A study of self-oscillation instability in varicap-based electrical networks: analytical and numerical approaches


  • V.A. Vasilchenko Lomonosov Moscow State University
  • M.O. Korpusov Lomonosov Moscow State University
  • D.V. Lukyanenko Lomonosov Moscow State University https://orcid.org/0000-0001-5140-3617
  • A.A. Panin Lomonosov Moscow State University




Sobolev-type equation, numerical diagnostics of solution’s blow-up


The blow-up of solutions is analytically and numerically studied for a certain Sobolev-type equation describing processes in varicap-based electrical networks. The energy method is used for the analytical study. For the numerical analysis, the original partial differential equation is approximated using a system of ordinary differential equations solved by the one-stage Rosenbrock scheme with a complex coefficient. The numerical diagnostics of solution’s blow-up is based on a posteriori asymptotically exact error estimation on sequentially condensed grids.

Author Biographies

V.A. Vasilchenko

M.O. Korpusov

D.V. Lukyanenko

A.A. Panin


  1. E. Mitidieri and S. I. Pokhozhaev, “A Priori Estimates and Blow-up of Solutions to Nonlinear Partial Differential Equations and Inequalities,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 234, 3-383 (2001) [Proc. Steklov Inst. Math. 234, 1-362 (2001)].
  2. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-up in Quasilinear Parabolic Equations (Nauka, Moscow, 1987; Gruyter, Berlin, 1995).
  3. V. A. Galaktionov and S. I. Pokhozhaev, “Third-Order Nonlinear Dispersive Equations: Shocks, Rarefaction, and Blowup Waves,” Zh. Vychisl. Mat. Mat. Fiz. 48 (10), 1819-1846 (2008) [Comput. Math. Math. Phys. 48 (10), 1784-1810 (2008)].
  4. H. A. Levine, “Some Nonexistence and Instability Theorems for Solutions of Formally Parabolic Equations of the Form Put = -Au + F(u),” Arch. Rational Mech. Anal. 51 (5), 371-386 (1973).
  5. H. A. Levine, “Instability and Nonexistence of Global Solutions to Nonlinear Wave Equations of the Form Putt = -Au + F(u),” Trans. Am. Math. Soc. 192, 1-21 (1974).
  6. V. K. Kalantarov and O. A. Ladyzhenskaya, “The Occurrence of Collapse for Quasilinear Equations of Parabolic and Hyperbolic Types,” Zap. Nauch. Semin. Leningr. Otd. Mat. Inst. Steklova 69, 77-102 (1977). [J. Math. Sci. 10 (1), 53-70 (1978)].
  7. A. G. Sveshnikov, A. B. Al’shin, M. O. Korpusov, and Yu. D. Pletner, Linear and Nonlinear Equations of Sobolev Type (Fizmatlit, Moscow, 2007) [in Russian].
  8. M. O. Korpusov, Blow-up in Nonclassical Wave Equations (LIBROCOM, Moscow, 2010) [in Russian].
  9. M. O. Korpusov, “Blow-up of Ion Acoustic Waves in a Plasma,” Mat. Sb. 202 (1), 37-64 (2011) [Sb. Math. 202 (1), 35-60 (2011)].
  10. M. O. Korpusov, A. G. Sveshnikov, and E. V. Yushkov, Methods of the Theory of Solution Blow-Up for Nonlinear Equations of Mathematical Physics (Moscow State Univ., Faculty of Physics, Moscow, 2014) [in Russian].
  11. D. V. Luk’yanenko and A. A. Panin, “Blow-up Phenomena in the Model of a Space Charge Stratification in Semiconductors: Numerical Analysis of Original Equation Reduction to a Differential-Algebraic System,” Vychisl. Metody Programm. 17, 437-446 (2016).
  12. M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, and E. V. Yushkov, “Blow-up for One Sobolev Problem: Theoretical Approach and Numerical Analysis,” J. Math. Anal. Appl. 442 (2), 451-468 (2016).
  13. M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, and E. V. Yushkov, “Blow-up Phenomena in the Model of a Space Charge Stratification in Semiconductors: Analytical and Numerical Analysis,” Math. Meth. Appl. Sci. 40 (7), 2336-2346 (2017).
  14. M. O. Korpusov, D. V. Lukyanenko, E. A. Ovsyannikov, and A. A. Panin, “Local Solvability and Decay of the Solution of an Equation with Quadratic Noncoercive Nonlinearity,” Vestn. Yuzhn. Ural. Gos. Univ. Ser. Mat. Model. Programm. 10 (2), 107-123 (2017).
  15. M. O. Korpusov and D. V. Lukyanenko, “Instantaneous Blow-up Versus Local Solvability for One Problem of Propagation of Nonlinear Waves in Semiconductors,” J. Math. Anal. Appl. 459 (1), 159-181 (2018).
  16. M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, and G. I. Shlyapugin, “On the Blow-up Phenomena for a One-Dimensional Equation of Ion-Sound Waves in a Plasma: Analytical and Numerical Investigation,” Math. Methods Appl. Sci. 41 (8), 2906-2929 (2018).
  17. M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, and E. V. Yushkov, “Blow-up of Solutions of a Full Non-Linear Equation of Ion-Sound Waves in a Plasma with Non-Coercive Non-Linearities,” Izv. Ross. Akad. Nauk, Ser. Mat. 82 (2), 43-78 (2018) [Izv. Math. 82 (2), 283-317 (2018)].
  18. M. O. Korpusov, D. V. Lukyanenko, and A. D. Nekrasov, “Analytic-Numerical Investigation of Combustion in a Nonlinear Medium,” Zh. Vychisl. Mat. Mat. Fiz. 58 (9), 1553-1563 (2018) [Comput. Math. Math. Phys. 58 (9), 1499-1509 (2018)].
  19. E. A. Alshina, N. N. Kalitkin, and P. V. Koryakin, “Diagnostics of Singularities of Exact Solutions in Computations with Error Control,” Zh. Vychisl. Mat. Mat. Fiz. 45 (10), 1837-1847 (2005) [Comput. Math. Math. Phys. 45 (10), 1769-1779 (2005)].
  20. N. N. Kalitkin, A. B. Al’shin, E. A. Al’shina, and B. V. Rogov, Calculations on Quasi-Uniform Grids (Fizmatlit, Moscow, 2005) [in Russian].
  21. A. B. Al’shin and E. A. Al’shina, “Numerical Diagnosis of Blow-up of Solutions of Pseudoparabolic Equations,” J. Math. Sci. 148 (1), 143-162 (2008).
  22. M. I. Rabinovich, “Self-Oscillations of Distributed Systems,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 17 (4), 477-510 (1974). [Radiophys. Quantum Electron. 17 (4), 361-385 (1974)].
  23. M. O. Korpusov and E. A. Ovsyannikov, “Blow-up Instability in Nonlinear Wave Models with Distributed Parameters,” Izv. Ross. Akad. Nauk, Ser. Mat. (in press).
  24. E. Hairer and G. Wanner, Solving Ordinary Differential Equations. Stiff and Differential-Algebraic Problems (Springer, Berlin, 2002).
  25. N. N. Kalitkin, “Numerical Methods for Solving Stiff Systems,” Mat. Model. 7 (5), 8-11 (1995).
  26. H. H. Rosenbrock, “Some General Implicit Processes for the Numerical Solution of Differential Equations,” Comput. J. 5 (4), 329-330 (1963).
  27. A. B. Al’shin, E. A. Al’shina, N. N. Kalitkin, and A. B. Koryagina, “Rosenbrock Schemes with Complex Coefficients for Stiff and Differential Algebraic Systems,” Zh. Vychisl. Mat. Mat. Fiz. 46 (8), 1392-1414 (2006) [Comput. Math. Math. Phys. 46 (8), 1320-1340 (2006)].



How to Cite

Васильченко В.А., Корпусов М.О., Лукьяненко Д.В., Панин А.А. A Study of Self-Oscillation Instability in Varicap-Based Electrical Networks: Analytical and Numerical Approaches // Numerical methods and programming. 2019. 20. 323-336. doi 10.26089/NumMet.v20r328



Section 1. Numerical methods and applications

Most read articles by the same author(s)

1 2 > >>