Abstract
In the present study, a new implicit absolute stable difference scheme (DS) for an approximate solution of the time-dependent source identification problem (SIP) for the telegraph equation (TE) is presented. The stability of difference problem is established. In applications of abstract results in a Hilbert space with a self-adjoint positive definite operator (SAPDO), theorems on stability estimates for the solution of DSs for approximate solutions of the multidimensional time-dependent SIPs for telegraph equations are obtained. Finally, these DSs are tested on stability in both two- and three-dimensional examples with different boundary conditions and some computational results are illustrated.
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References
Anikonov YE (1995) Analytical representations of solutions to multidimensional inverse problems for evolutionary equations. J Inverse Ill-Posed Probl 3(4):259–267
Anikonov YE, Neshchadim M (2012) On analytical methods in the theory of inverse problems for hyperbolic equations. II. J Appl Ind Math 6(1):6–11
Ashyraliyev M (2021) On hyperbolic-parabolic problems with involution and Neumann boundary condition. Int J Appl Math 34(2):363–376
Ashyralyev A, Al-Hammouri A (2021) Stability of the space identification problem for the elliptic-telegraph differential equation. Math Methods Appl Sci 44(1):945–959
Ashyralyev A, Cekic F (2016) On source identification problem for telegraph differential equations. Differ Differ Equ Appl 245:39–50
Ashyralyev A, Emharab F (2019) Source identification problems for hyperbolic differential and difference equations. J Inverse Ill-Posed Probl 27(3):301–315
Ashyralyev A, Modanli M (2015) An operator method for telegraph partial differential and difference equations. Bound Value Probl 2015(1):1–17
Ashyralyev A, Ashyraliyev M, Ashyralyyeva M (2020) Identification problem for telegraph-parabolic equations. Comput Math Math Phys 60(8):1294–1305
Ashyralyyev C (2017) Numerical solution to Bitsadze–Samarskii type elliptic overdetermined multipoint NBVP. Bound Value Probl 2017(74):1–22
Ashyralyev A, Sobolevskii PE (2004) New difference schemes for partial differential equations. Birkhauser Verlag, Basel, Boston, Berlin
Askerov I (2022) On a determination of the boundary function in the initial boundary value problem for the second order hyperbolic equation. J Contemp Appl Math 12(1):45–50
Aslefallah M, Rostamy D (2019) Application of the singular boundary method to the two-dimensional telegraph equation on arbitrary domains. J Eng Math 118(1):1–14
Avdonin S, Nicaise S (2015) Source identification problems for the wave equation on graphs. Inverse Probl 31(9):095007
Azizbayov EI (2020) On the unique recovery of time-dependent coefficient in a hyperbolic equation from nonlocal data. UPB Sci Bull Ser A 82:171–182
Chang S, Weston VH (1997) On the inverse problem of the 3D telegraph equation. Inverse Probl 13(5):1207–1222
Ferreira M, Rodrigues MM, Vieira N (2017) Fundamental solution of the multidimensional time fractional telegraph equation. Fract Calc Appl Anal 20(4):868–894
Hafez R, Youssri Y (2020) Shifted Jacobi collocation scheme for multidimensional time-fractional order telegraph equation. Iran J Numer Anal Optim 10(1):195–223
Isgandarova GN (2015) On an inverse boundary value problem with time non-local conditions for one-dimensional hyperbolic equation. Trans NAS Azerbaijan Issue Math 35(4):95–101
Kabanikhin SI, Krivorotko OI (2015) Identification of biological models described by systems of nonlinear differential equations. J Inverse Ill-Posed Probl 23(5):519–527
Kal’menov TS, Sadybekov MA (2017) On a Frankl-type problem for a mixed parabolic-hyperbolic equation. Sib Math J 58(2):227–231
Kozhanov A, Safiullova R (2017) Determination of parameter in telegraph equation. Ufa Math J 9:62–75
Novickij J, Stikonas A (2014) On the stability of a weighted finite difference scheme for wave equation with nonlocal boundary conditions. Nonlinear Anal Model Control 19(3):460–475
Saadatmandi A, Dehghan M (2010) Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method. Numer Methods Partial Differ Equ: Int J 26(1):239–252
Sadybekov M, Oralsyn G, Ismailov M (2018) Determination of a time-dependent heat source under not strengthened regular boundary and integral overdetermination conditions. Filomat 32:809–814
Sobolevskii PE (1975) Difference methods for the approximate solution of differential equations. Voronezh State University Press, Voronezh, Russia
Tuan NH, Au VV, Can NH (2018) Regularization of initial inverse problem 235 for strongly damped wave equation. Appl Anal 97(1):69–88
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The publication has been prepared with the support of the “RUDN University Program 5-100” and published under target program BR05236656 of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan.
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Ashyralyev, A., Al-Hazaimeh, H. & Ashyralyyev, C. Absolute stability of a difference scheme for the multidimensional time-dependently identification telegraph problem. Comp. Appl. Math. 42, 333 (2023). https://doi.org/10.1007/s40314-023-02478-5
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DOI: https://doi.org/10.1007/s40314-023-02478-5