String–Diffusion Systems of Equations

  • Oleg Novik
  • Feodor Smirnov
  • Maxim Volgin


In Chaps.  9,  10, and  11 we turn our attention to the mathematical background of the geophysical results. In Chap.  9, we show that the systems of partial differential equations (PDEs) applied by us for description of the seismo-electromagnetic interaction during preparation of earthquakes, being without a type according to classical typing of differential equations in mathematical physics, are not something rare and exotic, because systems of equations with a similar structure are used in different domains of physics when nonstationary interactions of physical fields of different natures are investigated. The general structure of these systems is described in Sect. 9.1 and may be characterized as coupling of the hyperbolic and parabolic subsystems by the operators of interaction of the physical fields of different natures that are activated during the earthquake preparation process. In Sects. 9.29.5, we describe examples of similar systems of PDEs applied in physics (including geophysics of seismically active regions) according to different assumptions about the properties of the nonideal (from the rheological viewpoint) geological medium under consideration (e.g., anisotropy, “memory” in the case of magneto-thermo-viscoelasticity, etc.).


Typing of differential equations String–diffusion systems of partial differential equations of precursory physics 


  1. Alfven, H. (1981). Cosmic plasma, Astrophysics and space science library (p. 122). New York: Springer.CrossRefGoogle Scholar
  2. Christensen, R. (1982). Theory of viscoelasticity. Amsterdam: Elsevier.Google Scholar
  3. Feynman, R. P., Leighton, R. B., & Sands, M. (2005). Feynman lectures on physics. Boston: Addison–Wesley. Accessed 08 Jan 2019.
  4. Hutter, K., & Ven, A. A. F. (1978). Field matter interactions in thermoelastic solids, Lecture notes in physics (Vol. 88, p. 230). New York: Springer.Google Scholar
  5. Iesan, D., & Quintanilla, R. (2013). On strain gradient theory of thermoviscoelasticity. Mechanics Research Communication, 48, 52–58.CrossRefGoogle Scholar
  6. Kashinzev, G. (2001). Geodynamics and magmatism of the initial stage of formation of Atlantic. Геотектоника (Geotectonics), 2, 64–77 (In Russian).Google Scholar
  7. Keylis-Borok, V. I., & Monin A. S. (1959). Magnetoeleastic waves and the boundary of the earth core. In Proceedings of Academy of Sciences of USSR (pp. 1530–1541). Ser. Geophysics. No 11. (In Russian. Кейлис-Борок В.И., Монин А.С. 1959. Магнитоупругие волны и граница земного ядра. Известия АН СССР. Сер. Геофизическая. 1959. № 11. С. 1530–1541).Google Scholar
  8. Knopoff, L. (1955). The interaction between elastic wave motion and a magnetic field in electrical conductors. Journal of Geophysical Research, 60(4), 441–446.CrossRefGoogle Scholar
  9. Ladyjenskaya, O. A. (1969). The mathematical theory of viscous incompressible flow. New York: Gordon and Breach, Science Publications. (Translated from Russian, Moscow: Gos. Izdat. Fiz-Mat Lit. 1961).Google Scholar
  10. Ladyjenskaya, O. A., Solonnikov, V. A., & Uraltseva, N. N. (1967). Linear and quasilinear equations of parabolic type. Translations of mathematical monographs (Vol. 23). Providence: American Mathematical Society. (Translated from Russian, Moscow: Izdat. Nauka 1964).Google Scholar
  11. Lakes, R. S. (2009). Viscoelastic materials. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  12. Landau, L. D., & Lifshitz, E. M. (1975). The classical theory of fields (Vol. 2, 4th ed.). Oxford: Butterworth–Heinemann.Google Scholar
  13. Landau, L. D., & Lifshitz, E. M. (1980). Statistical physics, part 1 (Vol. 5, 3rd ed.). Oxford: Butterworth–Heinemann.Google Scholar
  14. Landau, L. D., & Lifshitz, E. M. (1986). Theory of elasticity (Vol. 7, 3rd ed.). Oxford: Butterworth–Heinemann.Google Scholar
  15. Landau, L. D., & Lifshitz, E. M. (1987). Fluid mechanics (Vol. 6, 2nd ed.). Oxford, Butterworth–Heinemann.Google Scholar
  16. Landau, L. D., Lifshitz, E. M., & Pitaevskii, L. P. (1984). Electrodynamics of continuous media (Vol. 8, 1st ed.). Oxford: Butterworth–Heinemann.Google Scholar
  17. Lokshin, A. A., & Suvorova, Yu. V. (1982). Mathematical theory of propagation of waves in media with memory (p. 150). Moscow: Publishing House of Moscow State University. (In Russian. Локшин А.А., Суворова Ю.В. Математическая теория распространения волн в средах с памятью. М.: Изд-во МГУ, 1982).Google Scholar
  18. Maugin, G. (1988). Continuum mechanics of electromagnetic solids. North Holland: Elsevier.Google Scholar
  19. Mikhaylov, I. G., Solov’ev, V. A., & Syrnikov, Y. P. (1964). Osnovy molecularnoi acustiki (Foundations of molecular acoustics). Moscow: Nauka. (In Russian).Google Scholar
  20. Nemeth, M. D. (2011). An in-depth tutorial on constitutive equations for anisotropic materials (p. 582). NASA/TM-2011-217314. Accessed 08 Jan 2019.
  21. Novik, O. B. (1969). The Cauchy problem for the system of PDEs including the hyperbolic and parabolic operators. Journal of Computational Mathematics and Mathematical Physics, 1, 122–136. (In Russian).Google Scholar
  22. Novik, O. B. (1983a). Interaction of the geothermic field with the field of deformations of a layer. Geothermic investigations in Middle Asia and Kasahstan. Abstracts of reports of all-union conference. Ashgabat (In Russian).Google Scholar
  23. Novik, O. B. (1983b). Mathematical description of non-stationary fields of deformations and temperature. In Proceedings of high educational institutes (Известия вузов) (p. 24). Ser. Geology and Prospecting. Deposited in VINITI, № 6788-83 (In Russian).Google Scholar
  24. Novik, O. B. (1988). Finite difference schemes for the problems of thermo-mechanical effect in the oceanic lithosphere: Collection of scientific papers (pp. 89–97). Geothermic investigations on the bottoms of the water areas. Мoscow: Nauka (Наука) (In Russian).Google Scholar
  25. Novik, O. B., Mikhaylovskaya, I. B., & Ershov, S. V. (2000). Mathematical problems of designing the development of marine geothermal deposits. Collected volume “Earth’s thermal field and methods of its investigation” (pp. 306–310). Moscow: Publishing House of Peoples’ Friendship University of Russia. (In Russian: Новик О.Б., Михайловская И.Б., Ершов С.В. 2000. Математические задачи при проектировании разработки морских геотермальных мeсторождений. Сб. «Тепловое поле Земли и методы его изучения», стр. 306-310. Изд-во Рос. Универ. Дружбы Народов. Москва).Google Scholar
  26. Podstrigach, Y. C., Lomakin, V. A., & Kolyano, Y. M. (1984). Thermo-elasticity of bodies of non-uniform structure (p. 368). Moscow: Nauka. (In Russian).Google Scholar
  27. Shashkov, A.G., Bubnov, V.A., & Yanovsky, S.Y. (2004). Wave phenomena of heat conductivity: the system-structural approach (p. 289). Moscow: Editorial URSS. (In Russian).Google Scholar
  28. Shirong, M. (1992). Progress in earthquake prediction in China during 80’s. Journal of Earthquake Prediction Research, 1, 43–57.Google Scholar
  29. Tamm, I. E. (1979). Fundamentals of the theory of electricity. Moscow: Mir. (Translated from Russian by G. Leib).Google Scholar

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Authors and Affiliations

  • Oleg Novik
    • 1
  • Feodor Smirnov
    • 1
  • Maxim Volgin
    • 1
  1. 1.IZMIRAN of the Russian Academy of SciencesMoscowRussia

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