Computational Approach via HalfSweep Similarity and SOR Schemes for 2D Hyperbolic Telegraph Equations
Computational Approach via HalfSweep Similarity and SOR Schemes for 2D Hyperbolic Telegraph Equations 



© 2022 by IJETT Journal  
Volume70 Issue9 

Year of Publication : 2022  
Authors : Nur Afza Mat Ali, Jumat Sulaiman, Azali Saudi, Nor Syahida Mohamad 

DOI : 10.14445/22315381/IJETTV70I9P205 
How to Cite?
Nur Afza Mat Ali, Jumat Sulaiman, Azali Saudi, Nor Syahida Mohamad, "Computational Approach via HalfSweep Similarity and SOR Schemes for 2D Hyperbolic Telegraph Equations ," International Journal of Engineering Trends and Technology, vol. 70, no. 9, pp. 4756, 2022. Crossref, https://doi.org/10.14445/22315381/IJETTV70I9P205
Abstract
This paper uses the similarity transformation to get similar solutions to the twodimensional hyperbolic telegraph equation (2D HTE). The similarity solution of 2D HTE is then derived using a rotated fivepoint similarity finite difference (SFD) discretization scheme to obtain the rotated fivepoint SFD approximation equation. Gathering rotated approximation equations generate a linear system with largescale and sparse matrix characteristics. Since then, the linear system has been solved using the halfsweep similarity technique via halfsweep successive overrelaxation (HSSORSFD) iteration. Three numerical examples are presented in this paper to validate the performance of the HSSORSFD iteration in solving 2D HTE. The numerical findings showed that the version of HSSORSFD iteration is the best compared to the standard similarity methods: fullsweep GaussSeidel (FSGSSFD) and fullsweep SOR (FSSORSFD) iterations in terms of iteration number and computational time.
Keywords
Twodimensional hyperbolic telegraph equation, Similarity transformation, Similarity solution, Rotated similarity finite difference method, Similarity halfsweep 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 ThreeDimensional Telegraph Equation to CosmicRay 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. 928, 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. 16, 2013.
[5] P. M. Jordan, and A. Puri, “Digital Signal Propagation in Dispersive Media,” Journal of Applied Physics, vol. 85, no. 3, pp. 12731282, 1999.
[6] B. K. Singh, J. P. Shukla, and M. Gupta, “Study of OneDimensional Hyperbolic Telegraph Equation via a Hybrid Cubic BSpline Differential Quadrature Method,” International Journal of Applied and Computational Mathematics, vol. 7, no. 1, pp. 117, 2021.
[7] G. Arora, and V. Joshi, “Comparison of Numerical Solution of 1D Hyperbolic Telegraph Equation using BSpline and Trigonometric BSpline by Differential Quadrature Method,” Indian Journal of Science and Technology, vol. 9, no. 45, pp. 18, 2016.
[8] R. C. Mittal, and R. A. Bhatia, “Numerical Study of Twodimensional Hyperbolic Telegraph Equation by Modified BSpline Differential Quadrature Method,” Applied Mathematics and Computation, vol. 244, no. 1, pp. 97697, 2014.
[9] B. K. Singh, and P. Kumar, “An Algorithm Based on a New DQM with Modified Extended Cubic BSplines for Numerical Study of Two Dimensional Hyperbolic Telegraph Equation,” Alexandria Engineering Journal, vol. 57, no. 1, pp. 175191, 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. 1629, 2013.
[11] S. Jator, “Block Unification Scheme for Elliptic, Telegraph, and SineGordon Partial Differential Equations,” American Journal of Computational Mathematics, vol. 5, no. 1, pp. 175185, 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. 274294, 2018.
[13] A. N. Ceretani, and D. A.Tarzia, “Similarity Solution for a TwoPhase OneDimensional Stefan Problem with a Convective Boundary Condition and a Mushy Zone Model,” Computational and Applied Mathematics, vol. 37, no. 2, pp. 22012217, 2018.
[14] N.A.M. Ali, J. Sulaiman, A. Saudi, and N.S. Mohamad, “The Similarity Finite Difference Solutions for TwoDimensional Parabolic Partial Differential Equations via SOR Iteration,” In Computational Science and Technology, vol. 724, no. 1, pp. 515526, 2021.
[15] Y. Wang, L. Lin, and J. Li, “A similarity Solution for Undrained Expansion of a Cylindrical Cavity in K 0Consolidated Anisotropic Soils,” Geomechanics and Engineering, vol. 25, no. 4, pp. 303315, 2021.
[16] N.A.M. Ali, R. Rahman, J. Sulaiman, and K. Ghazali, “Numerical Solutions of Unsteady AdvectionDiffusion 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 2D and 3D using Generalized Finite Difference Method (GFDM),” Engineering Analysis with Boundary Elements, vol. 112, pp. 1324, 2020.
[18] M. Aslefallah, and D. Rostamy, “Application of the singular Boundary Method to the TwoDimensional Telegraph Equation on Arbitrary Domains,” Journal of Engineering Mathematics, vol. 118, no. 1, pp. 114, 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. 19261942, 2016.
[20] V. Devi, R.K. Maurya, S. Singh, and V.K. Singh, “Lagrange’s Operational Approach for the Approximate Solution of TwoDimensional 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. 2127, 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. 208233, 2007.
[23] H. Ding, and Y. Zhang, “A New FourthOrder Compact Finite Difference Scheme for the TwoDimensional SecondOrder Hyperbolic Equation,” Journal of Computational and Applied Mathematics,” vol. 230, no. 2, pp. 626632, 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,” SpringerVerlag, 1995.
[26] Y. Saad, “Iterative Methods for Sparse Linear Systems,” International Thomas Publishing, 1996.
[27] A. Saudi, and J. Sulaiman, “HalfSweep GaussSeidel (HSGS) Iterative Method for Robot Path Planning,” In the 3rd Int. Conf. on Informatics and Technology (Informatics09), Kuala Lumpur, pp. 2728, 2009.
[28] L. H. Ali, J. Sulaiman, and S.R.H. Hashim, “Numerical Solution of Fuzzy Fredholm Integral Equations of Second Kind using HalfSweep GaussSeidel Iteration,” Journal of Engineering Science and Technology, vol. 15, pp. 33033313, 2020.
[29] N.A.M. Ali, R. Rahman, J. Sulaiman, and K. Ghazali, “Solutions of ReactionDiffusion Equations using Similarity Reduction and HSSOR Iteration,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 16, pp. 14301438, 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. 395406, 2021.
[31] R. Rahman, N.A.M Ali, J. Sulaiman, and F. Muhiddin, “Caputo’s Finite Difference Solution of Fractional TwoPoint 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 SecondOrder Linear Fredholm IntegroDifferential Equations using the HalfSweep SOR Iterative Method,” International Journal of Engineering Trends and Technology, vol. 69, no. 6, pp. 136143, 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. 609623, 2017.
[34] F.A. Muhiddin, J. Sulaiman, and A. Sunarto, “A Class of Weighted Point Schemes for the Grünwald Implicit Finite Difference Solution of TimeFractional 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. 1824, 2020.
[36] A.A. Dahalan, M. S. Muthuvalu, and J. Sulaiman, “Numerical Solutions of TwoPoint Fuzzy Boundary Value Problem using HalfSweep Alternating Group Explicit Method,” American Institute of Physics, vol. 1557, no. 1, pp. 103107, 2013.
[37] A.A. Dahalan, M.S. Muthuvalu, and J. Sulaiman, “Numerical Solutions of TwoPoint Fuzzy Boundary Value Problem using HalfSweep Alternating Group Explicit Method,” In AIP Conference Proceedings, vol. 1557, no. 1, pp. 103107, 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. 12, pp. 6170, 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 PointEDGMSOR Iterative Method for Solution of 2D Helmholtz Equations,” In International Conference on Informatics Engineering and Information Science, Berlin, Heidelberg:Springer, pp. 218227, 2011.
[41] R. Rahman, N.A.M Ali, J. Sulaiman, and F. Muhiddin, “Application of the HalfSweep EGSOR Iteration for TwoPoint Boundary Value Problems of Fractional Order,” Advances in Science, Technology and Engineering Systems, vol. 4, no. 2, pp. 237243, 2019.
[42] N. F. A. Zainal, J. Sulaiman, and M.U. Alibubin, “Application of HalfSweep SOR Iteration with Nonlocal Arithmetic Discretization Scheme for Solving Burgers’ Equation,” ARPN J. Eng. Appl. Sci., vol. 14, pp. 616621, 2019.
[43] M. K. M. Akhir, M. Othman, J. Sulaiman, Z. A. Majid,andM. Suleiman, “Halfsweep Modified Successive Overrelaxation for Solving TwoDimensional Helmholtz Equations,” Australian Journal of Basic and Applied Sciences, vol. 5, no. 12, pp. 30333039, 2011.
[44] N.A.M. Ali, J. Sulaiman, A. Saudi, and N.S. Mohamad, “SOR Iterative Method with Wave Variable Transformation for Solving AdvectionDiffusion 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 ConvectionDiffusion Equation,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 23, no. 1, pp. 472478, 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, “FourthOrder Solutions of Nonlinear TwoPoint Boundary Value Problems by NewtonHSSOR Iteration,” AIP Conference Proceeding, vol. 1602, pp. 6975, 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. 6573, 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. 253263, 1994.
[51] J. Lin, F. Chen, Y. Zhang, and J. Lu, “An Accurate Meshless Collocation Technique for Solving TwoDimensional Hyperbolic Telegraph Equations in Arbitrary Domains,” Engineering Analysis with Boundary Elements, vol. 108, pp. 372384, 2019.
[52] J. Lin, F. Chen, Y. Zhang, and J. Lu, “An Accurate Meshless Collocation Technique for Solving TwoDimensional Hyperbolic Telegraph Equations in Arbitrary Domains,” Engineering Analysis with Boundary Elements, vol. 108, pp. 372384, 2019.
[53] B.K. Singh and P. Kumar, “An Algorithm Based on a New DQM with Modified Extended Cubic BSplines for Numerical Study of Two Dimensional Hyperbolic Telegraph Equation,” Alexandria Engineering Journal, vol. 57, no. 1, pp. 175191, 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, “QuarterSweep Iterative Alternating Decomposition Explicit Algorithm Applied to Diffusion Equations,” International Journal of Computer Mathematics, vol. 81, no. 12, pp. 15591565, 2004.
[56] J.V. Lung, and J. Sulaiman, “On QuarterSweep Finite Difference Scheme for OneDimensional Porous Medium Equations,” International Journal of Applied Mathematics, vol. 33, no. 3, pp. 439, 2020.