ON AN INVERSE PROBLEM WITH AN INTEGRAL OVERDETERMINATION CONDITION FOR THE BURGERS EQUATION
DOI:
https://doi.org/10.26577/JMMCS.2023.v117.i1.03Keywords:
Burgers equation, inverse problem, a priori estimates, Galerkin methodAbstract
In this paper we consider one inverse problem for the Burgers equation with an
integral overdetermination and periodic boundary conditions in a domain that is trapezoid. Using an integral overdetermination, boundary and initial conditions, we reduce the inverse problem to the study of an already direct initial boundary value problem
for the loaded Burgers equation. Next, we use a one-to-one transformation of independent variables to move from a trapezoid to a rectangular domain, where we study an auxiliary problem, for which the methods of Faedo-Galerkin, a priori estimates and functional analysis have been proved
a theorem on its unique solvability in Sobolev classes. Note that the obtained a priori estimates are uniform with respect to the summation index of the approximate solution and do not depend on time.
Further, on the basis of this theorem, due to the correspondence of spaces, theorems on the unique solvability of the original inverse problem are proved. Also, for the selected initial data, the paper presents graphs of the initial-boundary problem for the loaded Burgers equation and the desired function of the inverse problem, which together constitute the solution of the original inverse problem.
References
[2] C. Montoya, "Inverse source problems for the Korteweg-de Vries-Burgers equation with mixed boundary conditions J. Inverse Ill-Posed Probl. 27: 6 (2019), 1–18.
[3] D.J. Korteweg, G. de Vries, "On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves Philos. Mag. 39: 240 (1895), 422–443.
[4] E. Hopf, "The partial differential equation ut + uux + μuxx Commun. Pure Appl. Math. 3: 3 (1950), 201–230.
[5] E. Nane, N.H. Tuan, N.H. Tuan, "A random regularized approximate solution of the inverse problem for Burgers’ equation Stat. Probab. Lett. 132: (2018), 46–54.
[6] G. Berikelashvili, M. Mirianashvili, "On the convergence of difference schemes for the generalized BBM-Burgers equation Georgian Math. J. 26: 3 (2019), 341–349.
[7] G.B. Whitham, Linear and Nonlinear Waves, New York: John Wiley and Sons, 1975.
[8] H. Li, J. Zhou, "Direct and inverse problem for the parabolic equation with initial value and time-dependent
boundaries Appl. Anal. 95: 6 (2016), 1307–1326.
[9] I. Baglan, F. Kanca, "Two-dimensional inverse quasilinear parabolic problem with periodic boundary condition Appl.Anal. 98: 8 (2019), 1549–1565.
[10] I. Kukavica, "Log-Log convexity and backward uniqueness Proc. Am. Math. Soc. 135: 8 (2007), 2415–2421.
[11] J. Apraiz, A. Doubova, E. Fernandez-Cara, M. Yamamoto, "Some inverse problems for the Burgers equation and related systems Commun. Nonlinear Sci. Numer. Simul. 107 (2022), 106113.
[12] J. Cyranka, P. Zgliczynski, "Existence of globally attracting solutions for one-dimensional viscous Burgers equation with nonautonomous forcing-A computer assisted proof SIAM J. Appl. Dyn. Syst. 14: 2 (2015), 787–821.
[13] J.D. Cole, "On a quasilinear parabolic equation occuring in aerodinamics Q. Appl. Math. 9: 3 (1951), 225–236.
[14] J. Li, B.-Y. Zhang, Z. Zhang, "Well-posedness of the generalized Burgers equation on a finite interval Appl.Anal. 98: 16 (2019), 2802–2826.
[15] J.-L. Lions, E. Magenes, Problemes aux limites non homogenes et applications, vol. 1, Paris: Dunod; 1968.
