Stable difference scheme for the numerical solution of the source identification problem for hyperbolic equations

Allaberen Ashyralyev; Fathi Emharab

doi:10.3842/umzh.v76i5.7407

Allaberen Ashyralyev Bahcesehir University, Istanbul, Turkey and Institute of Mathematics and Mathematical Modelling, Almaty, Kazakhstan
Fathi Emharab Omar Al-Mukhtar University, El-Beida, Libyan Arab Jamahiriya

DOI: https://doi.org/10.3842/umzh.v76i5.7407

Keywords: Source identification problem, well-posedness, hyperbolic differential equations, difference scheme

Abstract

UDC 517.9

We present a stable difference scheme of the second order of accuracy for a one-dimensional hyperbolic equation. The well-posedness of the difference scheme is established. Numerical results are presented.

References

Yu. Ya. Belov, Inverse problems for partial differential equations, Inverse and Ill-posed Probl. Ser., VSP (2002).

G. Di Blasio, A. Lorenzi, Identification problems for parabolic delay differential equations with measurement on the boundary, J. Inverse and Ill-posed Probl., 15, № 7, 709–734 (2007).

V. T. Borukhov, P. N. Vabishchevich, Numerical solution of the inverse problem of reconstructing a distributed right-hand side of a parabolic equation, Comput. Phys. Commun., 126, № 1-2, 32–36 (2000).

Yu. A. Gryazin, M. V. Klibanov, T. R. Lucas, Imaging the diffusion coefficient in a parabolic inverse problem in optical tomography, Inverse Problems, 15, № 2, 373 (1999).

V. Isakov, Inverse problems for partial differential equations, 127, Springer, New York (2006).

N. I. Ivanchov, On the determination of unknown source in the heat equation with nonlocal boundary conditions, Ukr. Math. J., 47, № 10, 1647–1652 (1995).

S. I. Kabanikhin, O. I. Krivorotko, Identification of biological models described by systems of nonlinear differential equations, J. Inverse and Ill-posed Probl., 23, № 5, 519–527 (2015).

A. I. Prilepko, D. G. Orlovsky, I. A. Vasin, Methods for solving inverse problems in mathematical physics, CRC Press (2000).

A. Ashyralyev, On the problem of determining the parameter of a parabolic equation, Ukr. Math. J., 62, № 9, 1397–1408 (2011).

A. Ashyralyev, D. Agirseven, On source identification problem for a delay parabolic equation, Nonlinear Anal., Model. and Control, 19, № 3, 335–349 (2014).

A. Ashyralyev, Ch. Ashyralyyev, On the problem of determining the parameter of an elliptic equation in a Banach space, Nonlinear Anal. Model. and Control, 19, № 3, 350–366 (2014).

A. Ashyralyev, A. Erdogan, Well-posedness of the right-hand side identification problem for a parabolic equation, Ukr. Math. J., 66, № 2 (2014).

A. U. Sazaklioglu, A. Ashyralyev, A. Said Erdogan, Existence and uniqueness results for an inverse problem for semilinear parabolic equations, Filomat, 31, № 4, 1057–1064 (2017).

Ch. Ashyralyyev, High order of accuracy difference schemes for the inverse elliptic problem with Dirichlet condition, Boundary Value Problems, 2014, № 1 (2014).

Ch. Ashyralyyev, High order approximation of the inverse elliptic problem with Dirichlet–Neumann conditions, Filomat, 28, № 5, 947–962 (2014).

M. Ashyraliyeva, M. Ashyraliyev, On a second order of accuracy stable difference scheme for the solution of a source identification problem for hyperbolic-parabolic equations, AIP Conf. Proc., 1759, № 1, AIP Publ. (2016).

M. Choulli, M. Yamamoto, Generic well-posedness of a linear inverse parabolic problem with diffusion parameters, J. Inverse and Ill-posed Probl., 7, № 3, 241–254 (1999).

M. Dehghan, An inverse problem of finding a source parameter in a semilinear parabolic equation, Appl. Math. Model., 25, № 9, 743–754 (2001).

Y. S. Eidelman, The boundary value problem for differential equations with a parameter, Differents. Uravn., 14, № 7, 1335–1337 (1978).

Saitoh Saburou, Vu Kim Tuan, M. Yamamoto, Reverse convolution inequalities and applications to inverse heat source problems, J. Inequal. Pure and Appl. Math., 3, № 5, 80 (2002).

A. A. Samarskii, P. N. Vabishchevich, Numerical methods for solving inverse problems of mathematical physics, 52, Walter de Gruyter (2008).

F. A. Aliev et al., A method to determine the coefficient of hydraulic resistance in different areas of pump-compressor pipes, TWMS J. Pure and Appl. Math., 7, № 2, 211–217 (2016).

F. A. Aliev et al., Algorithm for calculating the parameters of formation of gas-liquid mixture in the shoe of gas lift well, Appl. and Comput. Math., 15, № 3, 370–376 (2016).

M. Grasselli, S. I. Kabanikhin, A. Lorenzi, An inverse hyperbolic integrodifferential problem arising in geophysics. I, Sib. Math. J., 33, № 3, 415–426 (1992).

M. Grasselli, S. I. Kabanikhin, A. Lorenzi, An inverse hyperbolic integrodifferential problem arising in geophysics II, Nonlinear Anal., 15, № 3, 283–298 (1990).

A. Ashyralyev, F. Emharab, Source identification problems for hyperbolic differential and difference equations, J. Inverse and Ill-posed Probl., 27, № 3, 301–315 (2019).

A. Ashyralyev, P. E. Sobolevskii, New difference schemes for partial differential equations, operator theory advances and applications, Birkhäuser-Verlag etc. (2004).

Nguyen Thi Van Anh, Bui Thi Hai Yen, Source identification problems for abstract semilinear nonlocal differential equations, Inverse Probl. and Imaging, 16, № 5, 1389–1428 (2022).

J. Janno, Determination of time-dependent sources and parameters of nonlocal diffusion and wave equations from final data, Fract. Calc. and Appl. Anal., 23, № 6, 1678–1701 (2020).