    April  2021, 8(2): 129-138. doi: 10.3934/jdg.2020029

## New solutions of hyperbolic telegraph equation

 1 Department of Mathematics, Firat University, Elazig, Turkey 2 Department of Mathematics, University of Bonab, Bonab, Iran 3 Dept. of Mathematical Engineering, Yildiz Technical Univ., Istanbul, Turkey 4 Poznan University of Technology, Poznan, Poland, IAM, Metu, Ankara, Turkey

* Corresponding author: mehmetaliakinlar@gmail.com (Mehmet Ali Akinlar)

Received  April 2020 Revised  July 2020 Published  April 2021 Early access  September 2020

We present a new method based on unification of fictitious time integration (FTI) and group preserving (GP) methods. The GP method is applied in numerically discretized ordinary differential equations obtained from application of FTI method to a given partial differential equation (PDE). The algorithm is applied to hyperbolic telegraph equation and utilizes the Cayley transformation and the Pade approximations in the Minkowski space. It avoids unauthentic solutions and ghost fixed points which is one of the advantages of the present method over other related numerical methods in the literature. The technique is tested on three specific examples for various parameter values appearing in the telegraph equation and discretization steps. Such solutions of the telegraph equation are obtained first time in this paper. Illustrative figures are provided. Efficiency of the method is determined by an error analysis which is achieved by comparing numerical solutions with exact solutions.

Citation: Mustafa Inc, Mohammad Partohaghighi, Mehmet Ali Akinlar, Gerhard-Wilhelm Weber. New solutions of hyperbolic telegraph equation. Journal of Dynamics and Games, 2021, 8 (2) : 129-138. doi: 10.3934/jdg.2020029
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##### References:
  S. Abbasbandy and M. Hashemi, Group preserving scheme for the Cauchy problem of the Laplace equation, Engineering Analysis with Boundary Elements, 35 (2011), 1003-1009.  doi: 10.1016/j.enganabound.2011.03.010.   A. Al-Fayadh and H. Khawwan, Variational iteration transform method for solving Burger and coupled Burger's equations, ARPN J. Eng. Appl. Sci., 12 (2017), 6926-6932. A. Al-Fayadh and N. Hazim, Implementation of wavelet based transform for numerical solutions of partial differential equations, IOSR J. Math., 13 (2017), 30-34. I. Baltas, A. Xepapadeas and A. N. Yannacopoulos, Robust portfolio decisions for financial institutions, Journal of Dynamics & Games, 5 (2018), 61-94.  doi: 10.3934/jdg.2018006.   M. Dehghan and A. Shokri, A numerical method for solving the hyperbolic telegraph equation, Numer. Methods Partial Differ. Equ., 24 (2008), 1080-1093.  doi: 10.1002/num.20306.   M. S. Hashemi, D. Baleanu, M. Partohaghighi and E. Darvishi, Solving the time fractional diffusion equation using Lie group integrator, Thermal Science, 19 (2015), S77–S83. doi: 10.2298/TSCI15S1S77H.  M. S. Hashemi, D. Baleanu and M. Partohaghighi, A lie group approach to solve the fractional Poisson equation, Rom. J. Phys., 60 (2015), 1289-1297. M. S. Hashemi, M. Inc, E. Karatas and E. Darvishi, Numerical treatment on one-dimensional hyperbolic telegraph equation by the method of line-group preserving scheme, Phys. J. Plus, 134 (2019), Article number: 153. doi: 10.1140/epjp/i2019-12500-y.  T. Hoheisel, M. Laborde and A. Oberman, A regularization interpretation of the proximal point method for weakly convex functions, Journal of Dynamics & Games, 7 (2020), 79-96.  doi: 10.3934/jdg.2020005.  M. Inc, A. I. Aliyu, A. Yusufa and D. Baleanu, Combined optical solitary waves and conservation laws for nonlinear Chen–Lee–Liu equation in optical fibers, Optik, 158 (2018), 297-304. C.-S. Liu, Solving an inverse Sturm-Liouville problem by a Lie-group method, Boundary Value Problems, 2008 (2008), Art. ID 749865, 18 pp. doi: 10.1155/2008/749865.   C.-S. Liu, The Fictitious time integration method to solve the space and time-fractional Burgers equations, CMC, 15 (2010), 221-240. C.-S. Liu, A group preserving scheme for Burgers equation with very large Reynolds number, CMES: Computer Modeling in Engineering & Sciences, 12 (2006), 197-211.  C.-S. Liu, An efficient backward group preserving scheme for the backward in time Burgers equation, CMES: Computer Modeling in Engineering & Sciences, 12 (2006), 55-65.  A. Meiappane, V. P. Venkataesan and M. J. Prabavadhi, On analytical methods for solving Poisson equation, Sch. J. Res. Math. Comput. Sci., 1 (2016), 37-43. S. T. Mohyud-Din, M. A. Noor and K. I. Noor, Modified variational iteration method for solving Sine-Gordon equations, World Appl. Sci. J., 6 (2009), 999-1004. M. Partohaghighi, M. Inc, D. Baleanu and S. P. Dmoshokoa, Fictitious time integration method for solving the time fractional gas dynamic equation, Thermal Science, (2019), 1–11. A. Saadatmandi and M. Dehghan, Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method, Numer. Methods Partial Differ. Equ., 26 (2010), 239-252.  doi: 10.1002/num.20442.   X. Wang and S. Atluri, A unification of the concepts of the variational iteration, Adomian decomposition and Picard iteration methods and a local variational iteration method, Tech Science Press, 111 (2016), 567–585. V. H. Weston and S. He, Wave splitting of the telegraph equation in R3 and its application to inverse scattering, Inverse Problems, 9 (1993), 789-812.  doi: 10.1088/0266-5611/9/6/013.    Exact and numerical solutions and error for $T = 0.05,\,\, m = n = 25,\,\, \zeta = 60000\,\,\kappa = 0.1$ and $u_i^j(0) = 0.1$ and $\Delta \xi = 3/10000000000$ for Ex.1. Exact and numerical solutions and error for $T = 0.05,\, m = m = 30,\, \zeta = 8000,\,\kappa = 0.1$ and $u_i^j(0) = 0.001$ and $\Delta \xi = 3/10000000000$ for Ex.2. Exact and numerical solutions and error for $T = 0.05,\, m = n = 40,\, \zeta = 7995,\,\kappa = 0.1$ and $u_i^j(0) = 0.001$ and $\Delta\xi = 3/10000000000$ for Ex.2. Exact solution, numerical solution and error for $T = 0.05,\, m = n = 25,\, \zeta = 5995,\,\kappa = 0.01$ and $u_i^j(0) = 0.01$ and $\Delta \xi = 3/10000000000$ for Ex. 3.
Solution and error values for $m=n=25$ and $u_i^j(0)=0.1$ for Ex.1
 (x, t) Numerical Exact Error (-2, 0) 0.1353 0.1353 8.0756e-14 (-1, 0.01) 0.3641 0.3641 2.5085e-13 (0, 0.02) 0.9773 0.9773 9.6537e-13 (1, 0.03) 2.6346 2.6346 7.8534e-12 (2, 0.04) 7.0875 7.0875 2.2148e-10
 (x, t) Numerical Exact Error (-2, 0) 0.1353 0.1353 8.0756e-14 (-1, 0.01) 0.3641 0.3641 2.5085e-13 (0, 0.02) 0.9773 0.9773 9.6537e-13 (1, 0.03) 2.6346 2.6346 7.8534e-12 (2, 0.04) 7.0875 7.0875 2.2148e-10
Numerical and exact solutions and error values for $m=n=30$, and $u_i^j(0)=0 .001$ for Ex.2
 (x, t) Numerical Exact Error (0, 0) 0 0 0 (2, 0.01) 0.9289 0.9289 1.9657e-13 (4.02) -0.6876 -0.6876 9.7778e-14 (6, 0.03) -0.4197 -0.4197 3.7708e-14 (8, 0.04) 0.9628 0.9628 2.2468e-13 (10, 0.05) 3.6693e-16 3.6693e-16 2.9738e-44
 (x, t) Numerical Exact Error (0, 0) 0 0 0 (2, 0.01) 0.9289 0.9289 1.9657e-13 (4.02) -0.6876 -0.6876 9.7778e-14 (6, 0.03) -0.4197 -0.4197 3.7708e-14 (8, 0.04) 0.9628 0.9628 2.2468e-13 (10, 0.05) 3.6693e-16 3.6693e-16 2.9738e-44
Solutions with error values for $m=n=40$ and $u_i^j(0)=0.001$ for Ex. 2
 (x, t) Numerical Exact Error (-10, 0) -3.6739e-16 -3.6739e-16 2.9813e-44 (-5, 0.01) 0.9350 0.9350 4.9081e-12 (0.02) -0.9924 -0.9924 1.3054e-11 (5, 0.03) -0.9346 -0.9346 9.7228e-12 (10, 0.04) 3.6708e-16 3.6708e-16 2.9763e-44
 (x, t) Numerical Exact Error (-10, 0) -3.6739e-16 -3.6739e-16 2.9813e-44 (-5, 0.01) 0.9350 0.9350 4.9081e-12 (0.02) -0.9924 -0.9924 1.3054e-11 (5, 0.03) -0.9346 -0.9346 9.7228e-12 (10, 0.04) 3.6708e-16 3.6708e-16 2.9763e-44
Solutions with error values for $m=n=25$, $u_i^j(0)=0.001$ for Ex.3
 (x, t) Numerical Exact Error (-2, 0) -0.9093 -0.9093 2.6858e-13 (-1, 0.01) -0.8328 -0.8328 6.0082e-12 (0, 0.02) 0 0 3.1758e-18 (1, 0.03) 0.8173 0.8173 1.3564e-11 (2, 0.04) 0.8740 0.8740 2.4814e-13
 (x, t) Numerical Exact Error (-2, 0) -0.9093 -0.9093 2.6858e-13 (-1, 0.01) -0.8328 -0.8328 6.0082e-12 (0, 0.02) 0 0 3.1758e-18 (1, 0.03) 0.8173 0.8173 1.3564e-11 (2, 0.04) 0.8740 0.8740 2.4814e-13
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