Computational Approach via Half-Sweep Similarity and SOR Schemes for 2D Hyperbolic Telegraph Equations

  IJETT-book-cover  International Journal of Engineering Trends and Technology (IJETT)          
  
© 2022 by IJETT Journal
Volume-70 Issue-9
Year of Publication : 2022
Authors : Nur Afza Mat Ali, Jumat Sulaiman, Azali Saudi, Nor Syahida Mohamad
DOI : 10.14445/22315381/IJETT-V70I9P205

How to Cite?

Nur Afza Mat Ali, Jumat Sulaiman, Azali Saudi, Nor Syahida Mohamad, "Computational Approach via Half-Sweep Similarity and SOR Schemes for 2D Hyperbolic Telegraph Equations ," International Journal of Engineering Trends and Technology, vol. 70, no. 9, pp. 47-56, 2022. Crossref, https://doi.org/10.14445/22315381/IJETT-V70I9P205

Abstract
This paper uses the similarity transformation to get similar solutions to the two-dimensional hyperbolic telegraph equation (2-D HTE). The similarity solution of 2-D HTE is then derived using a rotated five-point similarity finite difference (SFD) discretization scheme to obtain the rotated five-point SFD approximation equation. Gathering rotated approximation equations generate a linear system with large-scale and sparse matrix characteristics. Since then, the linear system has been solved using the half-sweep similarity technique via half-sweep successive over-relaxation (HSSOR-SFD) iteration. Three numerical examples are presented in this paper to validate the performance of the HSSOR-SFD iteration in solving 2-D HTE. The numerical findings showed that the version of HSSOR-SFD iteration is the best compared to the standard similarity methods: full-sweep Gauss-Seidel (FSGS-SFD) and full-sweep SOR (FSSOR-SFD) iterations in terms of iteration number and computational time.

Keywords
Two-dimensional hyperbolic telegraph equation, Similarity transformation, Similarity solution, Rotated similarity finite difference method, Similarity half-sweep SOR method.

Reference
[1] M. Dehghan, and A. Mohebbi, “The Combination of Collocation, Finite Difference, and Multigrid Methods for Solution of the Two‐ Dimensional Wave Equation,” Numerical Methods for Partial Differential Equations: An International Journal, vol. 24, no. 3, pp. 897- 910, 2008.
[2] R. C. Tautz, and I. Lerche, “Application of the Three-Dimensional Telegraph Equation to Cosmic-Ray Transport,” Research in Astronomy and Astrophysics, vol. 16, no. 10, pp. 162, 2016.
[3] J. Banasiak, and J.R. Mika, “Singularly Perturbed Telegraph Equations with Applications in the Random Walk Theory,” Journal of Applied Mathematics and Stochastic Analysis, vol. 11, no. 1, pp. 9-28, 1998.
[4] V. K. Srivastava, M. K. Awasthi, R. K. Chaurasia, and M. Tamsir, “The Telegraph Equation and its Solution by Reduced Differential Transform Method,” Modelling and Simulation in Engineering, vol. 2013, no. 1, pp. 1-6, 2013.
[5] P. M. Jordan, and A. Puri, “Digital Signal Propagation in Dispersive Media,” Journal of Applied Physics, vol. 85, no. 3, pp. 1273-1282, 1999.
[6] B. K. Singh, J. P. Shukla, and M. Gupta, “Study of One-Dimensional Hyperbolic Telegraph Equation via a Hybrid Cubic B-Spline Differential Quadrature Method,” International Journal of Applied and Computational Mathematics, vol. 7, no. 1, pp. 1-17, 2021.
[7] G. Arora, and V. Joshi, “Comparison of Numerical Solution of 1D Hyperbolic Telegraph Equation using B-Spline and Trigonometric B-Spline by Differential Quadrature Method,” Indian Journal of Science and Technology, vol. 9, no. 45, pp. 1-8, 2016.
[8] R. C. Mittal, and R. A. Bhatia, “Numerical Study of Twodimensional Hyperbolic Telegraph Equation by Modified B-Spline Differential Quadrature Method,” Applied Mathematics and Computation, vol. 244, no. 1, pp. 976-97, 2014.
[9] B. K. Singh, and P. Kumar, “An Algorithm Based on a New DQM with Modified Extended Cubic B-Splines for Numerical Study of Two Dimensional Hyperbolic Telegraph Equation,” Alexandria Engineering Journal, vol. 57, no. 1, pp. 175-191, 2018.
[10] M. Javidi, “Chebyshev Spectral Collocation Method for Computing Numerical Solution of Telegraph Equation,” Computational Methods for Differential Equations, vol. 1, no. 1, pp. 16-29, 2013.
[11] S. Jator, “Block Unification Scheme for Elliptic, Telegraph, and Sine-Gordon Partial Differential Equations,” American Journal of Computational Mathematics, vol. 5, no. 1, pp. 175-185, 2015.
[12] H. Zhou, G. Kong, H. Liu, and L. Laloui, “Similarity Solution for Cavity Expansion in Thermoplastic Soil,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 42, no. 2, pp. 274-294, 2018.
[13] A. N. Ceretani, and D. A.Tarzia, “Similarity Solution for a Two-Phase One-Dimensional Stefan Problem with a Convective Boundary Condition and a Mushy Zone Model,” Computational and Applied Mathematics, vol. 37, no. 2, pp. 2201-2217, 2018.
[14] N.A.M. Ali, J. Sulaiman, A. Saudi, and N.S. Mohamad, “The Similarity Finite Difference Solutions for Two-Dimensional Parabolic Partial Differential Equations via SOR Iteration,” In Computational Science and Technology, vol. 724, no. 1, pp. 515-526, 2021.
[15] Y. Wang, L. Lin, and J. Li, “A similarity Solution for Undrained Expansion of a Cylindrical Cavity in K 0-Consolidated Anisotropic Soils,” Geomechanics and Engineering, vol. 25, no. 4, pp. 303-315, 2021.
[16] N.A.M. Ali, R. Rahman, J. Sulaiman, and K. Ghazali, “Numerical Solutions of Unsteady Advection-Diffusion Equations by Using EG Iteration with Wave Variable Transformation,” Journal of Physics: Conference Series, vol. 1358, pp. 012049, 2019.
[17] F. Ureña, L. Gavete, J.J. Benito, A. García, and A.M. Vargas, “Solving the Telegraph Equation in 2-D and 3-D using Generalized Finite Difference Method (GFDM),” Engineering Analysis with Boundary Elements, vol. 112, pp. 13-24, 2020.
[18] M. Aslefallah, and D. Rostamy, “Application of the singular Boundary Method to the Two-Dimensional Telegraph Equation on Arbitrary Domains,” Journal of Engineering Mathematics, vol. 118, no. 1, pp. 1-14, 2019.
[19] E. Hesameddini, and E. Asadolahifard, “A New Spectral Galerkin Method for Solving the Two Dimensional Hyperbolic Telegraph Equation,” Computers and Mathematics with Applications, vol. 72, no. 7, pp. 1926-1942, 2016.
[20] V. Devi, R.K. Maurya, S. Singh, and V.K. Singh, “Lagrange’s Operational Approach for the Approximate Solution of Two-Dimensional Hyperbolic Telegraph Equation Subject to Dirichlet Boundary Conditions,” Applied Mathematics and Computation, vol. 367, pp. 124717, 2020.
[21] Y. Zhou, W. Qu, Y. Gu, and H. Gao, “A Hybrid Meshless Method for the Solution of the Second Order Hyperbolic Telegraph Equation in Two Space Dimensions,” Engineering Analysis with Boundary Elements, vol. 115, pp. 21-27, 2020.
[22] J.J. Benito, F. Urena, and L. Gavete, “Solving Parabolic and Hyperbolic Equations by the Generalized Finite Difference Method,” Journal of Computational and Applied Mathematics, vol. 209, no. 2, pp. 208-233, 2007.
[23] H. Ding, and Y. Zhang, “A New Fourth-Order Compact Finite Difference Scheme for the Two-Dimensional Second-Order Hyperbolic Equation,” Journal of Computational and Applied Mathematics,” vol. 230, no. 2, pp. 626-632, 2009.
[24] D. M. Young, “Iterative Solution of Large Linear Systems,” London: Academic Press, 1971.
[25] W. Hackbusch, “Iterative Solution of Large Sparse Systems of Equations,” Springer-Verlag, 1995.
[26] Y. Saad, “Iterative Methods for Sparse Linear Systems,” International Thomas Publishing, 1996.
[27] A. Saudi, and J. Sulaiman, “Half-Sweep Gauss-Seidel (HSGS) Iterative Method for Robot Path Planning,” In the 3rd Int. Conf. on Informatics and Technology (Informatics09), Kuala Lumpur, pp. 27-28, 2009.
[28] L. H. Ali, J. Sulaiman, and S.R.H. Hashim, “Numerical Solution of Fuzzy Fredholm Integral Equations of Second Kind using Half-Sweep Gauss-Seidel Iteration,” Journal of Engineering Science and Technology, vol. 15, pp. 3303-3313, 2020.
[29] N.A.M. Ali, R. Rahman, J. Sulaiman, and K. Ghazali, “Solutions of Reaction-Diffusion Equations using Similarity Reduction and HSSOR Iteration,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 16, pp. 1430-1438, 2019.
[30] N. S.Mohamad,J. Sulaiman, A.Saudi, and N.F.A. Zainal, “First Order Piecewise Collocation Solution of Fredholm Integral Equation Second Type using Sor Iteration,” In Computational Science and Technology, Singapore: Springer, pp. 395-406, 2021.
[31] R. Rahman, N.A.M Ali, J. Sulaiman, and F. Muhiddin, “Caputo’s Finite Difference Solution of Fractional Two-Point Boundary Value Problems using SOR Iteration,” In AIP Conference Proceedings, vol. 2013, no. 1, pp. 020034, 2018.
[32] M. M. Xu, J. Sulaiman, and L. H. Ali, “Linear Rational Finite Difference Approximation for Second-Order Linear Fredholm Integro-Differential Equations using the Half-Sweep SOR Iterative Method,” International Journal of Engineering Trends and Technology, vol. 69, no. 6, pp. 136-143, 2021.
[33] N.Z.F.M. Radzuan, M.N. Suardi, and J. Sulaiman, “KSOR Iterative Method with Quadrature Scheme for Solving System of Fredholm Integral Equations of Second Kind,” Journal of Fundamental and Applied Sciences, vol. 9, no, 5S, pp. 609-623, 2017.
[34] F.A. Muhiddin, J. Sulaiman, and A. Sunarto, “A Class of Weighted Point Schemes for the Grünwald Implicit Finite Difference Solution of Time-Fractional Parabolic Equations using KSOR Method,” In Journal of Physics: Conference Series, vol. 1298, no. 1, pp. 012001, 2019.
[35] A.A. Dahalan, A. Saudi, and J. Sulaiman, “Path Finding of Static Indoor Mobile Robot via AOR Iterative Method Using Harmonic Potentials,” ZULFAQAR International Journal of Defence Science, Engineering & Technology, vol. 3, no. 2, pp. 18-24, 2020.
[36] A.A. Dahalan, M. S. Muthuvalu, and J. Sulaiman, “Numerical Solutions of Two-Point Fuzzy Boundary Value Problem using HalfSweep Alternating Group Explicit Method,” American Institute of Physics, vol. 1557, no. 1, pp. 103-107, 2013.
[37] A.A. Dahalan, M.S. Muthuvalu, and J. Sulaiman, “Numerical Solutions of Two-Point Fuzzy Boundary Value Problem using Half-Sweep Alternating Group Explicit Method,” In AIP Conference Proceedings, vol. 1557, no. 1, pp. 103-107, 2013.
[38] A.R. Abdullah, “The four Point Explicit Decoupled Group (EDG) Method: A Fast Poisson Solver,” International Journal of Computer Mathematics, vol. 38, no. 1-2, pp. 61-70, 1991.
[39] M.K. Hasan, M. Othman, Z. Abbas, J. Sulaiman, and F. Ahmad, “Parallel Solution of High Speed Low Order FDTD on 2D Free Space Wave Propagation,” In International Conference on Computational Science and Its Applications, Berlin, Heidelberg: Springer, pp. 13- 24, 2007.
[40] M. K. M. Akhir, M. Othman, J. Sulaiman, Z.A. Majid, and M. Suleiman, “The Four Point-EDGMSOR Iterative Method for Solution of 2D Helmholtz Equations,” In International Conference on Informatics Engineering and Information Science, Berlin, Heidelberg:Springer, pp. 218-227, 2011.
[41] R. Rahman, N.A.M Ali, J. Sulaiman, and F. Muhiddin, “Application of the Half-Sweep EGSOR Iteration for Two-Point Boundary Value Problems of Fractional Order,” Advances in Science, Technology and Engineering Systems, vol. 4, no. 2, pp. 237-243, 2019.
[42] N. F. A. Zainal, J. Sulaiman, and M.U. Alibubin, “Application of Half-Sweep SOR Iteration with Nonlocal Arithmetic Discretization Scheme for Solving Burgers’ Equation,” ARPN J. Eng. Appl. Sci., vol. 14, pp. 616-621, 2019.
[43] M. K. M. Akhir, M. Othman, J. Sulaiman, Z. A. Majid,andM. Suleiman, “Half-sweep Modified Successive Overrelaxation for Solving Two-Dimensional Helmholtz Equations,” Australian Journal of Basic and Applied Sciences, vol. 5, no. 12, pp. 3033-3039, 2011.
[44] N.A.M. Ali, J. Sulaiman, A. Saudi, and N.S. Mohamad, “SOR Iterative Method with Wave Variable Transformation for Solving Advection-Diffusion Equations,” AIP Conference Proceeding, vol. 2013, pp. 020036, 2018.
[45] N.A.M. Ali, J. Sulaiman, A. Saudi, and N.S. Mohamad, “Performance of Similarity Explicit Group Iteration for Solving 2D Unsteady Convection-Diffusion Equation,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 23, no. 1, pp. 472-478, 2021.
[46] D.M. Young, “Iterative Solution of Large Linear Systems,” London: Academic Press, 1971.
[47] Y. Saad, “Iterative Methods for Sparse Linear Systems,” 2nd edn. Philadelphia, PA: SIAM, 2003.
[48] J. Sulaiman, M.K. Hasan, M. Othman, and S.A.A. Karim, “Fourth-Order Solutions of Nonlinear Two-Point Boundary Value Problems by Newton-HSSOR Iteration,” AIP Conference Proceeding, vol. 1602, pp. 69-75, 2014.
[49] A.M. Saeed and N.H.M. Ali, “On the Convergence of the Preconditioned Group Rotated Iterative Methods in the Solution of Elliptic PDEs,” Applied Mathematics and Information Sciences, vol. 5, no. 1, pp. 65-73, 2011.
[50] A. Ibrahim and A. R. Abdullah, “Solving the Two Dimensional Diffusion Equation by the Four Point Explicit Decoupled Group (EDG) Iterative Method,” International Journal of Computer Mathematics, vol. 58, pp. 253-263, 1994.
[51] J. Lin, F. Chen, Y. Zhang, and J. Lu, “An Accurate Meshless Collocation Technique for Solving Two-Dimensional Hyperbolic Telegraph Equations in Arbitrary Domains,” Engineering Analysis with Boundary Elements, vol. 108, pp. 372-384, 2019.
[52] J. Lin, F. Chen, Y. Zhang, and J. Lu, “An Accurate Meshless Collocation Technique for Solving Two-Dimensional Hyperbolic Telegraph Equations in Arbitrary Domains,” Engineering Analysis with Boundary Elements, vol. 108, pp. 372-384, 2019.
[53] B.K. Singh and P. Kumar, “An Algorithm Based on a New DQM with Modified Extended Cubic B-Splines for Numerical Study of Two Dimensional Hyperbolic Telegraph Equation,” Alexandria Engineering Journal, vol. 57, no. 1, pp. 175-191, 2018.
[54] A. Sunarto, and J. Sulaiman, “Implementation QSGS Iteration Applied to Fractional Diffusion Equation,” In Journal of Physics: Conference Series, vol. 1363, no. 1, pp. 012086, 2019.
[55] J. Sulaiman, M. Othman, and M.K. Hasan, “Quarter-Sweep Iterative Alternating Decomposition Explicit Algorithm Applied to Diffusion Equations,” International Journal of Computer Mathematics, vol. 81, no. 12, pp. 1559-1565, 2004.
[56] J.V. Lung, and J. Sulaiman, “On Quarter-Sweep Finite Difference Scheme for One-Dimensional Porous Medium Equations,” International Journal of Applied Mathematics, vol. 33, no. 3, pp. 439, 2020.