THEORETICAL FOUNDATIONS AND SPECIFICITY OF MATHEMATICAL MODELLING
ABOUT NEW NONLINEAR PROPERTIES OF THE PROBLEM OF NONLINEAR THERMAL CONDUCTIVITY
- ^{1} National University of Uzbekistan, Uzbekistan
Abstract
In this paper we are consider a problem of nonlinear heat conduction with double nonlinearity under action of a strong absorption. For which an exact analytical solution is found, analysis of which makes it possible to reveal a number of characteristic features of thermal processes in nonlinear media. The following nonlinear effects are established: an inertial effect of a finite velocity of propagation of thermal disturbances, spatial heat localization and finite time effect i.e. existence of a thermal structure in a medium with strong absorption.
Keywords
References
- Aripov M. Abdullaeva Z., On the bottom of the exact solution of a nonlinear problem with absorption or a source. Bulletin of the TATU, №4 2016, 107-113.
- Martinson L. K., Evolution of a heat pulse in a nonlinear medium with bulk heat absorption, high temperature Thermophysics, 1983, vol. 21, i. 4, 801-803.
- Mersaid Aripov, Shakhlo A. Sadullaeva. Properties of solutions to reaction-diffusion equation with double nonlinearity wih distributed parameters. Log SFU. Ser. Mat and Phys., 6: 2 (2013), 157-167.
- Zeldovich B. V., Raizer Yu. P. Physics of shock waves and high-temperature hydrodynamic phenomena. M.: Science, 1966. 686 s.
- C.Jin, J.Yin Critical exponent of a double degenerate parabolic equation in non-divergence form with nonlinear source. Chin. Ann. Math., Ser. A, 30(2009), 525-538.
- Wang M., Wei Y. Blow-up properties for a degenerate parabolic system with nonlinear localized sources // J. Math. Anal. Appl. 343 (2008), 621--635.
- Raimbekov J.R. The Properties of the Solutions for Cauchy Problem of Nonlinear Parabolic Equations in Non-Divergent Form with Density // Journal of Siberian Federal University. Mathematics & Physics 2015, 8(2), 192--200.
- Mersaid Aripov, Shakhlo A. Sadulaeva, ―To properties of solutions to reaction diffusion equation with double nonlinearity with distributed parameters‖, Jour. of Siberian Fed. Univer. Math. & Phys. 6(2013), pp. 157-167
- P. Cianci, A. V. Martynenko, and A. F. Tedeev, ―The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source,‖ Nonlinear Analysis: Theory, Methods & Applications A, vol. 73, no. 7, pp. 2310– 2323, 2010.
- Said Benachour, Razvan Iagar, Philippe Laurencot. Large time behavior for the fast diffusion equation with critical absorption. Journal of Differential Equations, Elsevier, 2016, 260, pp.8000- 8024.
- Chunhua J., Jingxue Y. Self-similar solutions for a class of non-divergence form equations // Nonlinear Differ. Equ. Appl. Nodea. – 2013. – Vol. 20, Issue 3. – P. 873--893.
- Friedman, A., McLeod, B.: Blow-up of solutions of nonlinear degenerate parabolic equations. Arch. Rational Mech. Anal.,96 , 55–80 (1986)
- Wiegner, M.: Blow-up for solutions of some degenerate parabolic equations. Differ. Integral Eqs. 7, 1641–1647 (1994)
- Bertsch, M., Bisegna, P.: Blow-up of solutions of a nonlinear parabolic equation in damage mechanics. Eur. J. Appl. Math. 8, 89–123 (1997)
- Angenent, S.: On the formation of singularities in the curve shortening flow. J.Differ. Geom.33, 601–633 (1991)
- Z.Wu, J.Zhao, J.Yun and F.Li, Nonlinear Diffusion Equations, New York, Singapore: World Scientific Publishing, 2001
- A. Gmira, On quasilinear parabolic equations involving measure date, Asymptotic Analysis North-Holland, 3, 1990 pp. 43-56.
- J. Yang and J. Zhao, A note to the evolutional P-Laplace equation with absorption, Acta. Aci. Nat.Jilin. 2, 1995, pp. 35-38
- J. Zhao, Source-type solutions of quasilinear degenerate parabolic equation with absorption, Chin. Ann. of Math., ISB1, 1994, pp. 89-104.
- J.Zhao and H.Yuan, The Cauchy problem of a class of doubly degenerate parabolic equation (in chinese), Chinese Ann. of Math. 16As2, 1995, pp. 1881-196.
- Y.Li and Ch. Xie, Blow-up for p-Laplace parabolic equations, E.J.D.E. (20)2003, pp. 1-12.