Using matrix stability for variable telegraph partial differential equation

Mahmut Modanli, Bawar Mohammed Faraj, Faraedoon Waly Ahmed

Abstract


The variable telegraph partial differential equation depend on initial boundary value problem has been studied. The coefficient constant time-space telegraph partial differential equation is obtained from the variable telegraph partial differential equation throughout using Cauchy-Euler formula. The first and second order difference schemes were constructed for both of coefficient constant time-space and variable time-space telegraph partial differential equation. Matrix stability method is used to prove stability of difference schemes for the variable and coefficient telegraph partial differential equation. The variable telegraph partial differential equation and the constant coefficient time-space telegraph partial differential equation are compared with the exact solution. Finally, approximation solution  has been found for both equations. The error analysis table presents the obtained numerical results.

Keywords


Time-space telegraph differential equations, matrix stability, first and second order difference schemes, approximation solution.

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References


Celik, C., & Duman, M.(2012). Crank- Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. Journal of computational physics, 231(4), 1743-1750.

Gorial, I. I. (2011). Numerical methods for fractional reaction-dispersion equation with Riesz space fractional derivative. Engineering and Technology Journal, 29(4), 709-715.

Jafari, H., & Daftardar-Gejji, V. (2006). Solv- ing linear and nonlinear fractional diffusion and wave equations by Adomian decomposition. Applied Mathematics and Computation, 180(2), 488-497.

Karatay, I., Bayramoglu, S¸R., & S¸ahin, A. (2011). Implicit difference approximation for the time fractional heat equation with the nonlocal condition. Applied Numerical Mathematics, 61(12), 1281-1288.

Su, L., Wang, W., & Yang, Z. (2009). Finite difference approximations for the fractional advection-diffusion equation. Physics Letters A, 373(48), 4405-4408.

Tadjeran, C., Meerschaert, M. M., Scheffler, H.P.(2006). A second order accurate numerical approximation for the fractional diffusion equation. Journal of computational physics, 213(1), 205-213.

Nouy, A. (2010). A priori model reduction through proper generalized decomposition for solving time-dependent partial differ- ential equations. Computer Methods in Applied Mechanics and Engineering, 199(23-24), 1603-1626.

Wu, J. (1996). Theory and applications of partial functional differential equations, Springer-Verlag, New York.

Pontryagin, L. S. (2018). Mathematical theory of optimal processes, Routledge, London.

He, J. H. (2008). Recent development of the homotopy perturbation method. Topological methods in nonlinear analysis, 31(2), 205-209.

Holmes, E. E., Lewis, M. A., Banks, J. E., & Veit, R. R. (1994). Partial differential equations in ecology: spatial interactions and population dynamics. Ecology, 75(1), 17-29.

Dehghan, M., Shokri, A. (2008). A numerical method for solving the hyperbolic telegraph equation. Numerical Methods for Partial Differential Equations, 24(4), 1080-1093.

Faraj, B., & Modanli, M. (2017). Using difference scheme method for the numerical solution of telegraph partial differential equation. Journal of Garmian University, 3, 157-163.

Ashyralyev, A., & Modanli, M. (2015). An operator method for telegraph partial differential and difference equations. Boundary Value Problems, 41(1), 1-17.

Ashyralyev, A., & Modanli, M. (2015). Nonlocal boundary value problem for telegraph equations. AIP Conference Proceedings, 1676(1), 020078-1 - 020078-4.

Ashyralyev A, & Modanli, M. (2014). A numerical solution for a telegraph equation. AIP Conference Proceedings, 1611(1), 300-304.

Faraj, B. M. (2018). Difference scheme methods for telegraph partial differential equations. MSc Thesis. Harran University.

Smith, G. D. (1985). Numerical solution of partial differential equations: finite difference methods. Oxford university press.

Richtmyer, R. D., & Morton, K. W. (1994). Difference methods for initial-value problems, Krieger




DOI: http://dx.doi.org/10.11121/ijocta.01.2020.00870

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Copyright (c) 2020 Bawar Mohammed Faraj, Mahmut Modanli, Faraedoon Waly Ahmed

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