Research article

Numerical solution of multi-term time fractional wave diffusion equation using transform based local meshless method and quadrature

  • The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles. In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The present papers deals with the approximation of one and two dimensional multi-term time fractional wave diffusion equations. In this work a numerical method which combines Laplace transform with local radial basis functions method is presented. The Laplace transform eliminates the time variable with which the classical time stepping procedure is avoided, because in time stepping methods the accuracy is achieved at a very small step size, and these methods face sever stability restrictions. For spatial discretization the local meshless method is employed to circumvent the issue of shape parameter sensitivity and ill-conditioning of collocation matrices in global meshless methods. The bounds of the stability for the differentiation matrix of our numerical scheme are derived. The method is tested and validated against 1D and 2D wave diffusion equations. The 2D equations are solved over rectangular, circular and complex domains. The computational results insures the stability, accuracy, and efficiency of the method.

    Citation: Jing Li, Linlin Dai, Kamran, Waqas Nazeer. Numerical solution of multi-term time fractional wave diffusion equation using transform based local meshless method and quadrature[J]. AIMS Mathematics, 2020, 5(6): 5813-5838. doi: 10.3934/math.2020373

    Related Papers:

    [1] Jiang Zhao, Dan Wu . The risk assessment on the security of industrial internet infrastructure under intelligent convergence with the case of G.E.'s intellectual transformation. Mathematical Biosciences and Engineering, 2022, 19(3): 2896-2912. doi: 10.3934/mbe.2022133
    [2] Roman Gumzej . Intelligent logistics systems in E-commerce and transportation. Mathematical Biosciences and Engineering, 2023, 20(2): 2348-2363. doi: 10.3934/mbe.2023110
    [3] Dawei Li, Enzhun Zhang, Ming Lei, Chunxiao Song . Zero trust in edge computing environment: a blockchain based practical scheme. Mathematical Biosciences and Engineering, 2022, 19(4): 4196-4216. doi: 10.3934/mbe.2022194
    [4] Xiang Nan, Kayo kanato . Role of information security-based tourism management system in the intelligent recommendation of tourism resources. Mathematical Biosciences and Engineering, 2021, 18(6): 7955-7964. doi: 10.3934/mbe.2021394
    [5] Yanxu Zhu, Hong Wen, Jinsong Wu, Runhui Zhao . Online data poisoning attack against edge AI paradigm for IoT-enabled smart city. Mathematical Biosciences and Engineering, 2023, 20(10): 17726-17746. doi: 10.3934/mbe.2023788
    [6] Li Yang, Kai Zou, Kai Gao, Zhiyi Jiang . A fuzzy DRBFNN-based information security risk assessment method in improving the efficiency of urban development. Mathematical Biosciences and Engineering, 2022, 19(12): 14232-14250. doi: 10.3934/mbe.2022662
    [7] Xiaodong Qian . Evaluation on sustainable development of fire safety management policies in smart cities based on big data. Mathematical Biosciences and Engineering, 2023, 20(9): 17003-17017. doi: 10.3934/mbe.2023758
    [8] Ridha Ouni, Kashif Saleem . Secure smart home architecture for ambient-assisted living using a multimedia Internet of Things based system in smart cities. Mathematical Biosciences and Engineering, 2024, 21(3): 3473-3497. doi: 10.3934/mbe.2024153
    [9] Shitharth Selvarajan, Hariprasath Manoharan, Celestine Iwendi, Taher Al-Shehari, Muna Al-Razgan, Taha Alfakih . SCBC: Smart city monitoring with blockchain using Internet of Things for and neuro fuzzy procedures. Mathematical Biosciences and Engineering, 2023, 20(12): 20828-20851. doi: 10.3934/mbe.2023922
    [10] Roman Gumzej, Bojan Rosi . Open interoperability model for Society 5.0's infrastructure and services. Mathematical Biosciences and Engineering, 2023, 20(9): 17096-17115. doi: 10.3934/mbe.2023762
  • The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles. In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The present papers deals with the approximation of one and two dimensional multi-term time fractional wave diffusion equations. In this work a numerical method which combines Laplace transform with local radial basis functions method is presented. The Laplace transform eliminates the time variable with which the classical time stepping procedure is avoided, because in time stepping methods the accuracy is achieved at a very small step size, and these methods face sever stability restrictions. For spatial discretization the local meshless method is employed to circumvent the issue of shape parameter sensitivity and ill-conditioning of collocation matrices in global meshless methods. The bounds of the stability for the differentiation matrix of our numerical scheme are derived. The method is tested and validated against 1D and 2D wave diffusion equations. The 2D equations are solved over rectangular, circular and complex domains. The computational results insures the stability, accuracy, and efficiency of the method.


    The rapid advancement of Smart Cities is crucial for the economic growth and sustainable development of urban areas. These cities rely on a vast network of sensors that collect enormous amounts of data, posing significant challenges in secure data collection, management, and storage. Artificial Intelligence (AI) has emerged as a powerful computational model, demonstrating notable success in processing large datasets, particularly in unsupervised settings. Deep Learning models provide efficient learning representations, enabling systems to learn features from data automatically.

    However, the rise in cyberattacks presents ongoing threats to data privacy and integrity in Smart Cities. Unauthorized access and data breaches are growing concerns, exacerbated by various network vulnerabilities and risks. This highlights the necessity for further research to address security issues, ensuring that Smart City operations remain secure, resilient, and dependable.

    The main aim of this Special Issue is to gather high-quality research papers and reviews focusing on AI-based solutions, incorporating other enabling technologies such as Blockchain and Edge Intelligence. These technologies address the challenges of data security, privacy, and network authentication in IoT-based Smart Cities. After a rigorous review process, 14 papers were accepted. These papers cover a broad scope of topics and offer valuable contributions to the field of AI-based security applications in Smart Cities. They provide innovative solutions for secure data management, advanced algorithms for threat detection and prevention, and techniques for ensuring data privacy and integrity.

    All accepted papers are categorized into six different dimensions: 1) Prediction and forecasting models, 2) Security and encryption techniques, 3) Edge computing and IoT, 4) Automotive and transportation systems, 5) Artificial intelligence and machine learning applications, and 6) 3D printing and object identification. The brief contributions of these papers are discussed as follows:

    In the prediction and forecasting models dimension, Tang et al. [1] proposed a ride-hailing demand prediction model named the spatiotemporal information-enhanced graph convolution network. This model addresses issues of inaccurate predictions and difficulty in capturing external spatiotemporal factors. By utilizing gated recurrent units and graph convolutional networks, the model enhances its perceptiveness to external factors. Experimental results on a dataset from Chengdu City show that the model performs better than baseline models and demonstrates robustness in different environments. Similarly, Chen et al. [2] constructed a novel BILSTM-SimAM network model for short-term power load forecasting. The model uses Variational Mode Decomposition (VMD) to preprocess raw data and reduce noise. It combines Bidirectional Long Short-Term Memory (BILSTM) with a simple attention mechanism (SimAM) to enhance feature extraction from load data. The results indicate an R² of 97.8%, surpassing mainstream models like Transformer, MLP, and Prophet, confirming the method's validity and feasibility

    In the security and encryption techniques dimension, Bao et al. [3] developed a Fibonacci-Diffie-Hellman (FIB-DH) encryption scheme for network printer data transmissions. This scheme uses third-order Fibonacci matrices combined with the Diffie-Hellman key exchange to secure data. Experiments demonstrate the scheme's effectiveness in improving transmission security against common attacks, reducing vulnerabilities to data leakage and tampering. Cai et al. [4] introduced a robust and reversible database watermarking technique to protect shared relational databases. The method uses hash functions for grouping, firefly and simulated annealing algorithms for efficient watermark location, and differential expansion for embedding the watermark. Experimental results show that this method maintains data quality while providing robustness against malicious attacks. Yu et al. [11] investigated Transport Layer Security (TLS) fingerprinting techniques for analyzing and classifying encrypted traffic without decryption. The study discusses various fingerprint collection and AI-based techniques, highlighting their pros and cons. The need for step-by-step analysis and control of cryptographic traffic is emphasized to use each technique effectively. Salim et al. [14] proposed a lightweight authentication scheme for securing IoT devices from rogue base stations during handover processes. The scheme uses SHA256 and modulo operations to enable quick authentication, significantly reducing communication overhead and enhancing security compared to existing methods.

    In the edge computing and IoT dimension, Yu et al. [5] proposed an edge computing-based intelligent monitoring system for manhole covers (EC-MCIMS). The system uses sensors, LoRa communication, and a lightweight machine learning model to detect and alert about unusual states of manhole covers, ensuring safety and timely maintenance. Tests demonstrate higher responsiveness and lower power consumption compared to cloud computing models. Zhu et al. [7] introduced an online poisoning attack framework for edge AI in IoT-enabled smart cities. The framework includes a rehearsal-based buffer mechanism and a maximum-gradient-based sample selection strategy to manipulate model training by incrementally polluting data streams. The proposed method outperforms existing baseline methods in both attack effectiveness and storage management. Firdaus et al. [10] discussed personalized federated learning (PFL) with a blockchain-enabled distributed edge cluster (BPFL). Combining blockchain and edge computing technologies enhances client privacy, security, and real-time services. The study addresses the issue of non-independent and identically distributed data and statistical heterogeneity, aiming to achieve personalized models with rapid convergence.

    In the automotive and transportation systems dimension, Douss et al. [6] presented a survey on security threats and protection mechanisms for Automotive Ethernet (AE). The paper introduces and compares different in-vehicle network protocols, analyzes potential threats targeting AE, and discusses current security solutions. Recommendations are proposed to enhance AE protocol security. Yang et al. [13] proposed a lightweight fuzzy decision blockchain scheme for vehicle intelligent transportation systems. The scheme uses MQTT for communication, DH and Fibonacci transformation for security, and the F-PBFT consensus algorithm to improve fault tolerance, security, and system reliability. Experimental results show significant improvements in fault tolerance and system sustainability.

    In the artificial intelligence and machine learning applications dimension, Pan et al. [8] focused on aerial image target detection using a cross-scale multi-feature fusion method (CMF-YOLOv5s). The method enhances detection accuracy and real-time performance for small targets in complex backgrounds by using a bidirectional cross-scale feature fusion sub-network and optimized anchor boxes. Wang et al. [9] reviewed various AI techniques for ground fault line selection in modern power systems. The review discusses artificial neural networks, support vector machines, decision trees, fuzzy logic, genetic algorithms, and other emerging methods. It highlights future trends like deep learning, big data analytics, and edge computing to improve fault line selection efficiency and reliability.

    In the 3D printing and object identification dimension, Shin et al. [12] presented an all-in-one encoder/decoder approach for the non-destructive identification of 3D-printed objects using terahertz (THz) waves. The method involves 3D code insertion into the object's STL file, THz-based detection, and code extraction. Experimental results indicate that this approach enhances the identification efficiency and practicality of 3D-printed objects.

    In conclusion, 14 excellent full-length research articles have been provided in this special issue on "Artificial Intelligence-based Security Applications and Services for Smart Cities." These papers offer valuable contributions to secure data management, threat detection, and data privacy in IoT-based Smart Cities. We would like to thank all the researchers for their contributions, the MBE editorial assistance, and all the referees for their support in making this issue possible.



    [1] K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Elsevier, 1974.
    [2] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Soliton. Fract., 7 (1996), 1461-1477. doi: 10.1016/0960-0779(95)00125-5
    [3] A. Carpinteri, F. Mainardi, Fractals and Fractional Calculas Continuum Mechanics, SpringerVerlag Wien, 1997.
    [4] F. Liu, M. M. Meerschaert, R. J. McGough, et al. Numerical methods for solving the multi-term time-fractional wave-diffusion equation, Fract. Calc. Appl. Anal., 16 (2013), 9-25.
    [5] H. Ding, C. Li, Numerical algorithms for the fractional diffusion-wave equation with reaction term, Abstr. Appl. Anal., 2013 (2013), 1-15.
    [6] K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229-248. doi: 10.1006/jmaa.2000.7194
    [7] H. Ye, F. Liu, V. Anh, et al. Maximum principle and numerical method for the multi-term timespace riesz-caputo fractional differential equations, Appl. Math. Model., 227 (2014), 531-540.
    [8] C. M. Chen, F. Liu, I. Turner, et al. A fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227 (2007), 886-897. doi: 10.1016/j.jcp.2007.05.012
    [9] M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), 7792-7804. doi: 10.1016/j.jcp.2009.07.021
    [10] F. Liu, C. Yang, K. Burrage, Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term, J. Comput. Appl. Math., 231 (2009), 160-176. doi: 10.1016/j.cam.2009.02.013
    [11] H. Ding, C. Li, High-order numerical methods for riesz space fractional turbulent diffusion equation, Fract. Calc. Appl. Anal., 19 (2016), 19-55.
    [12] A. Chen, C. Li, Numerical solution of fractional diffusion-wave equation, Numer. Func. Anal. Opt., 37 (2016), 19-39. doi: 10.1080/01630563.2015.1078815
    [13] M. Garg, P. Manohar, Numerical solution of fractional diffusion-wave equation with two space variables by matrix method, Fract. Calc. Appl. Anal., 13 (2010), 191-207.
    [14] M. Dehghan, M. Safarpoor, M. Abbaszadeh, Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations, J. Comput. Appl. Math., 290 (2015), 174-195. doi: 10.1016/j.cam.2015.04.037
    [15] J. Huang, Y. Tang, L. Vázquez, et al. Two finite difference schemes for time fractional diffusionwave equation, Numer. Algorithms, 64 (2013), 707-720. doi: 10.1007/s11075-012-9689-0
    [16] A. H. Bhrawy, E. H. Doha, D. Baleanu, et al. A spectral tau algorithm based on jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, J. Comput. Phys., 293 (2015), 142-156. doi: 10.1016/j.jcp.2014.03.039
    [17] M. H. Heydari, M. R. Hooshmandasl, F. M. M. Ghaini, et al, Wavelets method for the time fractional diffusion-wave equation, Phys. Lett. A, 379 (2015), 71-76. doi: 10.1016/j.physleta.2014.11.012
    [18] Y. N. Zhang, Z. Z. Sun, X. Zhao, Compact alternating direction implicit scheme for the twodimensional fractional diffusion-wave equation, SIAM J. Numer. Anal., 50 (2012),1535-1555. doi: 10.1137/110840959
    [19] J. Ren, Z. Z. Sun, Efficient numerical solution of the multi-term time fractional diffusion-wave equation, E. Asian J. Appl. Math., 5 (2015),1-28. doi: 10.4208/eajam.080714.031114a
    [20] J. Y. Yang, J. F. Huang, D. M.Liang, et al. Numerical solution of fractional diffusion-wave equation based on fractional multistep method, Appl. Math. Model., 38 (2014), 3652-3661. doi: 10.1016/j.apm.2013.11.069
    [21] Y. Yang, Y. Chen, Y. Huang, et al. Spectral collocation method for the time-fractional diffusionwave equation and convergence analysis, Comput. Math. Appl., 73 (2017), 1218-1232. doi: 10.1016/j.camwa.2016.08.017
    [22] T. Belytschko, Y. Y. Lu, L. Gu, Element free galerkin methods, Int. J. Numer. Meth. Eng., 37 (1994), 229-256. doi: 10.1002/nme.1620370205
    [23] W. K. Liu, Y. Chen, S. Jun, et al. Overview and applications of the reproducing kernel particle methods, Arch. Comput. Method. E., 3 (1996), 3-80. doi: 10.1007/BF02736130
    [24] W. Chen, Singular boundary method: a novel, simple, meshfree, boundary collocation numerical method, Chin. J. Solid Mech., 30 (2009), 592-599.
    [25] Z. Fu, W. Chen, C. Zhang, Boundary particle method for cauchy inhomogeneous potential problems, Inverse probl. Sci. En., 20 (2012), 189-207. doi: 10.1080/17415977.2011.603085
    [26] V. R. Hosseini, E. Shivanian, W. Chen, Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping, J. Comput. Phys., 312 (2016), 307- 332.
    [27] M. Dehghan, M. Abbaszadeh, A. Mohebbi, Analysis of a meshless method for the time fractional diffusion-wave equation, Numer. algorithms, 73 (2016), 445-476. doi: 10.1007/s11075-016-0103-1
    [28] Y. Gu, P. Zhuang, F. Liu, An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation, CMES-Comp. Model. Eng., 56 (2010), 303-334.
    [29] Q. Liu, Y. T. Gu, P. Zhuang, et al. An implicit rbf meshless approach for time fractional diffusion equations, Comput. Mech., 48 (2011), 1-12. doi: 10.1007/s00466-011-0573-x
    [30] M. Abbaszadeh, M. Dehghan, An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate, Numer. Algorithms, 75 (2017), 173-211. doi: 10.1007/s11075-016-0201-0
    [31] R. Salehi, A meshless point collocation method for 2-d multi-term time fractional diffusion-wave equation, Numer. Algorithms, 74 (2017), 1145-1168. doi: 10.1007/s11075-016-0190-z
    [32] M. Uddin, Kamran, A. Ali, A localized transform-based meshless method for solving time fractional wave-diffusion equation, Eng. Anal. Bound. Elem., 92 (2018), 108-113. doi: 10.1016/j.enganabound.2017.10.021
    [33] W. Chen, L. Ye, H. Sun, Fractional diffusion equations by the kansa method, Comput. Math. Appl., 59 (2010), 1614-1620. doi: 10.1016/j.camwa.2009.08.004
    [34] P. Zhuang, Y. T. Gu, F. Liu, et al. Time-dependent fractional advection-diffusion equations by an implicit mls meshless method, Int. J. Numer. Meth. Eng., 88 (2011), 1346-1362. doi: 10.1002/nme.3223
    [35] J. Y. Yang, Y. M. Zhao, N. Liu, et al. An implicit mls meshless method for 2-d time dependent fractional diffusion-wave equation, Appl. Math. Model., 39 (2015), 1229-1240. doi: 10.1016/j.apm.2014.08.005
    [36] G. J. Moridis, E. J. Kansa, The Laplace transform multiquadric method: a highly accurate scheme for the numerical solution of partial differential equations, J. Appl. sci. comput., 1 (1994) 375-475.
    [37] W. McLean, V. Thomee, Time discretization of an evolution equation via laplace transforms, IMA J. Numer. Anal., 24 (2004), 439 -463.
    [38] W. McLean, V. Thomée, Numerical solution via laplace transforms of a fractional order evolution equation, J. Integral Equ. Appl., 22 (2010), 57-94. doi: 10.1216/JIE-2010-22-1-57
    [39] B. A. Jacobs, High-order compact finite difference and Laplace transform method for the solution of time fractional heat equations with Dirichlet and Neumann boundary conditions, Numer. Meth. Part. D. E., 32 (2016), 1184-1199. doi: 10.1002/num.22046
    [40] Q. T. L. Gia, W. Mclean, Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions, Adv. Comput. Math., 40 (2014), 353-375. doi: 10.1007/s10444-013-9311-6
    [41] A. Talbot, The accurate numerical inversion of laplace transform, IMA J. Appl. Math., 23 (1979), 97-120. doi: 10.1093/imamat/23.1.97
    [42] D. G. Duffy, On the numerical inversion of laplace transforms: comparison of three new methods on characteristic problems from applications, ACM T. Math. Software, 19 (1993), 333-359. doi: 10.1145/155743.155788
    [43] J. A. C. Weideman, Optimizing talbot's contours for the inversion of the laplace transform, SIAM J. Numer. Anal., 44 (2006), 2342-2362. doi: 10.1137/050625837
    [44] C. Lubich, A. Schädle, Fast convolution for nonreflecting boundary conditions, SIAM J. Sci. Comput., 24 (2002), 161-182. doi: 10.1137/S1064827501388741
    [45] F. Zeng, I. Turner, K. Burrage, et al. A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equations, SIAM J. Sci. Comput., 40 (2018), A2986-A3011.
    [46] B. Dingfelder, J. Weideman, An improved talbot method for numerical laplace transform inversion, Numer. Algorithms, 68 (2015), 167-183. doi: 10.1007/s11075-014-9895-z
    [47] R. Schaback, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math., 3 (1995), 251-264. doi: 10.1007/BF02432002
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5060) PDF downloads(423) Cited by(8)

Article outline

Figures and Tables

Figures(13)  /  Tables(12)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog