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

by Davide Radi 1,2, Laerte Sorini 3 and Luciano Stefanini 3,*
1
Department of Economics and Management, University of Pisa, Via C. Ridolfi, 10, 56124 Pisa (PI), Italy
2
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
*
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.
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.