Data Collection in ResourceLimited Networks (WSNs, IoT, Sensor Cloud)
View this Special IssueResearch Article  Open Access
Yongtao Xuan, Rohul Amin, Fakhar Zaman, Zohaib Khan, Imad Ullah, Shah Nazir, "SecondOrder Delay Differential Equations to Deal the Experimentation of Internet of Industrial Things via Haar Wavelet Approach", Wireless Communications and Mobile Computing, vol. 2021, Article ID 5551497, 9 pages, 2021. https://doi.org/10.1155/2021/5551497
SecondOrder Delay Differential Equations to Deal the Experimentation of Internet of Industrial Things via Haar Wavelet Approach
Abstract
In this article, an efficient numerical approach for the solution of secondorder delay differential equations to deal with the experimentation of the Internet of Industrial Things (IIoT) is presented. With the help of the Haar wavelet technique, the considered problem is transformed into a system of algebraic equations which is then solved for the required results by using Gauss elimination algorithm. Some numerical examples for convergence of the proposed technique are taken from the literature. Maximum absolute and root mean square errors are calculated for various collocation points. The results show that the Haar wavelet method is an effective method for solving delay differential equations of second order. The convergence rate is also measured for various collocation points, which is almost equal to 2.
1. Introduction
Delay differential equations (DDEs) are type of DEs in which the solution of the unknown function is given in the previous time interval. A system whose performance does not depend directly on time is a timeinvariant delay system. This systems can be defined by constant coefficients of the th order ordinary DEs [1]. DDEs are used for modelling of various phenomenon such as modelling of systems with memory, electric circuits, and mechanical systems. The application of these systems in population dynamics [2] can be used in communication networks, mass transportation, remote controls, and biological systems. Many of the processes, both natural and artificial, in medicine, chemistry, engineering, economics, etc. involve time delays. The Internet of Things (IoT) contributes in facilitating the needs of daily life such as IoT for healthcare using effects of Mobile computing [3] and nonlinear delay integrodifferential equations for wireless sensor network and IoT [4].
There are numerous approaches available in the literature for the solution DDEs of second order. Seong and Majid [1] developed the Adams Moulton technique to solve the secondorder DDEs. Ibrahim [5], used 2hstep Spline method to solve the DDEs. Sedaghat [2], utilized the Chebyshev polynomials method to find the solution of DDEs. Ehigie et al. [6] implement a 3point block technique to solve DDEs of second order. Chebyshev wavelet technique was developed by Ghasemi and Kajan [7] to solve the DDEs. Ahmad et al. [8] solved the DDEs by a block hybrid method. Multistep methods was used by Okunuga and Ehigie [9] to solve the DDEs. Brown [10] used a method of implicit multistep to solve the DDEs. Ismail et al. [11] found the solution of DDEs by 3point block methods. Ehigie et al. [12] used a method of 2step continuous linear multistep to solve the secondorder DDEs. Some other wellknown methods are the following: RungeKutta [13], Shift Walsh matrix method [14], Hermite interpolation method [15], method of retarded initial value problems [16], onestep collocation method [17], coupled block technique [18], onestep block techniques [11], implicit block technique [19], Direct integration implicit variable method [19], predictorcorrector method [20], Taylor method [21], fuzzy mapping and control method [22], variational iteration method [23, 24], and Galerkin method [25]. A structure for the IIoT cloudfog hybrid network for industry data processing was proposed by Liu et al. [26]. Sahal et al. [27] studied the strong point and flaws of open source technologies for big data. Khan et al. [28] offered the idea of IIoT in a novel manner for supporting readers to comprehend the IIoT. Gulati and Kaur [29] analysed the main opportunities assimilated from the idea of IoT into industry with suggesting reference architecture. The use of Haar wavelet have wideranging applications in scientific computing. The Haar Collocation Technique (HCT) is used for fractional Riccati type differential equations [30], Birthmark based identification [31], delay FredholmVolterra integral equations [32], delay integrodifferential equations [4], systems of fractional differential equations [33], and fractional integrodifferential equations [34] in recent literature. This article studies the solutions of secondorder DDEs, that is, we develop numerical technique using Haar wavelet with constant delay.
In this paper, we discuss the solution of the secondorder DDEs using a HCT to deal with the experimentation of IIoT, consider linear DDEs as where is a control function, is the delay condition, and is a state function.
The paper is organized as some basic results and notions are given in Section 2. Section 3 provides the HCT solution for linear DDEs of second order. In Section 4, the HCT validation is given. The results are discussed in Section 5, and the conclusion is given in the last part of the paper.
2. Haar Wavelet
Scaling function on is [35]
Mother wavelet on is
The other terms can be written as where where , and . The number can be calculated as . If we take interval , then values of , , and are: , , . Any member in , is , we truncate this series is Let
Also, is We obtain
Generally,
Thus, [9],
The interval for HCT is discretized as The points defined in the above Eq. (10) are called collocation points (CPs).
3. Numerical Method
Here, we describe the proposed method for secondorder DDEs to deal with the experimentation of IIoT. Let , then
Integrating Eq. (11) from to ,
from Eq. (1), , so we get
Now, integrating Eq. (13) from to , the following relation yields:
from Eq. (1), , so we have
By putting Eq. (11), Eq. (13), and Eq. (15) in Eq. (1), we get
Discretizing this Eq. (16) at CPs, we have
The above system in matrix notations as given by where
Hence, is calculated as . This is a linear system of order , which is solved by the Gauss elimination technique. By putting these s in Eq. (15), we get the required solution of secondorder DDEs defined in (1).
4. Numerical Examples
Let be approximate and is exact solution for CPs and GPs, then maximum absolute error is . The root mean square errors at CPs is . In CPs, convergence rate is
Example 1. Consider DDE of second order [8] with delay condition and initial condition The analytical solution is .
Example 2. Consider the following secondorder DDE [1] with delay condition and initial condition The exact solution is .
Example 3. Consider the following secondorder DDE [1] with delay condition and initial condition The exact solution is .
Example 4. Consider DDE of second order as [36] with delay condition and initial condition The exact solution is .
Example 5. Consider the following secondorder DDE [8] with delay condition and initial condition The exact solution is .
5. Discussion
The secondorder derivative in DDE is expressed as Haar function and the value of the first derivative is obtained by the process of integration. By applying the HCT, we obtain a system of linear equations by substituting CPs. The method of Gauss elimination is used for this system. Finally, by utilizing these coefficients, the solution at CPs is obtained. and errors for different number of CPs are given in Tables. and CPU time (seconds) are also reported in tables for each example. For Example 1, and errors for different number of CPs are shown in Table 1. Table 2 shows the errors for different number of CPs for Example 2, Table 3 represents errors for different number of CPs for Example 3, Table 4 shows the errors for different number of CPs for Example 4, and Table 5 shows the errors for different number of CPs for Example 5. All errors are decreased by taking more CPs. is determined which is nearly equal to , supporting the results shown in [37] by Majak et al. The comparison of both numerical and analytical solution at CPs is also shown in figures. Figure 1 represents the comparison of approximate and exact solution for Example 1, Figure 2 represents the comparison of approximate and exact solution for Example 2, Figure 3 represents the comparison of approximate and exact solution for Example 3, Figure 4 represents the comparison of approximate and exact solution for Example 4, and Figure 5 represents the comparison of exact and approximate solution for Example 5.





6. Conclusion
HCT scheme is devoted for the solution of secondorder DDEs to deal with the experimentation of IIoT. The Haar technique is applied to linear DDEs for dealing with the experimentation of the Internet of Industrial Things. The Matlab software is utilized to experiment the Haar wavelet technique with the examples, and the numerical solution is compared with the exact solution. We compare the obtained solution with the exact solution and also we compute the and errors to show the accuracy of the Haar wavelet technique. We give some test problems for the illustration of our results. The experimental rate of convergence for different number of CPs is also calculated which is approximately equal to . The results show that HCT is efficient for solving secondorder DDEs. Our future work addresses to overcome the limitation of this study. Moreover, we will apply this technique to high order DDEs and system of DDEs.
Data Availability
No data is available.
Conflicts of Interest
There is no conflicting interest.
References
 H. Y. Seong and Z. A. Majid, “Solving second order delay differential equations using direct twopoint block method,” Ain Shams Engineering Journal, vol. 8, no. 1, pp. 59–66, 2017. View at: Publisher Site  Google Scholar
 S. Sedaghat, Y. Ordokhani, and M. Dehghan, “Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12, pp. 4815–4830, 2012. View at: Publisher Site  Google Scholar
 S. Nazir, Y. Ali, N. Ullah, and I. GarcíaMagariño, “Internet of things for healthcare using effects of mobile computing: a systematic literature review,” Wireless Communications and Mobile Computing, page, vol. 2019, article 5931315, pp. 1–20, 2019. View at: Publisher Site  Google Scholar
 R. Amin, S. Nazir, and I. GarcíaMagariño, “A collocation method for numerical solution of nonlinear delay integrodifferential equations for wireless sensor network and internet of things,” Sensor, vol. 20, no. 7, article 1962, 2020. View at: Publisher Site  Google Scholar
 M. A. Ibrahim, A. ElSafty, and S. M. AboHasha, “2hStep spline method for the solution of delay differential equations,” Computers & Mathematics with Applications, vol. 29, no. 8, pp. 1–6, 1995. View at: Publisher Site  Google Scholar
 J. O. Ehigie, S. A. Okunuga, and A. B. Sofoluwe, “3point block methods for direct integration of general secondorder ordinary differential equations,” Advances in Numerical Analysis, vol. 2011, Article ID 513148, 14 pages, 2011. View at: Publisher Site  Google Scholar
 M. Ghasemi and M. Tavassoli Kajani, “Numerical solutions of timevarying delay system by chebyshev wavlets,” Applied Mathematical Modelling, vol. 35, pp. 5235–5244, 2011. View at: Publisher Site  Google Scholar
 S. Z. Ahmad, F. Ismail, and N. Senu, “Solving oscillatory delay differential equations using block hybrid methods,” Journal of Mathematics, vol. 2018, Article ID 2960237, 7 pages, 2018. View at: Publisher Site  Google Scholar
 S. A. Okunuga and J. Ehigie, “A new derivation of continuous collocation multistep methods using power series as basis function,” Journal of the Nigerian Association of Mathematical Physics, vol. 3, pp. 43–50, 2009. View at: Google Scholar
 R. L. Brown, “Some characteristics of implicit multistep multiderivative integration formulas,” SIAM Journal on Numerical Analysis, vol. 14, pp. 982–993, 1977. View at: Publisher Site  Google Scholar
 F. Ismail, L. K. Yap, and O. Mohamad, “Explicit and implicit 3point block methods for solving special second order ordinary differential equations directly,” International Journal of Mathematical Analysis, vol. 3, no. 5, pp. 239–254, 2009. View at: Google Scholar
 J. O. Ehigie, S. A. Okunuga, and A. B. Sofoluwe, “On generalized 2step continuous linear multistep method of hybrid type for the integration of second order ordinary differential equations,” Archives of Applied Science Research, vol. 2, pp. 362–372, 2010. View at: Google Scholar
 J. B. Rosser, “A rungekutta for all seasons,” SIAM Review, vol. 9, no. 3, pp. 417–452, 1967. View at: Publisher Site  Google Scholar
 W. L. Chen and Y. P. Shi, “Shift walsh matrix and delay differential equations,” IEEE Transactions on Automatic Control, vol. 23, pp. 265–280, 1978. View at: Publisher Site  Google Scholar
 H. J. Oberle and H. J. Pesh, “Numerical treatment of delay differential equations by hermite interpolation,” Numerische Mathematik, vol. 37, no. 2, pp. 235–255, 1981. View at: Publisher Site  Google Scholar
 H. Arnt, “Numerical solution of retarded initialvalue problems: Local and global error and stepsize control,” Numerische Mathematik, vol. 43, no. 3, pp. 343–360, 1984. View at: Publisher Site  Google Scholar
 A. Bellen, “Onestep collocation for delay differential equations,” Journal of Computational and Applied Mathematics, vol. 10, no. 3, pp. 275–283, 1984. View at: Publisher Site  Google Scholar
 H. C. San, Z. A. Majid, and M. Othman, “Solving delay differential equations using coupled block method,” in 2011 Fourth International Conference on Modeling, Simulation and Applied Optimization, p. 11, Kuala Lumpur, Malaysia, April 2011. View at: Publisher Site  Google Scholar
 Z. A. Majid, N. A. Azmi, and M. Suleiman, “Solving second order ordinary differential equations using two point four step direct implicit block method,” European Journal of Scientific Research, vol. 31, pp. 29–36, 2009. View at: Google Scholar
 T. Allahviranloo, N. Ahmady, and E. Ahmady, “Numerical solution of fuzzy differential equations by predictorcorrector method,” Information Sciences, vol. 177, no. 7, pp. 1633–1647, 2007. View at: Publisher Site  Google Scholar
 S. Abbasbandy and T. Allahviranloo, “Numerical solutions of fuzzy differential equations by taylor method,” Computational Methods in Applied Mathematics, vol. 2, no. 2, pp. 113–124, 2002. View at: Publisher Site  Google Scholar
 S. L. Chang and L. A. Zadeh, “On fuzzy mapping and control,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 2, pp. 30–34, 1972. View at: Publisher Site  Google Scholar
 H. Jafari, M. Saeidy, and D. Baleanu, “The variational iteration method for solving nth order fuzzy differential equations,” Central European Journal of Physics, vol. 10, no. 1, pp. 76–85, 2012. View at: Publisher Site  Google Scholar
 Z. M. Odibat, “A study on the convergence of variational iteration method,” Mathematical and Computer Modelling, vol. 51, no. 910, pp. 1181–1192, 2010. View at: Publisher Site  Google Scholar
 C. Hwang and M. Y. Chen, “Analysis of timedelay systems using the Galerkin method,” International Journal of Control, vol. 44, no. 3, pp. 847–866, 1986. View at: Publisher Site  Google Scholar
 W. Liu, G. Huang, A. Zheng, and J. Liu, “Research on the optimization of iiot data processing latency,” Computer Communications, vol. 151, pp. 290–298, 2020. View at: Publisher Site  Google Scholar
 R. Sahal, J. G. Breslin, and M. I. Ali, “Big data and stream processing platforms for industry 4.0 requirements mapping for a predictive maintenance use case,” Journal of Manufacturing Systems, vol. 54, pp. 138–151, 2020. View at: Publisher Site  Google Scholar
 W. Z. Khan, M. H. Rehman, H. M. Zangoti, M. K. Afzal, N. Armi, and K. Salah, “Industrial internet of things: recent advances, enabling technologies and open challenges,” Computers and Electrical Engineering, vol. 81, p. 106522, 2020. View at: Publisher Site  Google Scholar
 N. Gulati and P. D. Kaur, “Towards socially enabled internet of industrial things: architecture, semantic model and relationship management,” Ad Hoc Networks, vol. 91, article 101869, 2019. View at: Publisher Site  Google Scholar
 M. M. Khashan, R. Amin, and M. I. Syam, “A new algorithm for fractional riccati type differential equations by using haar wavelet,” Mathematics, vol. 7, no. 6, p. 545, 2019. View at: Publisher Site  Google Scholar
 S. Nazir, S. Shahzad, R. Wirza et al., “Birthmark based identification of software piracy using haar wavelet,” Mathematics and Computers in Simulation, vol. 166, pp. 144–154, 2019. View at: Publisher Site  Google Scholar
 R. Amin, S. Nazir, and I. G. Magariño, “Efficient sustainable algorithm for numerical solution of nonlinear delay fredholmvolterra integral equations via haar wavelet for dense sensor networks in emerging telecommunications,” Transactions on Emerging Telecommunications Technologies, no. article e3877, 2020. View at: Publisher Site  Google Scholar
 T. Abdeljawad, R. Amin, K. Shah, Q. AlMdallal, and F. Jarad, “Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by haar wavelet collocation method,” Alexandria Engineering Journal, vol. 59, no. 4, pp. 2391–2400, 2020. View at: Publisher Site  Google Scholar
 R. Amin, K. Shah, M. Asif, I. Khan, and F. Ullah, “An efficient algorithm for numerical solution of fractional integrodifferential equations via Haar wavelet,” Journal of Computational and Applied Mathematics, vol. 381, p. 113028, 2021. View at: Publisher Site  Google Scholar
 I. Aziz and R. Amin, “Numerical solution of a class of delay differential and delay partial differential equations via haar wavelet,” Applied Mathematical Modelling, vol. 40, no. 2324, pp. 10286–10299, 2016. View at: Publisher Site  Google Scholar
 H. M. Radzi, Z. A. Majid, F. Ismail, and M. Suleiman, “Two and three point onestep block methods for solving delay differential equations,” Journal of Quality Measurement and Analysis, vol. 8, pp. 29–41, 2012. View at: Google Scholar
 J. Majak, B. S. Shvartsman, M. Kirs, M. Pohlak, and H. Herranen, “Convergence theorem for the Haar wavelet based discretization method,” Composite Structures, vol. 126, pp. 227–232, 2015. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2021 Yongtao Xuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.