Structures and Waves in a Nonlinear Heat-Conducting Medium

This paper is an overview of the main contributions of a Bulgarian team of researchers to the problem of finding the possible structures and waves in the open nonlinear heat-conducting medium, described by a reaction–diffusion equation. Being posed and actively worked out by the Russian school of A.A. Samarskii and S.P. Kurdyumov since the seventies of the last century, this problem still contains open and challenging questions.

[1]  V A Galaktionov,et al.  ON THE METHOD OF STATIONARY STATES FOR QUASILINEAR PARABOLIC EQUATIONS , 1990 .

[2]  A. A. Samarskii,et al.  Examples of the numerical calculation of temperature waves , 1963 .

[3]  R. Kohn,et al.  A rescaling algorithm for the numerical calculation of blowing-up solutions , 1988 .

[4]  Hiroshi Fujita,et al.  On the blowing up of solutions of the Cauchy problem for $u_1+\Delta u+u^{1+\alpha} , 1966 .

[5]  A. A. Samarskii,et al.  The burning of a nonlinear medium in the form of complex structures , 1977 .

[6]  C. Budd,et al.  Geometric integration and its applications , 2003 .

[7]  Robert D. Russell,et al.  Adaptivity with moving grids , 2009, Acta Numerica.

[8]  Howard A. Levine,et al.  The Role of Critical Exponents in Blowup Theorems , 1990, SIAM Rev..

[9]  S. Dimova,et al.  Numerical investigation of a new class of waves in an open nonlinear heat-conducting medium , 2013 .

[10]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[11]  Stefka Dimova,et al.  Numerical analysis of radically nonsymmetric blow-up solutions of a nonlinear parabolic problem , 1998 .

[12]  Victor A. Galaktionov,et al.  The problem of blow-up in nonlinear parabolic equations , 2002 .

[13]  E. Lazzaro,et al.  Reaction-Diffusion Problems in the Physics of Hot Plasmas , 2000 .

[14]  E. Zemlyanaya,et al.  Methods of computational physics for investigation of models of complex physical systems , 2007 .

[15]  A. A. Samarskii,et al.  ON APPROXIMATE SELF-SIMILAR SOLUTIONS OF A CLASS OF QUASILINEAR HEAT EQUATIONS WITH A SOURCE , 1985 .

[16]  V. Dorodnitsyn.  Applications of Lie Groups to Difference Equations , 2010 .

[17]  A. A. Samarskii,et al.  A quasilinear heat equation with a source: Peaking, localization, symmetry exact solutions, asymptotics, structures , 1988 .

[18]  Stefka Dimova,et al.  Numerical Investigation of Spiral Structure Solutions of a Nonlinear Elliptic Problem , 2010, NMA.

[19]  Stefka Dimova,et al.  Numerical analysis of the blow-up regimes of combustion of a two-component non-linear heat-conducting medium , 1995 .

[20]  Numerical analysis of the eigenfunctions for the combustion of a non-linear medium in the radial-symmetric case , 1991 .

[21]  Kenneth Eriksson,et al.  Galerkin Methods for Singular Boundary Value Problems in One Space Dimension , 1984 .

[22]  N. Riley,et al.  Scaling, self-similarity, and intermediate asymptotics , 1998 .

[23]  Jerrold Bebernes,et al.  Mathematical Problems from Combustion Theory , 1989 .

[24]  Vidar Thomée,et al.  A Lumped Mass Finite-element Method with Quadrature for a Non-linear Parabolic Problem , 1985 .

[25]  A. S. Kalashnikov.  Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations , 1987 .

[26]  S. Dimova,et al.  Numerical realization of blow‐up spiral wave solutions of a nonlinear heat‐transfer equation , 1994 .

[27]  A. P. Mikhailov,et al.  Thermal structures and fundamental length in a medium with nonlinear heat conduction and volumetric heat sources , 1976 .

[28]  Victor A. Galaktionov,et al.  Continuation of blowup solutions of nonlinear heat equations in several space dimensions , 1997 .

[29]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[30]  S. Dimova,et al.  LUMPED-MASS FINITE-ELEMENT METHOD WITH INTERPOLATION OF THE NONLINEAR COEFFICIENTS FOR A QUASILINEAR HEAT TRANSFER EQUATION , 1995 .

[31]  Non-stationary dissipative structures in a non-linear heat-conducting medium , 1983 .

[32]  S. P. Kurdiumov,et al.  Nonlinear processes in a dense plasma , 1976 .