Akram Muhammad, Ihsan Tayyaba, Allahviranloo Tofigh, Al-Shamiri Mohammed M Ali
Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan.
Faculty of Engineering and Natural Sciences, Istinye University, Istanbul, Turkey.
Math Biosci Eng. 2022 Aug 17;19(12):11868-11902. doi: 10.3934/mbe.2022554.
This study presents a new analytical method to extract the fuzzy solution of the fuzzy initial value problem (FIVP) of fourth-order fuzzy ordinary differential equations (FODEs) using the Laplace operator under the strongly generalized Hukuhara differentiability (SGH-differentiability). To this end, firstly the fourth-order derivative of the fuzzy valued function (FVF) according to the type of the SGH-differentiability is introduced, and then the relationships between the fourth-order derivative of the FVF and its Laplace transform are established. Furthermore, considering the types of differentiabilities and switching points, some fundamental theorems related to the Laplace transform of the fourth-order derivative of the FVF are stated and proved in detail and a method to solve FIVP by the fuzzy Laplace transform is presented in detail. An application of our proposed method in Resistance-Inductance circuit (RL circuit) is presented. Finally, FIVP's solution is graphically analyzed to visualize and support theoretical results.
本研究提出了一种新的分析方法，用于在强广义胡克哈拉可微性（SGH - 可微性）下，使用拉普拉斯算子提取四阶模糊常微分方程（FODEs）的模糊初值问题（FIVP）的模糊解。为此，首先根据SGH - 可微性的类型引入模糊值函数（FVF）的四阶导数，然后建立FVF的四阶导数与其拉普拉斯变换之间的关系。此外，考虑到可微性类型和切换点，详细阐述并证明了一些与FVF四阶导数的拉普拉斯变换相关的基本定理，并详细介绍了一种通过模糊拉普拉斯变换求解FIVP的方法。给出了我们提出的方法在电阻 - 电感电路（RL电路）中的应用。最后，对FIVP的解进行了图形分析，以直观呈现并支持理论结果。
