Abstract
Abstract Nowadays there are many approximate methods for thermal conductivity calculation, that lead to satisfactory outcomes in engineering practice. With the new approach, the solution of the nonlinear thermal conductivity equation was investigated. In addition to this, we reviewed the approximate solution of the nonlinear equation of thermal conductivity with cubic nonlinearity. To solve the problems of mathematical analysis, differential and integral equations, and boundary value problems of mathematical physics, difference, and interpolation are applied. Thus, for thinking of the effectiveness and reasonableness of these approaches, it is crucial for their theoretical investigation. The solution to these questions was found for each class of equation and each of its methods in their way and was often represented as significant difficulty, and in many cases was an obstacle for the current time. A natural approach to solving this issue is the use of the ideas of functional analysis. The variational principle initially was considered as a variational approach for solving linear functional equations and finding eigenvalues of linear operators. As in any variational approach, the problem of solving an equation will be brought to finding the extremum of the certain function of a special type, given over the entire space. It was found that the approach is useful in a way of minimizing functions of the more general type.