Salahshour S, Ahmadian A, Salimi M, Ferrara M, Baleanu D
Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran.
Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia.
Chaos. 2019 Aug;29(8):083110. doi: 10.1063/1.5096022.
Realizing the behavior of the solution in the asymptotic situations is essential for repetitive applications in the control theory and modeling of the real-world systems. This study discusses a robust and definitive attitude to find the interval approximate asymptotic solutions of fractional differential equations (FDEs) with the Atangana-Baleanu (A-B) derivative. In fact, such critical tasks require to observe precisely the behavior of the noninterval case at first. In this regard, we initially shed light on the noninterval cases and analyze the behavior of the approximate asymptotic solutions, and then, we introduce the A-B derivative for FDEs under interval arithmetic and develop a new and reliable approximation approach for fractional interval differential equations with the interval A-B derivative to get the interval approximate asymptotic solutions. We exploit Laplace transforms to get the asymptotic approximate solution based on the interval asymptotic A-B fractional derivatives under interval arithmetic. The techniques developed here provide essential tools for finding interval approximation asymptotic solutions under interval fractional derivatives with nonsingular Mittag-Leffler kernels. Two cases arising in the real-world systems are modeled under interval notion and given to interpret the behavior of the interval approximate asymptotic solutions under different conditions as well as to validate this new approach. This study highlights the importance of the asymptotic solutions for FDEs regardless of interval or noninterval parameters.
认识到在渐近情况下解的行为对于在控制理论和实际系统建模中的重复应用至关重要。本研究讨论了一种稳健且明确的方法，用于寻找具有阿坦加纳 - 巴莱努（A - B）导数的分数阶微分方程（FDEs）的区间近似渐近解。事实上，此类关键任务首先需要精确观察非区间情况的行为。在这方面，我们首先阐明非区间情况并分析近似渐近解的行为，然后，我们在区间算术下为FDEs引入A - B导数，并针对具有区间A - B导数的分数阶区间微分方程开发一种新的可靠近似方法以获得区间近似渐近解。我们利用拉普拉斯变换，基于区间算术下的区间渐近A - B分数阶导数来获得渐近近似解。这里开发的技术为在具有非奇异米塔格 - 莱夫勒核的区间分数阶导数下寻找区间近似渐近解提供了重要工具。在区间概念下对实际系统中出现的两种情况进行建模，并给出以解释在不同条件下区间近似渐近解的行为以及验证这种新方法。本研究强调了FDEs渐近解的重要性，而不考虑区间或非区间参数。
