Document Type : Original Paper

**Authors**

Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran

**Abstract**

In this article, we study the linear inverse problem for approximating a timewise-dependent

function in the second-order hyperbolic equation. To solve the problem, information such as Neumann

boundary conditions along with an integral condition and initial conditions at the initial moment and the

final instant have been provided. In the first step, we show that this problem has a unique solution. Then,

we change the main problem into a new one and then we present the spectral approximation based on

the Ritz-collocation method to recover the unknown functions. Discretization of the problem by using

the presented technique leads to a linear system of algebraic equations, which Tikhonov’s regularization

method is used to obtain stable solutions. The results of the numerical simulation confirm the high accuracy

and stability of the approximate solution.

function in the second-order hyperbolic equation. To solve the problem, information such as Neumann

boundary conditions along with an integral condition and initial conditions at the initial moment and the

final instant have been provided. In the first step, we show that this problem has a unique solution. Then,

we change the main problem into a new one and then we present the spectral approximation based on

the Ritz-collocation method to recover the unknown functions. Discretization of the problem by using

the presented technique leads to a linear system of algebraic equations, which Tikhonov’s regularization

method is used to obtain stable solutions. The results of the numerical simulation confirm the high accuracy

and stability of the approximate solution.

**Keywords**

- Inverse telegraph equation
- Existence and Uniqueness of solution
- Tikhonov regularization
- Spectral method
- Discretization

**Main Subjects**

[1] W. Alharbi, S. Petrovskii, Critical domain problem for the reaction-telegraph equation model of population dynamics, Mathematics, 6(4) (2018) 59, https://doi.org/10.3390/math6040059.

[2] N. Berwal, D. Panchal and C. L. Parihar, Haar wavelet method for numerical solution of telegraph equations, Italian Journal of Pure and Applied Mathematics, 33 (2013) 317-328.

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

[4] T. M. Elzaki, E. Hilal and J. S. Arabia, Analytical solution for telegraph equation by modified of Sumudu transform “Elzaki transform”, Mathematical Theory and Modeling, 2 (2012) 104-111.

[5] P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve , SIAM Review, 34 (1992) 561-580 .

[6] B. T. Johanson, D. Lesnic and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan problem, Applied Mathematical Modelling, 35 (2011) 4367-4378 .

[7] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, 2011.

[8] V. Isakov, Inverse Problems for Partial Differential Equations, Springer, 2006.

[9] A. I. Kozhanov, R. Safiullova, Determination of parameters in telegraph equation, Ufa Mathematical Journal, 9 (2017) 62-74.

[10] W. Liao, A computational method to estimate the unknown coefficient in a wave equation using boundary measurements, Inverse Problems in Science and Engineering, 19 (2011) 855-887.

[11] R. C. Mittal, R. Bhatia, Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method, Applied Mathematics and Computation, 220 (2011) 496-506.

[12] K. Rashedi, A spectral method based on Bernstein orthonormal basis functions for solving an inverse Roseneau equation, Computational & applied Mathematics, 41 (2022) (In Press).

[13] K. Rashedi, Reconstruction of a time-dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method, Mathematical Methods in the Applied Sciences, 46 (2023) 1752-1771.

[14] K. Rashedi, A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials, Mathematical Methods in the Applied Sciences, 44 (2021) 12980-12997.

[15] K. Rashedi and H. Adibi and M. Dehghan, Application of the Ritz-Galerkin method for recovering the spacewise-coefficients in the wave equation, Computer and Mathematics with Applications, 65 (2013) 1990-2008.

[2] N. Berwal, D. Panchal and C. L. Parihar, Haar wavelet method for numerical solution of telegraph equations, Italian Journal of Pure and Applied Mathematics, 33 (2013) 317-328.

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

[4] T. M. Elzaki, E. Hilal and J. S. Arabia, Analytical solution for telegraph equation by modified of Sumudu transform “Elzaki transform”, Mathematical Theory and Modeling, 2 (2012) 104-111.

[5] P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve , SIAM Review, 34 (1992) 561-580 .

[6] B. T. Johanson, D. Lesnic and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan problem, Applied Mathematical Modelling, 35 (2011) 4367-4378 .

[7] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, 2011.

[8] V. Isakov, Inverse Problems for Partial Differential Equations, Springer, 2006.

[9] A. I. Kozhanov, R. Safiullova, Determination of parameters in telegraph equation, Ufa Mathematical Journal, 9 (2017) 62-74.

[10] W. Liao, A computational method to estimate the unknown coefficient in a wave equation using boundary measurements, Inverse Problems in Science and Engineering, 19 (2011) 855-887.

[11] R. C. Mittal, R. Bhatia, Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method, Applied Mathematics and Computation, 220 (2011) 496-506.

[12] K. Rashedi, A spectral method based on Bernstein orthonormal basis functions for solving an inverse Roseneau equation, Computational & applied Mathematics, 41 (2022) (In Press).

[13] K. Rashedi, Reconstruction of a time-dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method, Mathematical Methods in the Applied Sciences, 46 (2023) 1752-1771.

[14] K. Rashedi, A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials, Mathematical Methods in the Applied Sciences, 44 (2021) 12980-12997.

[15] K. Rashedi and H. Adibi and M. Dehghan, Application of the Ritz-Galerkin method for recovering the spacewise-coefficients in the wave equation, Computer and Mathematics with Applications, 65 (2013) 1990-2008.

[16] K. Rashedi and F. Baharifard and A. Sarraf, Stable recovery of a space-dependent force function in a one-dimensional wave equation via Ritz collocation method, Journal of Mathematical Modeling, 10 (2022) 463-480.

[17] F. Torabi, R. Pourgholi, Numerical solution for solving inverse telegraph equation by extended cubic B-spline, International Journal of Nonlinear Analysis and Applications, (2022) In Press, https ://ijnaa.semnan.ac.ir/article7016.html

[18] T. Wei, M. Li, High order numerical derivatives for one-dimensional scattered noisy data, Applied Mathematics and Computation, 175 (2006) 1744-1759.

[19] J. Wen, M. Yamamoto, Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation, Inverse Problems in Science and Engineering, 17 (2009) 551–567.

[20] J. Wen and M. Yamamoto and T. Wei, Simultaneous determination of a time-dependent heat source and the initial temperature in an inverse heat conduction problem, Inverse Problems in Science and Engineering, 21 (2013) 485-499.

[21] S. A. Yousefi, Finding a control parameter in a onedimensional parabolic inverse problem by using the Bernstein Galerkin method, Inverse Problems in Science and Engineering, 17 (2009) 821-828.

[17] F. Torabi, R. Pourgholi, Numerical solution for solving inverse telegraph equation by extended cubic B-spline, International Journal of Nonlinear Analysis and Applications, (2022) In Press, https ://ijnaa.semnan.ac.ir/article7016.html

[18] T. Wei, M. Li, High order numerical derivatives for one-dimensional scattered noisy data, Applied Mathematics and Computation, 175 (2006) 1744-1759.

[19] J. Wen, M. Yamamoto, Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation, Inverse Problems in Science and Engineering, 17 (2009) 551–567.

[20] J. Wen and M. Yamamoto and T. Wei, Simultaneous determination of a time-dependent heat source and the initial temperature in an inverse heat conduction problem, Inverse Problems in Science and Engineering, 21 (2013) 485-499.

[21] S. A. Yousefi, Finding a control parameter in a onedimensional parabolic inverse problem by using the Bernstein Galerkin method, Inverse Problems in Science and Engineering, 17 (2009) 821-828.

July 2023

Pages 92-105

**Receive Date:**12 March 2023**Revise Date:**07 June 2023**Accept Date:**19 June 2023