Lie point symmetry analysis is performed for an unsteady nonlinear heat diffusion problem modeling thermal energy storage in a medium with a temperature-dependent power law thermal conductivity and subjected to a convective heat transfer to the surrounding environment at the boundary through a variable heat transfer coefficient. Large symmetry groups are admitted even for special choices of the constants appearing in the governing equation. We construct one-dimensional optimal systems for the admitted Lie algebras. Following symmetry reductions, we construct invariant solutions.

For many years,
considerable attention has been paid to the collection, storage, and use of
thermal energy to meet various energy demands. The use of solar energy to meet
the thermal demands of industries, electronics devices, and residential
establishments, and so forth, is fast growing in many countries of the world
[

Meanwhile, the solution of unsteady nonlinear heat
diffusion equations in rectangular, cylindrical, and spherical coordinates
remains a very important problem of practical relevance in the engineering
sciences [

In the present work, we study an unsteady nonlinear heat diffusion problem modeling thermal energy storage in a medium with power law temperature-dependent thermal conductivity and subjected to a convective heat transfer to the surrounding environment at the boundary. The mathematical formulation of the problem is established in Section two. In Section three, we introduce and apply some rudiments of Lie group techniques. In Section four, we construct the one-dimensional optimal systems, and perform reductions by one variable and construct invariant solutions in Section five. Some discussions and conclusions are presented in Section six.

Consider an unsteady thermal storage problem in a body
whose surface is subjected to heat transfer by convection to an external
environment having a heat transfer coefficient that varies with respect to
time. The energy equation in a rectangular, cylindrical, or spherical
coordinate system can be used to find the temperature distribution through a
region defined in an interval

Neglecting the bar symbol for clarity, the
dimensionless boundary value problem (BVP) becomes

In brief, a symmetry of a differential equation is an
invertible transformation of the dependent and independent variables
that does not change the original
differential equation. Symmetries depend continuously on a parameter and form a
group; the

The admitted Lie algebra is three-dimensional and
spanned by the base vectors

Extra symmetries may be obtained for the cases

Extra-admitted symmetries.

Constants | Symmetries |
---|---|

In this section, we determine nonequivalent
subalgebras of the symmetry algebra admitted by (

Commutator table.

0 | 0 | ||

0 | 0 | ||

0 | 0 | 0 |

The adjoint representation is constructed using
formula (

Adjoint representation for the base vectors
given in (

Ad | |||

In this subsection, we reduce the variables of the
governing BVP by one. We provide the invariant solution constructed using

Reductions by elements of the optimal systems.

Symmetry | Reductions |
---|---|

The time-dependent heat transfer coefficient may be
represented by

Note that the trivial solution to (

We consider cases (a) and (d) only as examples. For case (a) and in terms of original variables we obtain

The invariant solution (

Without loss of generality we let

Graphical representation of the invariant solution (

Graphical representation of the invariant solution (

Symmetry analysis may lead to extra solutions, if we
use the linear combinations of the admitted symmetries or elements of the
optimal systems as listed in Table

We have determined some examples of group invariant solutions which satisfy the realistic boundary conditions (it is a well-known fact that more often symmetries do not lead to solutions which satisfy the boundary conditions). In this manuscript, Lie group analysis resulted in some exotic admitted point symmetries. Furthermore, reduction by one variable of the governing equation has been performed using members of the optimal system.

The invariant solution (

We have given some exact (invariant) solutions to nonlinear heat diffusion equations with temperature-dependent conductivity and time-dependent heat transfer coefficient.

Raseelo J. Moitsheki wishes to thank the National Research Foundation of South Africa under Thuthuka program, for the generous financial support. The author is also grateful to the anonymous reviewers for their useful comments.