Heat distribution in the rod in presence of external non-stationary source

. This paper is devoted to the study of nonlinear heat distribution in a straight homogeneous rod in the presence of an external non-stationary source of heating or cooling in relation to the process of winter concreting of columns. Until now, to describe this nonlinear distribution, as a rule, the classical linear heat equation is used. However, these models do not adequately describe the real process, since the nonlinearity of the process and the presence of an external heat source are not taken into account. Using group analysis methods, we obtained a model that admits the widest group of Lie transformations compared to other basic models of the general model. For the differential equation defining this model, we have obtained all separable solutions and some invariant solutions. The set of these solutions depends on empirically determined parameters: one arbitrary smooth function and ten arbitrary constants. Each solution determines a new exact model for winter concreting of columns. For each solution, at certain values of the parameters on which this solution depends, the temperature distribution in the rod were obtained. The significance of the obtained solutions is as follows: 1) these solutions describe specific physical processes and can be used in practice, 2) these solutions can be used as test solutions in numerical calculations.


Introduction
The strength characteristics of products used in technology are largely related to the temperature of these products.This applies to rocketry, aircraft manufacturing, shipbuilding, construction of columns during winter concreting, and other areas.This led to active study of heat distribution models in the rod [1][2][3][4][5][6][7][8][9].One of the first papers that took into account the nonlinearity of the winter concreting process in the presence of an external heat source were the papers [10][11][12].In our paper, using the methods of group analysis [13][14][15][16], we study a nonlinear mathematical model of heat propagation in a rod in the presence of an external non-stationary source of heating or cooling, which describes the process of winter concreting of columns.This model is given by the following differential equation Here and in all subsequent formulas, the prime above a function denotes the derivative with respect to the variable on which this function depends.
Using the algorithm proposed in [16,17], the problem of group classification of equation (1) with condition (2) was solved.For the convenience of the reader, we do not present its results, so as not to clutter the paper with mathematical transformations.
Models that admits the widest group of Lie transformations, as shown by the experience of research (see, for example, [13][14][15][16][17], are the most effective from the point of view of mathematical research and from the point of view of obtaining the most complete information about the adequacy of these models to real processes.Therefore, further research is carried out for the model allowing the widest group.This model is given by the equation where 0   , since this equation describes the propagation of heat only for such  , , is an experimentally determined function, 1 k are 2 k arbitrary real numbers. Each solution to equation (3) defines a certain submodel of the model specified by this equation.
In the following formulas, the quantities  

Solutions with separable variables
These solutions have the form Substituting (4) into (3) and separating the variables leads to the equations where  is an arbitrary real number.
Solving these equations, we obtain all solutions to equation (3) with separable variables.
Equation (3) admits the following discrete transformations: 1) xx  , 2) uu  .With the accuracy to these transformations, we have: for

 
,     , equation (3) has one solution with separable variables and for 0   , another solution is added For some values of the parameters,   , we will show the temperature distributions given by formulas ( 5) and ( 6).

Let
. This means that the rod is cooling.The temperature at each fixed point x is monotonically decreasing by the law The rate of temperature change at each fixed point x has the form.
Fig. 1 shows a graph of this temperature distribution in the rod.This model can be used to study cooling of the heated column.
. This means that the rod is heating up.The temperature at each fixed point x is monotonically increases by law .
The rate of temperature change at each fixed point x has the form.
Fig. 2 shows a graph of this temperature distribution in the rod.This model can be used to study heating of the column.

Invariant solutions
The main group of equations ( 3) is generated by the operators In order to classify all invariant submodels of the model defined by equation ( 3) by the similarity criterion, it is necessary to classify all invariant solutions of this equation that are not connected by point transformations.For this purpose, an optimal system of subgroups of the Lie transformation group generated by the Lie algebra with basis (7) was constructed.It contains 14 one-parameter subgroups and 20 two-parameter subgroups.For each subgroup, a universal invariant in space   3 ,, R t x u was found.For the convenience of the reader, we do not present this optimal system and universal invariants of subgroups, so as not to clutter the paper with mathematical transformations.The obtained results made it possible to find all invariant solutions to equation (3), and, consequently, all invariant submodels of the model given by this equation.
We will present only solutions obtained in explicit form.We will indicate the physical meaning of the submodels that are defined by these solutions.Also we will indicate the possible application of these submodels in studying the winter concreting process.Each of these submodels is a new model for winter concreting columns.The fact that these solutions are indeed solutions to equation ( 3) can be verified by directly substituting these solutions into this equation.

Invariant solutions of rank 0
Invariant solutions of rank 0 are invariant under two-parameter subgroups.Among them, all solutions, except one, are solutions with separable variables, that is, they are contained among solutions ( 5) and ( 6).The only new solution is the following solution .. This means that the rod is heating up.The temperature at each fixed point x is monotonically increases by law The rate of temperature change at each fixed point x has the form . Fig. 3 shows a graph of this temperature distribution in the rod.This model can be used to study heating of the column.

Invariant solutions of rank 1
Invariant solutions of rank 1 are invariant under one-parameter subgroups.Among them, only the solutions given below are not solutions with separable variables.These solutions are not contained among solutions ( 5) and ( 6).
• A solution invariant under the subgroup generated by the operator 14 XX  This solution has the form Substituting ( 9) into (3) gives the following equation for the function For 6 0 c  , from (10) it follows that the solution of equation ( 3) is determined by the formula Fig. 4 shows the graph of solution (12) for .. This means that the rod is cooling.The temperature at each fixed point x is monotonically decreases by law .
The rate of temperature change at each fixed point x has the form • A solution invariant under the subgroup generated by the operator 14 XX  This solution has the form Substituting ( 13) into (3) gives equation (10) for the function Fig. 5 shows the graph of solution (12) for .
Fig. 5 shows a graph of this temperature distribution in the rod.This model can be used to study cooling of the heated column.

Conclusion
Models of heat distribution in the rod are widely used in modeling processes in construction during winter concreting of columns.However, these models do not always adequately describe real nonlinear processes.This is due to the fact that these models are usually linear and do not take into account the presence of a non-stationary external source.
In our paper, we studied a nonlinear model of heat distribution in a straight homogeneous rod in the presence of a nonstationary external source of heating or cooling.Using methods of group analysis of differential equations, we found a model that admits the widest group of Lie transformations compared to other basic models of the main model.
Models that admit the widest group, as the experience of conducted research shows, are the most effective from the point of view of mathematical research and from the point of view of obtaining the most complete information about the adequacy of these models to real processes.Therefore, further research was devoted to a model that admits the widest group of Lie transformations compared to other basic models.
For the differential equation defining this model, we have obtained all separable solutions and some invariant solutions.These solutions are determined by formulas (5), ( 6), ( 8), (11), (12) and (14).The set of these solutions depends on empirically determined parameters: one function and ten arbitrary constants.Each of the solutions obtained sets a new model for winter concreting of columns.For certain values of the parameters on which these solutions depend, graphs of the temperature distribution in the rod were obtained.Other choices of these parameters will allow these models to be used in processes of winter concreting of columns are different from those discussed in this paper.
follows from this equation that the function U  is implicitly determined by the transcendental equation

Fig. 4 Fig. 4 .
Fig.4shows a graph of this temperature distribution in the rod.This model can be used to study cooling of the heated column For