On the uniqueness of solutions in inverse problems for Burgers’ equation under a transverse diffusion

References

[1] H. Aref and P. K. Daripa, Note on finite difference approximations to Burgers’ equation, SIAM J. Sci. Statist. Comput. 5 (1984), no. 4, 856–864. 10.1137/0905060Search in Google Scholar

[2] A. V. Baev, Uniqueness of the solution of inverse initial value problems for the Burgers equation with an unknown source, Differ. Equ. 57 (2021), no. 6, 701–710. 10.1134/S001226612106001XSearch in Google Scholar

[3] J. Baker, A. Armaou and P. D. Christofides, Nonlinear control of incompressible fluid flow: application to Burgers’ equation and 2D channel flow, J. Math. Anal. Appl. 252 (2000), no. 1, 230–255. 10.1006/jmaa.2000.6994Search in Google Scholar

[4] H. Bateman, Some recent researches on the motion of fluids, Monthly Weather Rev. 43 (1915), 163–170. 10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2Search in Google Scholar

[5] J. Bec, R. Iturriaga and K. Khanin, Topological shocks in Burgers turbulence, Phys. Rev. Lett. 89 (2002), Article ID 024501. 10.1103/PhysRevLett.89.024501Search in Google Scholar

[6] J. Bec and K. Khanin, Forced Burgers equation in an unbounded domain, J. Stat. Phys. 113 (2003), 741–759. 10.1023/A:1027356518273Search in Google Scholar

[7] J. Bec and K. Khanin, Burgers turbulence, Phys. Rep. 447 (2007), no. 1–2, 1–66. 10.1016/j.physrep.2007.04.002Search in Google Scholar

[8] E. R. Benton and G. W. Platzman, A table of solutions of the one-dimensional Burgers equation, Quart. Appl. Math. 30 (1972), 195–212. 10.1090/qam/306736Search in Google Scholar

[9] I. A. Bogaevsky, Matter evolution in Burgulence, preprint (2004), https://arxiv.org/abs/math-ph/0407073. Search in Google Scholar

[10] M. Bonkile, A. Awasthi, C. Lakshmi, V. Mukundan and V. S. Aswin, A systematic literature review of Burgers equation with recent advances, Pramana J. Phys. 90 (2018), 10.1007/s12043-018-1559-4. 10.1007/s12043-018-1559-4Search in Google Scholar

[11] J. M. Burgers, Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion, Verh. Nederl. Akad. Wetensch. Afd. Natuurk. Sect. 1 17 (1939), no. 2, 1–53. Search in Google Scholar

[12] J. M. Burgers, A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, Academic Press, New York (1948), 171–199. 10.1016/S0065-2156(08)70100-5Search in Google Scholar

[13] D. Chaikovskii and Y. Zhang, Convergence analysis for forward and inverse problems in singularly perturbed time-dependent reaction-advection-diffusion equations, J. Comput. Phys. 470 (2022), Paper No. 111609. 10.1016/j.jcp.2022.111609Search in Google Scholar

[14] A. Chekhlov and V. Yakhot, Kolmogorov turbulence in a random-force-driven Burgers equation, Phys. Rev. E (3) 51 (1995), no. 5, R2739–R2742. 10.1103/PhysRevE.51.R2739Search in Google Scholar

[15] A. Chekhlov and V. Yakhot, Kolmogorov turbulence in a random-force-driven Burgers equation: Anomalous scaling and probability density functions, Phys. Rev. E (3) 52 (1995), no. 5, 5681–5684. 10.1103/PhysRevE.52.5681Search in Google Scholar PubMed

[16] J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951), 225–236. 10.1090/qam/42889Search in Google Scholar

[17] A. M. Denisov, Elements of the Theory of Inverse Problems, VSP, Utrecht, 1999. 10.1515/9783110943252Search in Google Scholar

[18] A. M. Denisov, Problems of determining the unknown source in parabolic and hyperbolic equations, Comput. Math. Math. Phys. 55 (2015), no. 5, 829–833. 10.1134/S0965542515050085Search in Google Scholar

[19] A. M. Denisov, Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation, Comput. Math. Math. Phys. 56 (2016), no. 10, 1737–1742. 10.1134/S0965542516100067Search in Google Scholar

[20] A. M. Denisov, Approximate solution of inverse problems for the heat equation with a singular perturbation, Comput. Math. Math. Phys. 61 (2021), no. 12, 2004–2014. 10.1134/S0965542521120071Search in Google Scholar

[21] C. A. J. Fletcher, Generating exact solutions of the two-dimensional Burgers’ equations, Internat. J. Numer. Methods Fluids 3 (1983), 213–216. 10.1002/fld.1650030302Search in Google Scholar

[22] R. J. Gelinas, S. K. Doss and K. Miller, The moving finite element method: Applications to general partial differential equations with multiple large gradients, J. Comput. Phys. 40 (1981), no. 1, 202–249. 10.1016/0021-9991(81)90207-2Search in Google Scholar

[23] V. Gurarie and A. Migdal, Instantons in the Burgers equation, Phys. Rev. E 54 (1996), 10.1103/PhysRevE.54.4908. 10.1103/PhysRevE.54.4908Search in Google Scholar PubMed

[24] S. N. Gurbatov, A. I. Saichev and S. F. Shandarin, A model description of the development of the large-scale structure of the Universe, Sov. Phys. Dokl. 20 (1984), 921–923. Search in Google Scholar

[25] R. S. Hirsh, Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys. 19 (1975), no. 1, 90–109. 10.1016/0021-9991(75)90118-7Search in Google Scholar

[26] E. Hopf, The partial differential equation u t + u ⁢ u x = μ ⁢ u x ⁢ x , Comm. Pure Appl. Math. 3 (1950), 201–230. 10.1002/cpa.3160030302Search in Google Scholar

[27] P. Jain and D. Holla, Numerical solution of coupled Burgers’ equations, Int. J. Nonlin. Mech. 13 (1978), 213–222. 10.1016/0020-7462(78)90024-0Search in Google Scholar

[28] G. Kreiss and H.-O. Kreiss, Convergence to steady state of solutions of Burgers’ equation, Appl. Numer. Math. 2 (1986), no. 3–5, 161–179. 10.1016/0168-9274(86)90026-7Search in Google Scholar

[29] A. Kudryavtsev and O. Sapozhnikov, Determination of the exact solutions to the inhomogeneous Burgers equation with the use of the Darboux transformation, Acoust. Phys. 57 (2011), 311–319. 10.1134/S1063771011030080Search in Google Scholar

[30] S. Leibovich and A. R. Seebas, Nonlinear Waves, Cornell University, Ithaca, 1974. Search in Google Scholar

[31] B. M. Levitan, Inverse Sturm–Liouville Problems, VNU Science, Utrecht, 1987. 10.1515/9783110941937Search in Google Scholar

[32] W. Liao, An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation, Appl. Math. Comput. 206 (2008), no. 2, 755–764. 10.1016/j.amc.2008.09.037Search in Google Scholar

[33] W. Liao, A fourth-order finite-difference method for solving the system of two-dimensional Burgers’ equations, Internat. J. Numer. Methods Fluids 64 (2010), no. 5, 565–590. 10.1002/fld.2163Search in Google Scholar

[34] D. V. Lukyanenko, A. A. Borzunov and M. A. Shishlenin, Solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection type with data on the position of a reaction front, Commun. Nonlinear Sci. Numer. Simul. 99 (2021), Paper No. 105824. 10.1016/j.cnsns.2021.105824Search in Google Scholar

[35] D. V. Lukyanenko, V. B. Grigorev, V. T. Volkov and M. A. Shishlenin, Solving of the coefficient inverse problem for a nonlinear singularly perturbed two-dimensional reaction-diffusion equation with the location of moving front data, Comput. Math. Appl. 77 (2019), no. 5, 1245–1254. 10.1016/j.camwa.2018.11.005Search in Google Scholar

[36] D. V. Lukyanenko, I. V. Prigorniy and M. A. Shishlenin, Some features of solving an inverse backward problem for a generalized Burgers’ equation, J. Inverse Ill-Posed Probl. 28 (2020), no. 5, 641–649. 10.1515/jiip-2020-0078Search in Google Scholar

[37] J. D. Murray, On Burgers’ model equations for turbulence, J. Fluid Mech. 59 (1973), 263–279. 10.1017/S0022112073001564Search in Google Scholar

[38] D. E. Panayotounakos and D. Drikakis, On the closed-form solutions of the wave, diffusion and Burgers equations in fluid mechanics, Z. Angew. Math. Mech. 75 (1995), no. 6, 437–447. 10.1002/zamm.19950750604Search in Google Scholar

[39] L. G. Reyna and M. J. Ward, On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math. 48 (1995), no. 2, 79–120. 10.1002/cpa.3160480202Search in Google Scholar

[40] E. Y. Rodin, On some approximate and exact solutions of boundary value problems for Burgers’ equation, J. Math. Anal. Appl. 30 (1970), 401–414. 10.1016/0022-247X(70)90171-XSearch in Google Scholar

[41] R. Sinuvasan, K. M. Tamizhmani and P. G. L. Leach, Algebraic resolution of the Burgers equation with a forcing term, Pramana J. Phys. 88 (2017), 10.1007/s12043-017-1382-3. 10.1007/s12043-017-1382-3Search in Google Scholar

[42] E. Varoḡlu and W. D. L. Finn, Space-time finite elements incorporating characteristics for the Burgers equation, Internat. J. Numer. Methods Engrg. 16 (1980), 171–184. 10.1002/nme.1620160112Search in Google Scholar

[43] G. B. Whitham, Linear and Nonlinear Waves, Pure Appl. Math., John Wiley & Sons, New York, 1974. Search in Google Scholar

[44] D. V. Widder, The Laplace Transform, Princeton Math. Ser. 6, Princeton University, Princeton, 1941. Search in Google Scholar

[45] G. M. Zaslavsky, Chaos in Dynamic Systems, Harwood Academic, Chur, 1985. Search in Google Scholar

[46] T. Zhanlav, O. Chuluunbaatar and V. Ulziibayar, Higher-order accurate numerical solution of unsteady Burgers’ equation, Appl. Math. Comput. 250 (2015), 701–707. 10.1016/j.amc.2014.11.013Search in Google Scholar