Nonlinear parabolic problems with an unknown source function and their applications for modelling and control of filtration processes




parabolic equations, boundary value problems, boundary control with boundary observation, conjugate problem, filtration processes


The work is connected with study of  nonlinear  parabolic systems arising in the modelling and control of nonstationary filtration processes in underground hydrodynamics. One of such statements is formulated as a system that involves the boundary value problem of the second kind for a quasilinear parabolic equation with an unknown source function in the right-hand side and, moreover, involves an additional equation for a time dependence of this function. In the other statement we consider control of this system controlled by the boundary regime. These statements essentially differ from usual boundary value problems and control problems for parabolic equations, where all the input data must be given. The obtained results have not only the theoretical interest but they are also important for investigation of various filtration processes. Some examples of such applications connected with fluid flow in the fractured porous media are discussed.

Author Biography

Nataliya L. Gol'dman


  1. N. Gol’dman, “Nonlinear Boundary Value Problems for a Parabolic Equation with an Unknown Source Function,” AIMS Math. 4 (5), 1508-1522 (2019).
    doi 10.3934/math.2019.5.1508.
  2. N. L. Gol’dman, “Study of Some Mathematical Models of Nonstationary Filtration Processes,” Vychisl. Metody Program. 21 (1), 1-12 (2020).
    doi 10.26089/NumMet.v21r101.
  3. J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer, Berlin, 1971; Mir, Moscow, 1972).
  4. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; AMS Press, Providence, 1968).
  5. A. Friedman, Partial Differential Equations of Parabolic Type (Prentice Hall, Englewood Cliffs, 1964; Mir, Moscow, 1968).
  6. N. L. Gol’dman, Inverse Stefan Problems (Kluwer, Dordrecht, 1997).
  7. N. L. Gol’dman, Inverse Stefan Problems. Theory and Methods of Solution (Mosk. Gos. Univ., Moscow, 1999) [in Russian].
  8. N. L. Gol’dman, “Boundary Value Problems for a Quasilinear Parabolic Equation with an Unknown Coefficient,” J. Differ. Equations 266 (8), 4925-4952 (2019).
    doi 10.1016/j.jde.2018.10.015.
  9. S. M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems (Nauka, Moscow, 1969; Springer, New York, 1975).
  10. L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1977; Pergamon Press, New York, 1982).
  11. F. P. Vasil’ev, Optimization Methods , Vols. 1, 2 (MTsNMO, Moscow, 2011) [in Russian].
  12. A. N. Tikhonov, A. V. Goncharskii, V. V. Stepanov, and A. G. Yagola, Regularizing Algorithms and a Priori Information (Nauka, Moscow, 1983) [in Russian].
  13. S. F. Gilyazov and N. L. Gol’dman, Regularization of Ill-Posed Problems by Iteration Methods (Kluwer, Dordrecht, 2000).
  14. G. I. Barenblatt, V. M. Entov, and V. M. Ryzhik, Theory of Fluid Flows through Natural Rocks (Nedra, Moscow, 1984; Kluwer, Dordrecht, 1990).
  15. K. S. Basniev, I. N. Kochina, and V. M. Maksimov, Underground Hydrodynamics (Nedra, Moscow, 1993) [in Russian].
  16. D. A. Gubaidullin and R. V. Sadovnikov, “Application of Parallel Algorithms for Solving the Problem of Fluid Flow to Wells with Complicated Configurations in Fractured Porous Reservoirs,” Vychisl. Metody Program. 8 (3), 244-251 (2007).
  17. M. H. Khairullin, A. I. Abdullin, P. E. Morozov, and M. N. Shamsiev, “Numerical Solution of the Coefficient Inverse Problem for a Deformable Fractured Porous Reservoir,” Mat. Model. 20 (11), 35-40 (2008).



How to Cite

Гольдман Н. Л. Nonlinear Parabolic Problems With an Unknown Source Function and Their Applications for Modelling and Control of Filtration Processes // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2022. 23. 207-229. doi 10.26089/NumMet.v23r313



Methods and algorithms of computational mathematics and their applications