A note on a symmetry analysis and exact solutions of a nonlinear fin equation

Abstract

A similarity analysis of a nonlinear fin equation has been carried out by M. Pakdemirli and A.Z. Sahin [Similarity analysis of a nonlinear fin equation, Appl. Math. Lett. (2005) (in press)]. Here, we consider a further group theoretic analysis that leads to an alternative set of exact solutions or reduced equations with an emphasis on travelling wave solutions, steady state type solutions and solutions not appearing elsewhere.

Transient heat transfer in longitudinal fins of various profiles with temperature-dependent thermal conductivity and heat transfer coefficient

“…Subsequently, symmetry analysts considered the problem in [12] to determine all forms of thermal conductivity and heat transfer coefficients for which the governing equation admits extra symmetries [13][14][15][16]. However, only general solutions were constructed.…”

Symmetry classification for the nonlinear heat conductivity equation

“…According to [14], v is a point infinitesimal generator of Eq. (1.1) if and only if Y (2) [Eq. (1.1)] = 0 when Eq.…”

“…Now by applying Propositions 4 and 4 for the optimal system (4.47), we want to find all nonequivalent equations in the form of Eq. (1.1) admitting E-extensions of the principal Lie algebra L E , by one dimension, i.e, equations of the form (1.1) such that they admit, together with the one basic operator ∂ ∂t of L 1 , also a second operator X (2) . In each case which this extension occurs, we indicate the corresponding coefficients E, H and the additional operator X (2) .…”

Exact solutions of a remarkable fin equation

“…was investigated with the symmetry point of view in a number of papers [3,19,20,24]. Here u is treated as the dimensionless temperature, t and x the dimensionless time and space variables, D the thermal conductivity, h = −N 2 f (x), N the fin parameter and f the heat transfer coefficient.…”