ODE, a global second-order approximation is obtained. A similar construction is then applied to interval-valued and (level-wise) fuzzy differential equations in the setting of generalized differentiability (gH-derivative). Properties of the new method are analyzed and a computational section illustrates the performance of the obtained procedures, in comparison with well-known efficient algorithms.
Keywords: F-transform; initial-value ODE; numerical ODE solver; interval differential equations; gH-Derivative; fuzzy differential equations F-transform; initial-value ODE; numerical ODE solver; interval differential equations; gH-Derivative; fuzzy differential equations
MDPI and ACS Style

Radi, D.; Sorini, L.; Stefanini, L. On the Numerical Solution of Ordinary, Interval and Fuzzy Differential Equations by Use of F-Transform. Axioms 2020, 9, 15.

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