Exact solutions of a remarkable fin equation

Abstract

A model 'remarkable' fin equation is singled out from a class of nonlinear (1 + 1)-dimensional fin equations. For this equation a number of exact solutions are constructed by means of using both classical Lie algorithm and different modern techniques (functional separation of variables, generalized conditional symmetries, hidden symmetries etc).

Symmetry classification for the nonlinear heat conductivity equation

“…Symmetry analysis for equations u t = [E(x, u)u x ] x + H(x, u) rather than previous results on special cases of this equation equation [19,20], is carried out exhaustively. Also, equivalence classification is given of the equation admitting an extension by one of the principal Lie algebra of the equation.…”

“…In this study, we generalize the study of a class of the nonlinear heat conductivity equations (HCEs) which has been recently studied in some special cases [13,19,20]. We deal with the class of nonlinear heat conductivity equations of the general form HCE : u t = [E(x, u)u x ] x + H(x, u), (1.1) in which we assumed that E, H are sufficiently smooth functions, E x , E u , H x , H u = 0, u is treated as the dimensionless temperature, t and x are the dimensionless time and space variables and E is the thermal conductivity.…”

Heat transfer enhancement in annular fins using functionally graded material

“…Any response to this question leads to the study of fin performance and its thermal optimization. Therefore, many different works have been performed in order to analyze heat transfer performance in the finned surfaces []. Some of the previous investigations have focused on the fin profile [] as one of the effective parameters on the heat transfer of the fin, while some of the others have considered thermal characteristics of fins [] or surrounding fluid [].…”

“…Heat Transfer-Asian Research, 42 (7), 2013 surfaces [1][2][3][4][5][6][7][8][9][10]. Some of the previous investigations have focused on the fin profile [11][12][13] as one of the effective parameters on the heat transfer of the fin, while some of the others have considered thermal characteristics of fins [14][15][16][17][18] or surrounding fluid [19,20].…”

Conservation laws and associated Lie point symmetries admitted by the transient heat conduction problem for heat transfer in straight fins

“…Fin problems have attracted some interest from the symmetry analysts. Lie symmetry analysis of a nonlinear fin equation in which the thermal conductivity is an arbitrary function of the temperature and the heat transfer coefficient is an arbitrary function of a spatial variable was performed, for example, by [7][8][9][10][11]. Recently, the study of fins in boiling liquids has been increasing enormously and it has been found that the heat transfer coefficient may not only be given by a constant but also depends on the temperature distribution between the heated surface and its adjacent fluid [12], see also [13].…”