| Summary: | An interesting property of the inverse F-transform <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> </semantics> </math> </inline-formula> of a continuous function <i>f</i> on a given interval <inline-formula> <math display="inline"> <semantics> <mrow> <mo>[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> says that the integrals of <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> </semantics> </math> </inline-formula> and <i>f</i> on <inline-formula> <math display="inline"> <semantics> <mrow> <mo>[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> coincide. Furthermore, the same property can be established for the restrictions of the functions to all subintervals <inline-formula> <math display="inline"> <semantics> <mrow> <mo>[</mo> <mi>a</mi> <mo>,</mo> <msub> <mi>p</mi> <mi>k</mi> </msub> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> of the fuzzy partition of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mover> <mi>x</mi> <mo>·</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> via F-transform, then computing (an approximation of) the solution <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> 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.
|
|---|
