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

An interesting property of the inverse F-transform fˆ of a continuous function f on a given interval [a,b] says that the integrals of fˆ and f on [a,b] coincide. Furthermore, the same property can be established for the restrictions of the functions to all subintervals [a,pk] 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.