In the present paper, a time-dependent source identification problem for a one dimensional delay hyperbolic equation with Dirichlet condition is studied. Operator-functions generated by the positive operator are considered. Theorems on the stability estimates for the solution of this problem are established. The first order of accuracy difference scheme for this source identification problem is presented. Numerical analysis and discussions are presented.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 35
Issue: 3
Year: 2022

DOI: 10.12732/ijam.v35i3.9

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  1. [1] G. Di. Blasio and A. Lorenzi, Identification problems for parabolic delay differential equations with measurement on the boundary, Journal of Inverse and Ill-Posed Problems, 15, No 7 (2007), 709-734.
  2. [2] I. Orazov and M.A. Sadybekov, On a class of problems of determining the temperature and density of heat source given initial and final temperature, Siberian Mathematical Journal, 53 (2012), 146-151.
  3. [3] M.A. Sadybekov, G. Dildabek and M.B. Ivanova, On an inverse problem of reconstructing a heat conduction process from nonlocal data, Advances in Mathematical Physics (2018), #8301656.
  4. [4] M.A. Sadybekov, G. Oralsyn and M. Ismailov, Determination of a timedependent heat source under not strengthened regular boundary and integral overdetermination conditions, Filomat, 32, No 3 (2018), 809-814.
  5. [5] S. Saitoh, V.K. Tuan and M. Yamamoto, Reverse convolution inequalities and applications to inverse heat source problems, J. of Inequalities in Pure and Applied Mathematics, 5 (2002), 80-91.
  6. [6] K. Sakamoto and M. Yamamoto, Initial-boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.
  7. [7] S.I. Kabanikhin, Method for solving dynamic inverse problems for hyperbolic equations, J. Inverse Problems, 12 (2014), 493-517.
  8. [8] A.A. Samarskii and P.N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Ser. Inverse and Ill-Posed, Problems, Walter de Gruyter, Berlin-New York (2007).
  9. [9] A. Ashyralyev and D. Agirseven, On source identification problem for a delay parabolic equation, Nonlinear Analysis: Modelling and Control, 19, No 3 (2014), 335-349.
  10. [10] A. Ashyralyev and C. Ashyralyyev, On the problem of determining the parameter of an elliptic equation in a Banach space, Nonlinear Analysis Modelling and Control, 3 (2014), 350-366.
  11. [11] A. Ashyralyev, D. Agirseven and R.P. Agarwal, Stability estimates for delay parabolic differential and difference equations, Appl. Comput. Math., 19 (2020), 175-204.
  12. [12] A. Ashyralyev and A. Al-Hammouri, Stability of the space identification problem for the elliptic-telegraph differential equation, Mathematical Methods in the Applied Sciences, 44, No 1 (2020), 945-959.
  13. [13] A. Ashyralyev and F. Emharab, Source identification problems for hyperbolic differential and difference equations, Journal of Inverse and Ill-posed Problems, 27, No 3 (2019), 301-315.
  14. [14] F. Emharab, Source Identification Problems for Hyperbolic Differential and Difference Equations, PhD Thesis, Near East University, Nicosia, 135 (2019).
  15. [15] A. Ashyralyev, A. Al-Hammouri and C. Ashyralyyev, On the absolute stable difference scheme for the space-wise dependent source identification problem for elliptic-telegraph equation, Numerical Methods for Partial Differential Equations, 37, No 2 (2021), 962-986.
  16. [16] Ahmad Mohammad Salem Al-Hammauri, The Source Identification Problem for Elliptic-Telegrah Equations, PhD Thesis, Near East University, Nicosia (2020).
  17. [17] A. Ashyralyev and M. Urun, Time-dependent source identification Schrodinger type problem, International Journal of Applied Mathematics, 34, No 2 (2021), 297-310; DOI: 10.12732/ijam.v34i2.7.
  18. [18] A.S. Erdogan, Numerical Solution of Parabolic Inverse Problem with an Unknown Source Function, PhD Thesis, Y´yld´yz Technical Universty, Istanbul, 112 (2010).
  19. [19] C. Ashyralyyev, Stability estimates for solution of Neumann-type overdetermined elliptic problem, Numerical Functional Analysis and Optimization, 38, No 10 (2017), 1226-1243.
  20. [20] M. Ashyraliyev, M.A. Ashyralyyeva and A. Ashyralyev, A note on the hyperbolic-parabolic identification problem with involution and Dirichlet boundary condition, Bulletin of the Karaganda University-Mathematics, 99, No 3 (2020), 120-129.
  21. [21] M. Ashyraliyev, On hyperbolic-parabolic problems with involution and Neumann boundary condition, International Journal of Applied Mathematics, 34, No 2 (2021), 363-376; DOI: 10.12732/ijam.v34i2.12.
  22. [22] A. Ashyralyev, M. Ashyraliyev and M.A. Ashyralyyeva, A note on the hyperbolic-parabolic identification problem with involution and Dirichlet boundary condition, Computational Mathematics and Mathematical Physics, 60, No 8 (2020), 1294-1305.
  23. [23] A. Ashyralyev and A.S. Erdogan, Well-posedness of the right-hand side identification problem for a parabolic equation, Ukrainian Mathematical Journal, 2 (2014), 165-177.
  24. [24] A. Ashyralyev and M. Urun, On the Crank-Nicholson difference scheme for the time-dependent source identification problem, Bulletin of the Karaganda University Mathematics, 99, No 2 (2021), 35-40.
  25. [25] R.R. Ashurov and M.D. Shakarova, Time-dependent source identification problem for fractional Shr¨odinger type equations, Lobachevskii Journal of Mathematics, 43 (2022), 1053-1064.
  26. [26] A.N. Al-Mutib, Stability properties of numerical methods for solving delay differential equations, J. Comput. and Appl. Math., 10, No 1 (1984), 71-79.
  27. [27] A. Ashyralyev and H. Akca, Stability estimates of difference schemes for neutral delay differential equations, Nonlinear Analysis: Theory, Methods and Applications, 44, No 4 (2001), 443-452.
  28. [28] A. Ashyralyev and P.E. Sobolevskii, On the stability of the delay differential and difference equations, Abstract and Applied Analysis, 6, No 5 (2001), 267-297.
  29. [29] L. Torelli, Stability of numerical methods for delay differential equations, J. Comput. and Appl. Math., 25 (1989), 15-26.
  30. [30] H. Musaev, The Cauchy problem for degenerate parabolic convolution equation, TWMS J. Pure Appl. Math., 12 (2021), 278-288.
  31. [31] A. Bellen, Z. Jackiewicz and M. Zennaro, Stability analysis of one-step methods for neutral delay-differential equations, Numer. Math., 52, No 6 (1988), 605-619.
  32. [32] A. F. Yenicerioglu and S. Yalcinbas, On the stability of the second-order delay differential equations with variable coefficients, Applied Mathematics and Computation, 152, No 3 (2004), 667-673.
  33. [33] A. F. Yenicerioglu, Stability properties of second order delay integrodifferential equations, Computers and Mathematics with Applications, 56, No 12 (2008), 309-311.
  34. [34] A. Ashyralyev and D. Agirseven, On the stable difference scheme for the Schrodinger equation with time delay, Computational Methods in Applied Mathematics, 20, No 1 (2020), 27-38.
  35. [35] D. Agirseven, On the stability of the Schrodinger equation with time delay, Filomat, 32, No 3 (2018), 759-766.