On the Numerical Solution of Ordinary, Interval and Fuzzy Differential Equations by Use of F-Transform
On the Numerical Solution of Ordinary, Interval and Fuzzy Differential Equations by Use of F-Transform
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利用F变换数值求解常微分方程、区间方程和模糊微分方程
DOI：
10.3390/axioms9010015
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发表时间：
2020
期刊：
影响因子：
2
通讯作者：
Luciano Stefanini
中科院分区：
文献类型：
--
作者：
D. Radi;Laerte Sorini;Luciano Stefanini
连续函数 f 在给定区间 [ a , b ] 上的 F 逆变换 f ^ 的一个有趣属性表明 f ^ 和 f 在 [ a , b ] 上的积分一致。此外，对于用于定义F变换的[a，b]的模糊划分的所有子区间[a，p k ]的函数的限制，可以建立相同的性质。基于这一事实，我们提出了一种常微分方程（初值常微分方程（ODE））数值解的新方法，通过 F 变换逼近导数 x · ( t )，然后通过精确积分计算解 x ( t )（的近似）。对于 ODE，获得全局二阶近似。然后，在广义可微性（gH 导数）的设置中，将类似的构造应用于区间值和（水平）模糊微分方程。分析了新方法的特性，并通过计算部分说明了所获得程序的性能，并与众所周知的高效算法进行了比较。
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.