Amin Rohul, Nazir Shah, García-Magariño Iván
Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa 25120, Pakistan.
Department of Computer Science, University of Swabi, Khyber Pakhtunkhwa 23430, Pakistan.
Sensors (Basel). 2020 Mar 31;20(7):1962. doi: 10.3390/s20071962.
Wireless sensor network and industrial internet of things have been a growing area of research which is exploited in various fields such as smart home, smart industries, smart transportation, and so on. There is a need of a mechanism which can easily tackle the problems of nonlinear delay integro-differential equations for large-scale applications of Internet of Things. In this paper, Haar wavelet collocation technique is developed for the solution of nonlinear delay integro-differential equations for wireless sensor network and industrial Internet of Things. The method is applied to nonlinear delay Volterra, delay Fredholm and delay Volterra-Fredholm integro-differential equations which are based on the use of Haar wavelets. Some examples are given to show the computational efficiency of the proposed technique. The approximate solutions are compared with the exact solution. The maximum absolute and mean square roots errors for distant number of collocation points are also calculated. The results show that Haar method is efficient for solving these equations for industrial Internet of Things. The results are compared with existing methods from the literature. The results exhibit that the method is simple, precise and efficient.
无线传感器网络和工业物联网一直是一个不断发展的研究领域，在智能家居、智能工业、智能交通等各个领域都有应用。对于物联网的大规模应用，需要一种能够轻松解决非线性延迟积分微分方程问题的机制。本文针对无线传感器网络和工业物联网的非线性延迟积分微分方程，开发了哈尔小波配置技术。该方法应用于基于哈尔小波的非线性延迟沃尔泰拉、延迟弗雷德霍姆和延迟沃尔泰拉 - 弗雷德霍姆积分微分方程。给出了一些例子来说明所提技术的计算效率。将近似解与精确解进行了比较。还计算了不同数量配置点的最大绝对误差和均方根误差。结果表明，哈尔方法对于解决工业物联网中的这些方程是有效的。将结果与文献中的现有方法进行了比较。结果表明该方法简单、精确且高效。
